· 640a chapter 12 rational expressions and equations pacing suggestions for the entire year can...

640A Chapter 12 Rational Expressions and Equations

Pacing suggestions for the entire year can be found on pages T20–T21.

Rational Expressionsand EquationsChapter Overview and Pacing

Rational Expressions and EquationsChapter Overview and Pacing

PACING (days)Regular Block

Basic/ Basic/ Average Advanced Average Advanced

Inverse Variation (pp. 642–647) 1 1 0.5 0.5• Graph inverse variations.• Solve problems involving inverse variation.

Rational Expressions (pp. 648–654) 1 2 0.5 1.5• Identify values excluded from the domain of a rational expression. (with 11-2• Simplify rational expressions. Follow-Up)Follow-Up: Use a graphing calculator to check simplified rational expressions.

Multiplying Rational Expressions (pp. 655–659) 1 1 0.5 0.5• Multiply rational expressions.• Use dimensional analysis with multiplication.

Dividing Rational Expressions (pp. 660–664) 1 2 0.5 1• Divide rational expressions.• Use dimensional analysis with division.

Dividing Polynomials (pp. 666–671) 1 2 0.5 1• Divide a polynomial by a monomial.• Divide a polynomial by a binomial.

Rational Expressions with Like Denominators (pp. 672–677) 1 1 0.5 0.5• Add rational expressions with like denominators.• Subtract rational expressions with like denominators.

Rational Expressions with Unlike Denominators (pp. 678–683) 2 2 1 1• Add rational expressions with unlike denominators.• Subtract rational expressions with unlike denominators.

Mixed Expressions and Complex Fractions (pp. 684–689) 2 2 1 1• Simplify mixed expressions.• Simplify complex fractions.

Solving Rational Equations (pp. 690–695) 2 2 1 1• Solve rational equations.• Eliminate extraneous solutions.

Study Guide and Practice Test (pp. 696–701) 1 1 0.5 0.5Standardized Test Practice (pp. 702–703)

Chapter Assessment 1 1 0.5 0.5

TOTAL 14 17 7 9

LESSON OBJECTIVES

*Key to Abbreviations: GCS � Graphing Calculator and Speadsheet Masters,SC � School-to-Career Masters, SM � Science and Mathematics Lab Manual

Study Guide and Intervention, Skills Practice, Practice, and Parent and Student Study Guide Workbooks are also available in Spanish.

ELL

Chapter 12 Rational Expressions and Equations 640B

Materials

705–706 707–708 709 710 91 12-1 12-1

711–712 713–714 715 716 SM 81–84 92 12-2 12-2 33 (Follow-Up: graphingcalculator)

717–718 719–720 721 722 773 93 12-3 12-3

723–724 725–726 727 728 GCS 46 94 12-4 12-4

729–730 731–732 733 734 773, 775 95 12-5 12-5 algebra tiles, product mat

735–736 737–738 739 740 96 12-6 12-6

741–742 743–744 745 746 774 17–18 SC 23 97 12-7 12-7

747–748 749–750 751 752 SC 24 98 12-8 12-8

753–754 755–756 757 758 774 GCS 45 99 12-9 12-9

759–772, 100776–778

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640C Chapter 12 Rational Expressions and Equations

Mathematical Connections and BackgroundMathematical Connections and Background

Inverse VariationPreviously, students have learned that situations

in which y increases as x increases are known as directvariations. However, there are also situations where ydecreases as x increases, or vice versa. These are inversevariations, which can be represented by equations of theform xy � k, where k � 0. To graph an inverse variation,find k and make a table of values for x and y.

The product rule for inverse variations states thatif (x1, y1) and (x2, y2) are solutions of an inverse variation,then x1y1 � x2y2 because both x1y1 and x2y2 equal k. Youcan use the equation x1y1 � x2y2 to solve for missing values of x and y.

Rational ExpressionsRational expressions are simply fractions that

have polynomials for numerators and denominators. Allproperties that apply to rational numbers also apply torational expressions, including the fact that the denomi-nator cannot equal zero. Because certain values of vari-ables in the denominator may produce a value of zerofor the denominator, these values are excluded from thedomain of the expression. To find theses excluded values,apply the Zero Product Property to the factors of thedenominator.

Simplifying rational expressions is just like simpli-fying rational numbers. To simplify rational expressionsin which both the numerator and denominator aremonomials, divide each by the GCF. To simplify rationalexpressions in which both the numerator and denomi-nator are polynomials, first factor each polynomial, andthen eliminate any common factors.

Multiplying Rational ExpressionsThe process of multiplying rational expressions

is just like that of multiplying rational numbers. Simplymultiply the numerators, and multiply the denominators.If rational expressions can be factored, do so before multi-plying so that simplifying will be easier. Multiplyingrational expressions with unit measures to convertbetween units is a process known as dimensional analysis.

Dividing Rational ExpressionsRecall that division is the inverse function of

multiplication. To divide rational expressions, multiplyby the reciprocal of the divisor, just like dividing ration-al numbers. Follow all the rules learned for multiplyingand simplifying rational expressions. Dimensionalanalysis can also be accomplished by dividing rationalexpressions involving units.

Prior KnowledgePrior KnowledgeIn Chapter 5, students solved problemsinvolving direct variation. In previous courses,they have simplified, added, subtracted, andmultiplied fractions and mixed numbers.

Students performed operations with monomials and polynomials in

Chapter 8.

This Chapter

Future ConnectionsFuture ConnectionsStudents will need to be able to solve rationalequations in many real-world problem situa-tions. Therefore, understanding all of theintricacies of rational expressions is necessarybefore delving into solving rational equations.Specifically, in Chapter 14, students will userational expressions and equations whenfinding permutations and combinations.

Continuity of InstructionContinuity of Instruction

This ChapterStudents are introduced to inverse variationproblems and compare these to direct vari-ation problems. Through inverse variationproblems, students are introduced to rationalexpressions. After being introduced torational expressions, students learn to add,subtract, multiply, and divide rational expres-sions. Students go on to learn how to dividepolynomials and simplify mixed expressionsand complex fractions. The chapter ends witha lesson on how to solve rational equations.

Chapter 12 Rational Expressions and Equations 640D

Dividing PolynomialsIn the process of simplifying rational expres-

sions or solving rational equations, it may be neces-sary to divide polynomials. To divide a polynomialby a monomial, simply divide each term of the poly-nomial by the monomial. To divide a polynomial by abinomial, first try to factor the polynomial to see ifthere is perhaps a common binomial factor. If factor-ing is not possible, perform the division by long divi-sion. The procedure is similar to long division of inte-gers. If a power is skipped in the polynomial, writethe power with a coefficient of 0 to hold its place.

Rational Expressions with Like DenominatorsWhen rational expressions have like denomi-

nators, add them by adding the numerators and writ-ing the sum over the common denominator. If thedenominators are polynomials, they must be exactlyalike in all ways. Some denominators are alike, butdo not appear to be. They may be inverse denomi-nators, such as x � 3 and 3 � x. Rewrite the seconddenominator as �(x � 3), and move the negative signto the numerator or change the operation from addi-tion to subtraction.

Subtract rational expressions with like denom-inators by subtracting the numerators, then write thedifference over the common denominator. Be sure tosubtract each term, not just the first one. When sub-tracting expressions, it is often easier to think of thesubtraction in terms of adding the additive inverse ofthe expression.

Rational Expressions withUnlike DenominatorsAs students learned in Lesson 12-6, rational

expressions that have like denominators can be easilyadded or subtracted. When rational expressions haveunlike denominators, a common denominator mustbe found before they can be added or subtracted. Theleast common denominator (LCD) is usually the easi-est to find. The LCD is the least common multiple ofthe two denominators. To find the LCD, find the primefactorization of both denominators, and then use eachfactor the greatest number of times it appears in eitherof the factorizations. Once the LCD is found, changeeach rational expression into an equivalent expres-sion with the LCD as the denominator. Then add orsubtract. Be sure to simplify whenever possible.

www.algebra1.com/key_concepts

Additional mathematical information and teaching notesare available in Glencoe’s Algebra 1 Key Concepts:Mathematical Background and Teaching Notes, which is available at www.algebra1.com/key_concepts. The lessons appropriate for this chapter are as follows.

• Simplifying Rational Expressions (Lesson 35)• Multiplying Rational Expressions (Lesson 36)• Dividing Rational Expressions (Lesson 37)• Rational Expressions with Unlike Denominators

(Lesson 38)

Mixed Expressions andComplex FractionsA mixed expression contains the sum of a

monomial and a rational expression. To simplifymixed expressions, change them into a single rationalexpression by rewriting the monomial as a rationalexpression with the same denominator as the givenrational expression.

A complex fraction is a fraction with a fractionin the numerator, denominator, or both. You can re-write complex fractions as division sentences. Dividethe numerator of the fraction by the denominator,and then write the quotient as a simple fraction.

Solving Rational EquationsAll that students have learned about rational

expressions can be applied to solving rational equa-tions. Rational equations are equations that containrational expressions. If both sides of a rational equa-tion are single fractions, then cross products may beused to solve the equation. Or, one can multiply eachside of the equation by the LCD of the fractions toeliminate the fractions, and then solve the resultingequation.

When solving rational equations, there may betwo solutions. Sometimes, both solutions are correct.Other times, an incorrect one may be introducedwhen multiplying both sides of the equation by theLCD. Always check all solutions in the original equa-tion to make sure they work. If a particular solutiondoes not work, then it is an extraneous solution.

http://www.algebra1.com/key_concepts

640E Chapter 12 Rational Expressions and Equations

TestCheck and Worksheet BuilderThis networkable software has three modules for interventionand assessment flexibility:• Worksheet Builder to make worksheet and tests• Student Module to take tests on screen (optional)• Management System to keep student records (optional)

Special banks are included for SAT, ACT, TIMSS, NAEP, and End-of-Course tests.

Key to Abbreviations: TWE = Teacher Wraparound Edition; CRM = Chapter Resource Masters

Ongoing Prerequisite Skills, pp. 641, 647,659, 664, 671, 683, 689

Practice Quiz 1, p. 659Practice Quiz 2, p. 677

AlgePASS: Tutorial Pluswww.algebra1.com/self_check_quizwww.algebra1.com/extra_examples

5-Minute Check TransparenciesPrerequisite Skills Workbook, pp. 17–18Quizzes, CRM pp. 773–774Mid-Chapter Test, CRM p. 775Study Guide and Intervention, CRM pp. 705–706,

711–712, 717–718, 723–724, 729–730, 735–736,741–742, 747–748, 753–754

MixedReview

Cumulative Review, CRM p. 776 pp. 647, 653, 659, 664, 671,677, 683, 689, 695

ErrorAnalysis

Find the Error, TWE pp. 657, 674, 686Unlocking Misconceptions, TWE pp. 649, 693

Find the Error, pp. 657, 674, 686Common Misconceptions, p. 673

StandardizedTest Practice

TWE pp. 702–703Standardized Test Practice, CRM pp. 777–778

Standardized Test Practice CD-ROM

www.algebra1.com/standardized_test

pp. 646, 647, 653, 659, 664,671, 676, 680, 681, 683, 688,695, 701, 702–703

Open-EndedAssessment

Modeling: TWE pp. 659, 677, 689Speaking: TWE pp. 647, 671, 683Writing: TWE pp. 653, 664, 695Open-Ended Assessment, CRM p. 771

Writing in Math, pp. 646, 653,658, 664, 671, 676, 683, 688,695

Open Ended, pp. 645, 651, 657,662, 669, 674, 681, 686, 694

Standardized Test, p. 703

ChapterAssessment

Multiple-Choice Tests (Forms 1, 2A, 2B), CRM pp. 759–764

Free-Response Tests (Forms 2C, 2D, 3), CRM pp. 765–770

Vocabulary Test/Review, CRM p. 772

TestCheck and Worksheet Builder(see below)

MindJogger Videoquizzes www.algebra1.com/

vocabulary_reviewwww.algebra1.com/chapter_test

Study Guide, pp. 696–700Practice Test, p. 701

Additional Intervention Resources

The Princeton Review’s Cracking the SAT & PSATThe Princeton Review’s Cracking the ACTALEKS

and Assessmentand AssessmentA

SSES

SMEN

TIN

TER

VEN

TIO

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Type Student Edition Teacher Resources Technology/Internet

http://www.algebra1.com/self_check_quiz

http://www.algebra1.com/extra_examples

http://www.algebra1.com/standardized_test

http://www.algebra1.com/vocabulary_review

http://www.algebra1.com/chapter_test

Chapter 12 Rational Expressions and Equations 640F

Algebra 1Lesson

AlgePASS Lesson

12-2 33 Simplifying Rational Expressions

ALEKS is an online mathematics learning system thatadapts assessment and tutoring to the student’s needs.Subscribe at www.k12aleks.com.

For more information on Reading and Writing inMathematics, see pp. T6–T7.

Intervention at HomeParent and Student Study Guide Parents and students may work together to reinforce theconcepts and skills of this chapter. (Workbook, pp. 91–100 or log on to www.algebra1.com/parent_student)

Intervention TechnologyAlgePASS: Tutorial Plus CD-ROM offers a complete, self-paced algebra curriculum.

Reading and Writingin Mathematics

Reading and Writingin Mathematics

Glencoe Algebra 1 provides numerous opportunities toincorporate reading and writing into the mathematics classroom.

Student Edition

• Foldables Study Organizer, p. 641• Concept Check questions require students to verbalize

and write about what they have learned in the lesson.(pp. 645, 651, 657, 662, 669, 674, 681, 686, 694)

• Reading Mathematics, p. 665 • Writing in Math questions in every lesson, pp. 646, 653,

658, 664, 671, 676, 683, 688, 695• WebQuest, pp. 652, 695

Teacher Wraparound Edition

• Foldables Study Organizer, pp. 641, 696• Study Notebook suggestions, pp. 645, 651, 657, 662,

665, 669, 681, 686, 693 • Modeling activities, pp. 659, 677, 689• Speaking activities, pp. 647, 671, 683• Writing activities, pp. 653, 664, 695• Differentiated Instruction, (Verbal/Linguistic), p. 674• Resources, pp. 640, 646, 652, 658, 663, 665,

670, 674, 676, 682, 688, 694, 696

Additional Resources

• Vocabulary Builder worksheets require students todefine and give examples for key vocabulary terms asthey progress through the chapter. (Chapter 12 ResourceMasters, pp. vii-viii)

• Reading to Learn Mathematics master for each lesson(Chapter 12 Resource Masters, pp. 709, 715, 721, 727,733, 739, 745, 751, 757)

• Vocabulary PuzzleMaker software creates crossword,jumble, and word search puzzles using vocabulary liststhat you can customize.

• Teaching Mathematics with Foldables provides suggestions for promoting cognition and language.

• Reading and Writing in the Mathematics Classroom• WebQuest and Project Resources• Hot Words/Hot Topics Sections 1.4, 6.4, 6.5, 8.2

ELL

For more information on Intervention andAssessment, see pp. T8–T11.

Log on for student study help.

• For each lesson in the Student Edition, there are ExtraExamples and Self-Check Quizzes.www.algebra1.com/extra_exampleswww.algebra1.com/self_check_quiz

• For chapter review, there is vocabulary review, test practice, and standardized test practice.www.algebra1.com/vocabulary_reviewwww.algebra1.com/chapter_testwww.algebra1.com/standardized_test

http://www.k12aleks.com

http://www.algebra1.com/parent_student






Have students read over the listof objectives and make a list ofany words with which they arenot familiar.

Point out to students that this isonly one of many reasons whyeach objective is important.Others are provided in theintroduction to each lesson.

Rational Expressionsand Equations

• inverse variation (p. 642)• rational expression (p. 648)• excluded values (p. 648)• complex fraction (p. 684)• extraneous solutions (p. 693)

Key Vocabulary• Lesson 12-1 Solve problems involving inverse

variation.

• Lessons 12-2, 12-3, 12-4, 12-6, and 12-7Simplify, add, subtract, multiply, and dividerational expressions.

• Lesson 12-5 Divide polynomials.

• Lesson 12-8 Simplify mixed expressions andcomplex fractions.

• Lesson 12-9 Solve rational equations.

Performing operations on rational expressions is an importantpart of working with equations. For example, knowing how todivide rational expressions and polynomials can help you simplifycomplex expressions. You can use this process to determine thenumber of flags that a marching band can make from a givenamount of material. You will divide rational expressions and

polynomials in Lessons 12-4 and 12-5.

640 Chapter 12 Rational Expressions and Equations640 Chapter 12 Rational Expressions and Equations

640 Chapter 12 Rational Expressions and Equations

NotesNotes

NCTM LocalLesson Standards Objectives

12-1 1, 2, 6, 8, 9, 10

12-2 1, 2, 6, 8, 9, 10

12-2 2, 8Follow-Up

12-3 1, 2, 6, 8, 9, 10

12-4 1, 2, 6, 8, 9, 10

12-5 1, 2, 6, 8, 9, 10

12-6 1, 2, 6, 8, 9, 10

12-7 1, 2, 6, 8, 9, 10

12-8 1, 2, 6, 8, 9, 10

12-9 1, 2, 6, 8, 9, 10

Key to NCTM Standards: 1=Number & Operations, 2=Algebra,3=Geometry, 4=Measurement, 5=Data Analysis & Probability, 6=ProblemSolving, 7=Reasoning & Proof,8=Communication, 9=Connections,10=Representation

Vocabulary BuilderThe Key Vocabulary list introduces students to some of the main vocabulary termsincluded in this chapter. For a more thorough vocabulary list with pronunciations ofnew words, give students the Vocabulary Builder worksheets found on pages vii andviii of the Chapter 12 Resource Masters. Encourage them to complete the definition of each term as they progress through the chapter. You may suggest that they addthese sheets to their study notebooks for future reference when studying for theChapter 12 test.

ELL

This section provides a review ofthe basic concepts needed beforebeginning Chapter 12. Pagereferences are included foradditional student help.

Additional review is provided inthe Prerequisite Skills Workbook,pp. 17–18.

Prerequisite Skills in the GettingReady for the Next Lesson sectionat the end of each exercise setreview a skill needed in the nextlesson.

Chapter 12 Rational Expressions and Equations 641Chapter 12 Rational Expressions and Equations 641

Make this Foldable to help you organize information aboutrational expressions and equations. Begin with a sheet of

plain 8�12

�" by 11" paper.

Label each tab as shown.

RationalExpressions

RationalEquationsOpen. Cut along the

second fold to make two tabs.

Fold thetop to thebottom.

Fold in half lengthwise.

Reading and Writing As you read and study the chapter, write notes and examplesunder each tab. Use this Foldable to apply what you learned about simplifying rational expressions and solving rational equations in Chapter 12.

Fold in Half

Cut

Fold Again

Label

Prerequisite Skills To be successful in this chapter, you’ll need to master these skills and be able to apply them in problem-solving situations. Review these skills before beginning Chapter 12.

For Lesson 12-1 Solve Proportions

Solve each proportion. (For review, see Lesson 3-6.)

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For Lesson 12-2 Greatest Common Factor

Find the greatest common factor for each pair of monomials. (For review, see Lesson 9-1.)

9. 30, 42 6 10. 60r2, 45r3 15r2 11. 32m2n3, 12m2n 12. 14a2b2, 18a3b 2a2b

For Lessons 12-3 through 12-8 Factor Polynomials

Factor each polynomial. (For review, see Lessons 9-2 and 9-3.) 16. (x � 5)(x � 9)13. 3c2d � 6c2d2 3c2d(1 � 2d)14. 6mn � 15m2 3m(2n � 5m)15. x2 � 11x � 24 (x � 3)(x � 8)16. x2 � 4x � 45 17. 2x2 � x � 21 18. 3x2 � 12x � 9

For Lesson 12-9 Solve Equations

Solve each equation. (For review, see Lessons 3-4, 3-5, and 9-3.) 20. no solution 21. ��146

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Chapter 12 Rational Expressions and Equations 641

For PrerequisiteLesson Skill

12-2 Finding Greatest Common Factors p. 647

12-3 Using Conversion Factors, p. 653

12-4 Factoring Polynomials, p. 659

12-5 Dividing Monomials, p. 664

12-6 Adding Polynomials, p. 671

12-7 Finding Least Common Multiples, p. 677

12-8 Dividing Rational Expressions, p. 683

12-9 Solving Equations, p. 689

Organization of Data and Expository Writing Ask students touse their notes to write expositions (explanations) for operationswith rational expressions and solving rational equations. They shouldpresent the information in such a manner that someone who didnot know about or understand rational expressions and equationswould understand them after reading what the students havewritten. Explain that textbooks are examples of expository writing.

TM

For more informationabout Foldables, seeTeaching Mathematicswith Foldables.

5-Minute CheckTransparency 12-1 Use as

a quiz or review of Chapter 11.

Mathematical Background notesare available for this lesson on p. 640C.

Building on PriorKnowledge

In Lesson 5-2 students learnedhow slope can be used to writeand graph direct variation equa-tions. In this lesson, students willdifferentiate between direct andinverse variation, and learn howto graph and solve inversevariation equations.

is inverse variationrelated to the gears on

a bicycle?Ask students:• Which gear ratio would you

want to use for pedaling on alevel surface? Explain. Youwould want to use the 117.8 gearratio because it requires theslowest pedaling rate. In otherwords, you have to pedal less tomaintain your speed.

• Which gear ratio would youwant to use for pedaling up asteep hill? Explain. You wouldwant to use the 40.5 ratio becausethis requires a much faster pedalingrate to maintain your speed.

Inverse Variationy varies inversely as x if there is some nonzero constant k such that xy � k.

GRAPH INVERSE VARIATION Recall that some situations in which y increases as x increases are direct variations. If y varies directly as x, we can representthis relationship with an equation of the form y � kx, where k � 0. However, in the application above, as one value increases the other value decreases. When theproduct of two values remains constant, the relationship forms an .We say y varies inversely as x or y is inversely proportional to x.

inverse variation

Vocabulary• inverse variation• product rule

Inverse Variation


The number of revolutions of the pedals made when riding a bicycle at a constant speed varies inversely as the gear ratio of the bicycle. In other words, as the gear ratio decreases, the revolutions per minute (rpm) increase. This is why when pedaling up a hill, shifting to a lower gear allows you to pedal with less difficulty.

is inverse variation related tothe gears on a bicycle?is inverse variation related tothe gears on a bicycle?

• Graph inverse variations.

• Solve problems involving inverse variation.

Pedaling Rates to MaintainSpeed of 10 mph

117.8

108.0

92.6

76.2

61.7

49.8

40.5

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97.8

114.0

138.6

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212.0

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RateGear Ratio

Look BackTo review directvariation, see Lesson 5-2.

Study Tip

Inverse VariationProblemsNote that to solve some inverse variationproblems, there are twosteps: first finding thevalue of k, and thenusing this value to find aspecific value of x or y.

Study Tip

Graph an Inverse VariationDRIVING The time t it takes to travel a certain distance varies inversely as the rate r at which you travel. The equation rt � 250 can be used to represent aperson driving 250 miles. Complete the table and draw a graph of the relation.

Solve for r � 5.

rt � 250 Original equation

5t � 250 Replace r with 5.

t � �2550

� Divide each side by 5.

t � 50 Simplify.

Solve the equation for the other values of r.

Example 1Example 1

5 10 15 20 25 30 35 40 45 50

50 25 16.67 12.5 10 8.33 7.14 6.25 5.56 5

r (mph)

t (hours)

5 10 15 20 25 30 35 40 45 50r (mph)

t (hours)

LessonNotes

1 Focus1 Focus

Chapter 12 Resource Masters• Study Guide and Intervention, pp. 705–706• Skills Practice, p. 707• Practice, p. 708• Reading to Learn Mathematics, p. 709• Enrichment, p. 710

Parent and Student Study GuideWorkbook, p. 91

5-Minute Check Transparency 12-1Answer Key Transparencies

TechnologyInteractive Chalkboard

Workbook and Reproducible Masters

Resource ManagerResource Manager

Transparencies

11

22

In-Class ExamplesIn-Class Examples

GRAPH INVERSEVARIATION

MANUFACTURING Theowner of Superfast ComputerCompany has calculated thatthe time t it takes to build aparticular model of computervaries inversely with thenumber of people p workingon the computer. The equationpt � 12 can be used to repre-sent the people building acomputer. Complete a tableand draw a graph of therelation.

Teaching Tip Ask students whynegative values are included forthe inverse variation in Example 2but not for Example 1. Negativerate and time values are notrealistic in Example 1.

Graph an inverse variation inwhich y varies inversely as x,and y � 1 as x � 4.

x

y

O 4 8

8

4

p

t

O

p 2 4 6 8 10 12

t 6 3 2 1.5 1.2 1

Lesson 12-1 Inverse Variation 643

Next, graph the ordered pairs: (5, 50), (10, 25), (15, 16.67), (20, 12.5), (25, 10), (30, 8.33), (35, 7.14), (40, 6.25), (45, 5.56), and (50, 5).

The graph of an inverse variation is not a straight line like the graph of a direct variation. As the rater increases, the time t that it takes to travel the same distance decreases.

www.algebra1.com/extra_examples

Graph an Inverse VariationGraph an inverse variation in which y varies inversely as x and y � 15 when x � 6.

Solve for k.

xy � k Inverse variation equation

(6)(15) � k x � 6, y � 15

90 � k The constant of variation is 90.

Choose values for x and y whose product is 90.

y

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Graphs of inverse variations can also be drawn using negative values of x.

x y

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2 45

3 30

6 15

9 10

USE INVERSE VARIATION If (x1, y1) and (x2, y2) are solutions of an inversevariation, then x1y1 � k and x2y2 � k.

x1y1 � k and x2y2 � kx1y1 � x2y2 Substitute x2y2 for k.

The equation x1y1 � x2y2 is called the for inverse variations. You canuse this equation to form a proportion.

x1y1 � x2y2 Product rule for inverse variations

�xx

1

2

yy

1

1

� � �xx

2

2

yy

2

1

� Divide each side by x2y1.

�xx

1

2

� � �yy

2

1

� Simplify.

You can use the product rule or a proportion to solve inverse variation problems.

product rule

ProportionsNotice that the proportionfor inverse variations isdifferent from theproportion for direct

variation, �xx

1

2� � �

yy

1

2�.

Study Tip

t

rO 10 20 30 40 50

10

20

30

40

50


2 Teach2 Teach

PowerPoint®

InteractiveChalkboard

PowerPoint®

Presentations

This CD-ROM is a customizable Microsoft® PowerPoint®presentation that includes:• Step-by-step, dynamic solutions of each In-Class Example

from the Teacher Wraparound Edition• Additional, Your Turn exercises for each example• The 5-Minute Check Transparencies• Hot links to Glencoe Online Study Tools


33

44

55

In-Class ExamplesIn-Class ExamplesUSE INVERSE VARIATION

If y varies inversely as x andy � 5 when x � 12, find xwhen y � 15. 4

If y varies inversely as x andy � 8 when x � 6, find ywhen x � 4. 12

PHYSICAL SCIENCE Usingthe same lever and fulcrumproblem from Example 5 inthe Student Edition, how farshould a 2-kilogram weightbe from the fulcrum if a 6kilogram weight is 3.2 metersfrom the fulcrum? 9.6 m

Answers (p. 645)

2. Sample answer: Direct variationequations are in the form y � kxand inverse variation equationsare in the form xy � k. The graphof a direct variation is linear whilethe graph of an inverse variationis nonlinear.

4.

5.

11.

xO 50 100�50

�50

�100

50

100

�100

y

xy � � 192

xO

y

4 8

4

8

�4�8

�4

�8

xy � 12

xO 50 100�50

�50

�100

50xy � 192

100

�100

yInverse variation is often used in real-world situations.


Solve for xIf y varies inversely as x and y � 4 when x � 7, find x when y � 14.

Let x1 � 7, y1 � 4, and y2 � 14. Solve for x2.

Method 1 Use the product rule.


7 � 4 � x2 � 14 x1 � 7, y1 � 4, and y2 � 14

�2184� � x2 Divide each side by 14.

2 � x2 Simplify.

Method 2 Use a proportion.

�xx

1

2

� � �yy

2

1

� Proportion for inverse variations

�x7

2

� � �144� x1 � 7, y1 � 4, and y2 � 14

28 � 14x2 Cross multiply.

2 � x2 Divide each side by 14.

Both methods show that x � 2 when y � 14.

Example 3Example 3

LeversA lever is a bar with a pivot point called the fulcrum. For a leverto balance, the lesserweight must bepositioned farther from the fulcrum.

Study Tip

Solve for yIf y varies inversely as x and y � �6 when x � 9, find y when x � 6.

Use the product rule.


9 � (�6) � 6y2 x1 � 9, y1 � �6, and x2 � 6

��

654� � y2 Divide each side by 6.

�9 � y2 Simplify.

Thus, y � �9 when x � 6.

Example 4Example 4

Use Inverse Variation to Solve a ProblemPHYSICAL SCIENCE When two objects are balanced on a lever, their distances from the fulcrum are inversely proportional to their weights. In other words, the greater the weight, the less distance it should be from the fulcrum in order to maintain balance. If an 8-kilogram weight is placed 1.8 meters from the fulcrum, how far should a 12-kilogram weight be placed from the fulcrum in order to balance the lever?

Let w1 � 8, d1 � 1.8, and w2 � 12. Solve for d2.

w1d1 � w2d2 Original equation

8 � 1.8 � 12d2 w1 � 8, d1 � 1.8, and w2 � 12

�1142.4� � d2 Divide each side by 12.

1.2 � d2 Simplify.

The 12-kilogram weight should be placed 1.2 meters from the fulcrum.

w 1d 1 � w 2d 2

fulcrum

lever

d1

w1

d2

w2

Example 5Example 5

TEACHING TIPEncourage students toestimate that a reasonableanswer must be less than1.8 since the 8-kg weightmust be farther from thefulcrum than the 12-kgweight.


Kinesthetic Borrow a fulcrum, lever, and weights from a science teacherto recreate the experiment in Example 5. After some experimentation,challenge students to calculate where to place the weights for the leverto balance.

Differentiated Instruction

PowerPoint®


1. OPEN ENDED Write an equation showing an inverse variation with a constant of 8. Sample answer: xy � 8

2. Compare and contrast direct variation and indirect variation equations and graphs. See margin.

3. Determine which situation is an example of inverse variation. Explain.

a. Emily spends $2 each day for snacks on her way home from school. The total amount she spends each week depends on the number of days schoolwas in session.

b. A business donates $200 to buy prizes for a school event. The number of prizes that can be purchased depends upon the price of each prize.

Graph each variation if y varies inversely as x. 4–5. See margin.4. y � 24 when x � 8 5. y � �6 when x � �2

Write an inverse variation equation that relates x and y. Assume that y variesinversely as x. Then solve.

6. If y � 12 when x � 6, find y when x � 8. xy � 72; 97. If y � �8 when x � �3, find y when x � 6. xy � 24; 48. If y � 2.7 when x � 8.1, find x when y � 5.4. xy � 21.87; 4.059. If x � �

12

� when y � 16, find x when y � 32. xy � 8; �14

�

10. MUSIC The length of a violin string varies inversely as the frequency of itsvibrations. A violin string 10 inches long vibrates at a frequency of 512 cycles per second. Find the frequency of an 8-inch string. 640 cycles per second

Concept Check

3. b; Sample answer:As the price increases, the numberpurchased decreases.

Guided Practice

Application

www.algebra1.com/self_check_quiz

GUIDED PRACTICE KEYExercises Examples

4, 5 1, 26–9 3, 410 5

Practice and ApplyPractice and Apply

indicates increased difficulty�

Graph each variation if y varies inversely as x. 11–16. See margin.11. y � 24 when x � �8 12. y � 3 when x � 4

13. y � 5 when x � 15 14. y � �4 when x � �12

15. y � 9 when x � 8 16. y � 2.4 when x � 8.1


17. If y � 12 when x � 5, find y when x � 3. xy � 60; 2018. If y � 7 when x � �2, find y when x � 7. xy � �14; �219. If y � 8.5 when x � �1, find x when y � �1. xy � �8.5; 8.520. If y � 8 when x � 1.55, find x when y � �0.62. xy � 12.4; �2021. If y � 6.4 when x � 4.4, find x when y � 3.2. xy � 28.16; 8.822. If y � 1.6 when x � 0.5, find x when y � 3.2. xy � 0.8; 0.2523. If y � 4 when x � 4, find y when x � 7. xy � 16; �

176�

24. If y � �6 when x � �2, find y when x � 5. xy � 12; �152�

25. Find the value of y when x � 7 if y � 7 when x � �23

�. xy � �134�; �

23

�

26. Find the value of y when x � 32 if y � 16 when x � �12

�. xy � 8; �14

�

27. If x � 6.1 when y � 4.4, find x when y � 3.2. xy � 26.84; 8.387528. If x � 0.5 when y � 2.5, find x when y � 20. xy � 1.25; 0.0625

Homework HelpFor See

Exercises Examples11–16 1, 217–28 3, 429–37 5

Extra PracticeSee page 846.

�

�

�

�


3 Practice/Apply3 Practice/Apply

Study NotebookStudy NotebookHave students—• add the definitions/examples of

the vocabulary terms to theirVocabulary Builder worksheets forChapter 12.

• include any other item(s) that theyfind helpful in mastering the skillsin this lesson.

About the Exercises …Organization by Objective• Graph Inverse Variation:

11–16• Use Inverse Variation: 17–37

Odd/Even AssignmentsExercises 11–28 are structuredso that students practice thesame concepts whether theyare assigned odd or evenproblems.

Assignment GuideBasic: 11–23 odd, 31–33, 38–60

Average: 11–31 odd, 34–60

Advanced: 12–30 even, 34–54(optional: 55–60)

12. 13. 14.

xO 10 20�10

�10

�20

10

20

�20

y

xy � 48

xO 10 20�10

�20

�40

20

40

�20

y

xy � 75

xO

y

4 8

4

8

�4�8

�4

�8

xy � 12

15.

16.

xO 10 20�10�20

�10

�20

10

20

xy � 19.44

y

xO 4 8�4

�4

�8

4

8

�8

y

xy � 72


Study Guide and Intervention

Inverse Variation

NAME ______________________________________________ DATE ____________ PERIOD _____

12-112-1

Less

on

12-

1

Graph Inverse Variation Situations in which the values of y decrease as the values ofx increase are examples of inverse variation. We say that y varies inversely as x, or y isinversely proportional to x.

Inverse Variation Equation an equation of the form xy � k, where k � 0

Suppose you drive 200 miles without stopping. The time it takes to travel a distance variesinversely as the rate at which you travel.Let x � speed in miles per hour and y � time in hours. Graph the variation.The equation xy � 200 can be used torepresent the situation. Use various speedsto make a table.

x

y

O 20 40 60

30

20

10

x y

10 20

20 10

30 6.7

40 5

50 4

60 3.3

Graph an inversevariation in which y varies inversely asx and y � 3 when x � 12.Solve for k.

xy � k Inverse variation equation

12(3) � k x � 12 and y � 3

36 � k Simplify.

Choose values for x and y whose product is 36.

x

y

O 12 24

24

12

x y

�6 �6

�3 �12

�2 �18

2 18

3 12

6 6

Example 1Example 1 Example 2Example 2

ExercisesExercises

Graph each variation if y varies inversely as x.

1. y � 9 when x � �3 2. y � 12 when x � 4 3. y � �25 when x � 5

4. y � 4 when x � 5 5. y � �18 when x � �9 6. y � 4.8 when x � 5.4

x

y

O 3.6�3.6�7.2 7.2

7.2

3.6

�3.6

�7.2

x

y

O 18�18�36 36

36

18

�18

�36

x

y

O 10�10�20 20

20

10

�10

�20

x

y

O 50�50�100 100

100

50

�50

�100

x

y

O 16�16�32 32

32

16

�16

�32

x

y

O 12�12�24 24

24

12

�12

�24

Study Guide and Intervention, p. 705 (shown) and p. 706

Graph each variation if y varies inversely as x.

1. y � �2 when x � �12 2. y � �6 when x � �5 3. y � 2.5 when x � 2


4. If y � 124 when x � 12, find y when x � �24. xy � 1488; �62

5. If y � �8.5 when x � 6, find y when x � �2.5. xy � �51; 20.4

6. If y � 3.2 when x � �5.5, find y when x � 6.4. xy � �17.6; �2.75

7. If y � 0.6 when x � 7.5, find y when x � �1.25. xy � 4.5; �3.6

8. If y � 6 when x � , find x when y � 4. xy � 3;

9. If y � 8 when x � , find x when y � �12. xy � 2; �

10. If y � 4 when x � �2, find x when y � �10. xy � �8;

11. If y � �7 when x � 4, find x when y � �6. xy � �28;

EMPLOYMENT For Exercises 12 and 13, use the following information.The manager of a lumber store schedules 6 employees to take inventory in an 8-hour workperiod. The manager assumes all employees work at the same rate.

12. Suppose 2 employees call in sick. How many hours will 4 employees need to takeinventory? 12 h

13. If the district supervisor calls in and says she needs the inventory finished in 6 hours,how many employees should the manager assign to take inventory? 8

14. TRAVEL Jesse and Joaquin can drive to their grandparents’ home in 3 hours if theyaverage 50 miles per hour. Since the road between the homes is winding andmountainous, their parents prefer they average between 40 and 45 miles per hour.How long will it take to drive to the grandparents’ home at the reduced speed?between 3 h 20 min and 3 h 45 min

14�3

4�5

1�6

1�4

3�4

1�2

x

y

Ox

y

O 12�12�24 24

24

12

�12

�24

x

y

O 8�8�16 16

16

8

�8

�16

Practice (Average)

Inverse Variation

NAME ______________________________________________ DATE ____________ PERIOD _____

12-112-1Skills Practice, p. 707 and Practice, p. 708 (shown)

Reading to Learn Mathematics

Inverse Variation

NAME ______________________________________________ DATE ____________ PERIOD _____

12-112-1

Pre-Activity How is inverse variation related to the gears on a bicycle?

Read the introduction to Lesson 12-1 at the top of page 642 in your textbook.

Given the data in the table, a bicyclist will pedal

(faster/slower) when shifting to a lower gear and

(faster/slower) when shifting to a higher gear.

Reading the Lesson

1. Write direct variation, inverse variation, or neither to describe the relationship between xand y described by each equation.

a. y � 3x b. xy � 5 c. y � �8x

direct variation inverse variation direct variation

d. y � e. x � f. y � 7x � 1

inverse variation inverse variation neither

2. Why does the equation xy � 0 not describe an inverse variation? The product of xand y must be a nonzero constant.

3. Suppose you want to graph an inverse variation in which y � 12 when x � 9. What twothings should you do before you sketch the graph? First, find the value of k. To dothis, you can multiply 9 and 12 to get 108. Second, make a table of valuesin which the product of x and y values is 108.

4. For each problem, assume that y varies inversely as x. Use the Product Rule to write anequation you could use to solve the problem. Then write a proportion you could use tosolve the problem.

Problem Product Rule Proportion

a. If y � 8 when x � 12, find y when x � 4.

8 � 12 � 4y �

b. If x � 50 when y � 6, find x when y � 30.

50 � 6 � x � 30 �

Helping You Remember

5. To remember how to set up a proportion to solve a problem involving inverse variation,write a sentence describing the form the proportion should have. Sample answer:The first x value divided by the second x value equals the second y valuedivided by the first y value.

30�6

50�x

y�8

12�4

10�y

2�x

slowerfaster

Reading to Learn Mathematics, p. 709

Direct or Indirect VariationFill in each table below. Then write inversely, or directly tocomplete each conclusion.

1. 2.

For a set of rectangles with a width For a car traveling at 55 mi/h, the

of 4, the area varies distance covered varies as the length. as the hours driven.

3. 4.

Th b f i f t b A j b i 128 h f k Th

Hours of Work 128 128 128

People Working 2 4 8

Hours per Person 64 32 16

Oat Bran �13

� cup �23

� cup 1 cup

Water 1 cup 2 cups 3 cups

Servings 1 2 3

directlydirectly

Hours 2 4 5 6

Speed 55 55 55 55

Distance 165 220 275 330

l 2 4 8 16 32

w 4 4 4 4 4

A 8 16 32 64 128

Enrichment

NAME ______________________________________________ DATE ____________ PERIOD _____

12-112-1Enrichment, p. 710

29. GEOMETRY A rectangle is 36 inches wide and 20 inches long. How wide is arectangle of equal area if its length is 90 inches? 8 in.

30. MUSIC The pitch of a musical note varies inversely as its wavelength. If thetone has a pitch of 440 vibrations per second and a wavelength of 2.4 feet, findthe pitch of a tone that has a wavelength of 1.6 feet. 660 vibrations per second

31. COMMUNITY SERVICE Students at Roosevelt High School are collectingcanned goods for a local food pantry. They plan to distribute flyers to homes in the community asking for donations. Last year, 12 students were able todistribute 1000 flyers in nine hours. How long would it take if 15 students hand out the same number of flyers this year? 7.2 h

TRAVEL For Exercises 32 and 33, use the following information.The Zalinski family can drive the 220 miles to their cabin in 4 hours at 55 miles perhour. Son Jeff claims that they could save half an hour if they drove 65 miles perhour, the speed limit.

32. How long will it take the family if they drive 65 miles per hour?

33. How much time would be saved driving at 65 miles per hour? about 40 min

CHEMISTRY For Exercises 34–36, use the following information.Boyle’s Law states that the volume of a gas V varies inversely with applied pressure P.

34. Write an equation to show this relationship. PV � k or P1V1 � P2V2

35. Pressure on 60 cubic meters of a gas is raised from 1 atmosphere to3 atmospheres. What new volume does the gas occupy? 20 m3

36. A helium-filled balloon has a volume of 22 cubic meters at sea level where the air pressure is 1 atmosphere. The balloon is released and rises to a pointwhere the air pressure is 0.8 atmosphere. What is the volume of the balloon at this height? 27.5 m3

37. ART Anna is designing a mobile to suspend from a gallery ceiling. A chain isattached eight inches from the end of a bar that is 20 inches long. On the shorterend of the bar is a sculpture weighing 36 kilograms. She plans to place anotherpiece of artwork on the other end of the bar. How much should the second pieceof art weigh if she wants the bar to be balanced? 24 kg

CRITICAL THINKING For Exercises 38 and 39, assume that y varies inversely as x.

38. If the value of x is doubled, what happens to the value of y? It is half.

39. If the value of y is tripled, what happens to the value of x? It is one third.

40. Answer the question that was posed at the beginning ofthe lesson. See margin.

How is inverse variation related to the gears on a bicycle?

Include the following in your answer:

• an explanation of how shifting to a lower gear ratio affects speed and thepedaling rate on a certain bicycle if a rider is pedaling 73.4 revolutions perminute while traveling at 15 miles per hour, and

• an explanation why the gear ratio affects the pedaling speed of the rider.

41. Determine the constant of variation if y varies inversely as x and y � 4.25 when x � �1.3. B

�3.269 �5.525 �0.306 �2.950DCBA

WRITING IN MATH


ArtAmerican sculptorAlexander Calder was the first artist to usemobiles as an art form.Source: www.infoplease.com


�

about 3 h 20 min


ELL

http://www.infoplease.com

Open-Ended Assessment

Speaking Ask students for anexample of inverse variation.Ask students to suggest valuesthat would be likely to occur.

Getting Ready for Lesson 12-2PREREQUISITE SKILL Studentswill learn how to simplify rationalexpressions in Lesson 12-2. Inorder to simplify rational expres-sions, students must be able tofind greatest common factors.Use Exercises 55–60 to determineyour students’ familiarity withfinding greatest common factors.

Answers

51.

52.

53.

54. y

xO

5x � 8y � �8

x � 4y � 4

3x � 2y � �16

y

xO

x � y � �3

x � y � 1

y � 0

y

xO

2y � �5x � 14

y � 2x � 3

y

xO

y � �x � 1

y � 3x � 5

Maintain Your SkillsMaintain Your Skills


42. Identify the graph of xy � k if x � �2 when y � �4. A

y

xO

4

4 8

8

�4

�4

�8

�8

Dy

xO

4

4 8

8

�4

�4

�8

�8

C

y

xO

4

4 8

8

�4

�4

�8

�8

B

xO

4

4 8

8

�4

�4

�8

�8

yA

Mixed Review

Getting Ready forthe Next Lesson

For each triangle, find the measure of the indicated angle to the nearest degree.(Lesson 11-7)

43. 41° 44. 56° 45. 73°

For each set of measures given, find the measures of the missing sides if �ABC � �DEF. (Lesson 11-6)

46. a � 3, b � 10, c � 9, d � 12 47. b � 8, c � 4, d � 21, e � 28e � 40, f � 36 a � 6, f � 14

48. MUSIC Two musical notes played at the same time produce harmony. Theclosest harmony is produced by frequencies with the greatest GCF. A, C, and C sharp have frequencies of 220, 264, and 275, respectively. Which pair of thesenotes produce the closest harmony? (Lesson 9-1) A and C sharp

Solve each equation. (Lesson 8-6)

49. 7(2y � 7) � 5(4y � 1) �9 50. w(w � 2) � 2w(w � 3) � 16 4

Solve each system of inequalities by graphing. (Lesson 7-5) 51–54. See margin.51. y � 3x � 5 52. y 2x � 3

y �x � 1 2y �5x � 14

53. x � y � 1 54. 3x � 2y �16x � y � �3 x � 4y � 4y 0 5x � 8y � �8

PREREQUISITE SKILL Find the greatest common factor for each set of monomials.(To review greatest common factors, see Lesson 9-1.)

55. 36, 15, 45 3 56. 48, 60, 84 12 57. 210, 330, 150 3058. 17a, 34a2 17a 59. 12xy2, 18x2y3 6xy2 60. 12pr2, 40p4 4p

3

10

?

12 10

?

8 7

?


4 Assess4 Assess

Answer

40. Sample answer: When the gear ratio is lower, the pedaling revolutionsincrease to keep a constant speed. Answers should include the following.

• Shifting gears will require that the rider increase pedaling revolutions.

• Lower gears at a constant rate will cause a decrease in speed, whilehigher gears at a constant rate will cause an increase in speed.


a quiz or review of Lesson 12-1.



In Chapter 9, students learnedhow to find the greatest commonfactor (GCF) and how to factormonomials and polynomialsusing the GCF. In this lesson,students will use the GCF tosimplify rational expressions.

can a rationalexpression be used in

a movie theater?Ask students:• How does the equation given

in this example differ from theinverse variations studied inLesson 12-1? There is a squaredterm and the variables are ondifferent sides of the equationinstead of on the same side.

• As the projector is moved awayfrom the screen, what happensto the image? The image willbecome larger but also dimmer.

EXCLUDED VALUES OF RATIONAL EXPRESSIONS The expression

�dk2� is an example of a rational expression. A is an algebraic

fraction whose numerator and denominator are polynomials.

Because a rational expression involves division, the denominator may not have avalue of zero. Any values of a variable that result in a denominator of zero must beexcluded from the domain of that variable. These are called of therational expression.

excluded values

rational expression

Vocabulary• rational expression• excluded values

Rational Expressions


• Identify values excluded from the domain of a rational expression.

• Simplify rational expressions.

can a rational expression be used in a movie theater?can a rational expression be used in a movie theater?

One Excluded ValueState the excluded value of �5

mm

��

63

�.

Exclude the values for which m � 6 � 0.

m � 6 � 0 The denominator cannot equal 0.

m � 6 Add 6 to each side.

Therefore, m cannot equal 6.

Example 1Example 1

Multiple Excluded ValuesState the excluded values of �

x2 �x2

5�x

5� 6

�.

Exclude the values for which x2 � 5x � 6 � 0.

x2 � 5x � 6 � 0 The denominator cannot equal zero.

(x � 2)(x � 3) � 0 Factor.

Use the Zero Product Property to solve for x.

x � 2 � 0 or x � 3 � 0

x � 2 x � 3

Therefore, x cannot equal 2 or 3.

Example 2Example 2

To determine the excluded values of a rational expression, you may be able tofactor the denominator first.

The intensity I of an image on a movie screen is inversely proportional to the square of the distance d between the projector and the screen. Recall from Lesson 12-1 that this can be represented by the equation

I � �dk2�, where k is a constant.

d

LessonNotes

1 Focus1 Focus



Science and Mathematics Lab Manual, pp. 81–84

5-Minute Check Transparency 12-2Real-World Transparency 12Answer Key Transparencies

TechnologyAlgePASS: Tutorial Plus, Lesson 33Interactive Chalkboard



Transparencies

44

In-Class ExampleIn-Class Example

Lesson 12-2 Rational Expressions 649

SIMPLIFY RATIONAL EXPRESSIONS Simplifying rational expressions issimilar to simplifying fractions with numbers. To simplify a rational expression, youmust eliminate any common factors of the numerator and denominator. To do this,

use their greatest common factor (GCF). Remember that �aabc� � �

aa

� � �bc

� and �aa

� � 1. So,

�aabc� � 1 � �

bc

� or �bc

�.


Use Rational ExpressionsLANDSCAPING Kenyi is helping hisparents landscape their yard and needs to move some large rocks. He plans to use a 6-foot bar as a lever. He positions it as shown at the right.

a. The mechanical advantage of a lever

is �LL

R

E�, where LE is the length of the

effort arm and LR is the length of the resistance arm. Calculate the mechanical advantage of the lever Kenyi is using.

Let b represent the length of the bar and e represent the length of the effort arm.Then b � e represents the length of the resistance arm.

Use the expression for mechanical advantage to write an expression for themechanical advantage in this situation.

�LL

R

E� � �b �

ee

� LE � e, LR � b � e

� �6 �

55

� e � 5, b � 6

� 5 Simplify.

The mechanical advantage is 5.

b. The force placed on the rock is the product of the mechanical advantage and the force applied to the end of the lever. If Kenyi can apply a force of 180 pounds, what is the greatest weight he can lift with the lever?

Since the mechanical advantage is 5, Kenyi can lift 5 � 180 or 900 pounds withthis lever.

Example 3Example 3

Expression Involving MonomialsSimplify ��

271aa

2

5bb

3�.

��271aa

2

5bb

3� � �

((77aa

2

2bb))((�3a

b3

2

))

� The GCF of the numerator and denominator is 7a2b.

1

� �((77aa

2

2bb))((�3a

b3

2

))

� Divide the numerator and denominator by 7a2b.

1

� ��3a

b3

2� Simplify.

Example 4Example 4

Online ResearchFor information about a career as a landscape architect,visit:www.algebra1.com/careers

You can use rational expressions to solve real-world problems.

LandscapeArchitectLandscape architects planthe location of structures,roads, and walkways aswell as the arrangement offlowers, trees, and shrubsin a variety of settings. Source: U.S. Bureau of Labor

and Statistics

fulcrum

rock5 feet

pinch bar

effort arm

resistance arm


2 Teach2 Teach

11

22

33


EXCLUDED VALUES OFRATIONAL EXPRESSIONS

State the excluded value of

. b cannot equal �7.

State the excluded values of

.

a cannot equal �3 or 4.

LANDSCAPING Refer toExample 3 in the StudentEdition. Suppose Kenyi findsa rock that he cannot movewith a 6 foot bar, so he getsan 8 foot bar. But this time,he places the fulcrum so thatthe effort arm is 6 feet long,and the resistance arm is 2 feet long.

a. Explain whether he has moreor less mechanical advantagewith his new setup. Eventhough the bar is longer, becausehe moved the fulcrum he has amechanical advantage of 3, sohis mechanical advantage is lessthan before.

b. If Kenyi can apply a force of180 pounds, what is thegreatest weight he can liftwith the longer bar? 540 lb

SIMPLIFY RATIONALEXPRESSIONS

Simplify .8x4�y5

32x5y2

�4xy7

5a2 � 2��a2 � a � 12

3b � 2�b � 7

Unlocking Misconceptions

Students may assume that excluded values are determined from thesimplified expression. This may or may not be true. Stress that excludedvalues must always be determined from the original denominator.

PowerPoint®

PowerPoint®

http://www.algebra1.com/careers


55

66


Teaching Tip Refer students toChapter 9 if they need to reviewhow to factor polynomials.

Simplify .

Simplify . State

the excluded values of x.

; The excluded values are

�4 and 9.

Concept CheckSimplify . State the

excluded values of x. Theexpression is in simplest form, andthere are no excluded values for x.

Answers

3. Sample answer: You need todetermine excluded values beforesimplifying. One or more factorsmay have been canceled in thedenominator.

24. ; 0, 0

25. ; 0, 0

26. ; 0, 0, 0

27. ; 0, 0, 0

28. ; � b2, 0, 0

29. ; 3n, 0, 0

30. x � 4; �5

31. z � 8; �2

32. ; �4, �2

33. ; �5, 2

34. ; �1, 6

35. ; �9, 3a � 3�a � 9

m � 6�m � 1

2�y � 5

4�x � 4

mn��12n � 4m

7�3

a2b��3a � 7b2

3x�8z

4r�q

a2�3b

5z�2y

2x � 4��x2 � 2x � 2

4�x � 9

4x � 16��x2 � 5x � 36

x � 2�x � 5

x2 � 9x � 14��x2 � 2x � 35

It is important to determine the excluded values of a rational expression using theoriginal expression rather than the simplified expression.


Expressions Involving PolynomialsSimplify �x

x

2

2��

2xx

��

1125

�.

�xx

2

2��

2xx

��

1125

� � �((xx

��

33))((xx

��

54))

� Factor.

1

� �((xx

��

33))((xx

��

54))

� Divide the numerator and denominator by the GCF, x � 3.

1

� �xx

��

54

� Simplify.

Example 5Example 5

Excluded ValuesSimplify �

x23�x

7�x

1�5

10�. State the excluded values of x.

�x2

3�x

7�x

1�5

10� � �

(x �3(x

2)�(x

5�)

5)� Factor.

1

� �(x �

3(x2)

�(x

5�)

5)� Divide the numerator and denominator by the GCF, x � 5.

1

� �x �

32

� Simplify.

Exclude the values for which x2 � 7x � 10 equals 0.

x2 � 7x � 10 � 0 The denominator cannot equal zero.

(x � 5)(x � 2) � 0 Factor.

x � 5 or x � 2 Zero Product Property

CHECK Verify the excluded values by substituting them into the originalexpression.

�x2

3�x

7�x

1�5

10� � �

523�(5

7)(�5)

1�5

10� x � 5

� �25

1�5

3�5

1�5

10� Evaluate.

� �00

� Simplify.

�x2

3�x

7�x

1�5

10� � �

223�(2

7)(�2)

1�5

10� x � 2

� �4 �

61�4

1�5

10� Evaluate.

� ��09� Simplify.

The expression is undefined when x � 5 and x � 2. Therefore, x � 5 and x � 2.

Example 6Example 6

Simplest FormWhen a rationalexpression is in simplestform, the numerator anddenominator have nocommon factors otherthan 1 or �1.

Study Tip

You can use the same procedure to simplify a rational expression in which thenumerator and denominator are polynomials.


Interpersonal Place students in pairs to work through Examples 5 and6. Have one student factor the numerator and the other student factorthe denominator. Then, ask them to compare their factors to identify theGCF in order to simplify the expression.


PowerPoint®


Guided Practice

Application

24–41. See margin.




Concept Check


Exercises Examples16–23 1, 224–27 428–41 5, 642–54 3


1. Describe how you would determine the values to be excluded from the

expression �x2 �

x �5x

3� 6

�.

2. OPEN ENDED Write a rational expression involving one variable for which the excluded values are �4 and �7. Sample answer: �(x � 4)

1(x � 7)�

3. Explain why �2 may not be the only excluded value of a rational expression

that simplifies to �xx

��

32

�. See margin.

State the excluded values for each rational expression.

4. �3

4�a

a� �3 5. �

2xx

2 ��

96

� �3 6. �n2 �

n �n �

520

� �5, 4

Simplify each expression. State the excluded values of the variables.

7. �7

5

0

6

xx3

2

yy2� �

54xy�; 0, 0 8. �

xx

2 ��

479

� x � 7; �7 9. �x2 �

x8�x

4� 16� �

x �1

4�; �4

10. �xx2

2

��

72xx

��

132

� �xx

��

14

�; 3, 4 11. �aa

2

2��

42aa��

182

� 12. �22xx2

2

��

15xx��

2218

�

13. Simplify �b2b2

��

133bb

��

436

�. State the excluded values of b. �bb

��

19

�; 4, 9

AQUARIUMS For Exercises 14 and 15, use the following information.Jenna has guppies in her aquarium. One week later, she adds four neon fish.

14. Write an expression that represents the fraction of neon fish in the aquarium. �4 �

4g

�

15. Suppose that two months later the guppy population doubles, she still has fourneons, and she buys 5 different tropical fish. Write an expression that shows thefraction of neons in the aquarium after the other fish have been added. �

9 �4

2g�


16. �mm

��

32

� 2 17. �b

3�b

5� �5 18. �

3nn2 �

�3168

� �6, 6 19. �2xx2 �

�2150

� �5, 5

20. �aa

2

2��

22aa

��

13

� 21. �xx2

2

��

26xx

��

195

� 22. �n2

n�

2 �n �

3630

� 23. �x2 �

251�2x

x�

2

35�

�3, 1 �5, 3 �6, 5 �7, �5Simplify each expression. State the excluded values of the variables.

24. �3

1

5

4

yyz2z

2

� 25. �1442aa

3

bb3

2� 26. �

6

1

4

6

qqr2r

2ss

�

27. �2

9

4

xx

2

yyzz2� 28. �

21a2b7a

�

3b4

2

9ab3� 29. �36mn

33m�

2n1

3

2m2n2�

30. �x2 �

x �x �

520

� 31. �z2 �

z1�0z

2� 16

� 32. �x2

4�x

6�x

8� 8

�

33. �y2 �

2y3

�

y �

4

10� 34. �

m2m�

2 �5m

3�6

6� 35. �

a2 �a2

6�a �

927

�

36. �xx2

2

��

3xx

��

22

� 37. �b2b2

��

220bb

��

864

� 38. �x3

x�

2 �10

xx2

��

2204x

�

39. �n3

n�

2 �12

8nn2��

1326n

� 40. �42xx

2

2��

68xx

��

48

� 41. �43mm

2

2

��

192mm

��

68

�


4–6 1, 27, 8 4

9–13 5, 614, 15 3

�xx

��

34

�;

�72

�, 4

1. Sample answer:Factor the denominator, set each factor equal to 0, and solve for x.

�aa

��

64

�; �4, 2





• include examples of how to factorrational expressions, and how toidentify excluded values.


About the Exercises …Organization by Objective• Excluded Values of

Rational Expressions: 16–23• Simplify Rational

Expressions: 24–41



Average: 17–41 odd, 46–52,55–80


Answers

36. ; 1, 2

37. ; 4, 16

38. ; �6, �4, 0

39. ; 0, 6

40. ; 2

41. ; �2, �13�4

2x � 1�x � 2

n � 2�n (n � 6)

x � 5�x (x � 6)

(b � 4)(b � 2)��(b � 4)(b � 16)

x � 2�x � 2




NAME ______________________________________________ DATE ____________ PERIOD _____

12-212-2

Less

on

12-

2

Excluded Values of Rational Expressions

Rational an algebraic fraction whose numerator and Expression denominator are polynomials

Example:

Because a rational expression involves division, the denominator cannot equal zero. Anyvalue of the denominator that results in division by zero is called an excluded value of thedenominator.

x2 � 1�

y2

State the excluded

value of .

Exclude the values for which m � 2 � 0.m � 2 � 0 The denominator cannot equal 0.

m � 2 � 2 � 0 � 2 Subtract 2 from each side.

m � �2 Simplify.

Therefore, m cannot equal �2.

4m � 8�m � 2

State the excluded

values of .

Exclude the values for which x2 � 9 � 0.x2 � 9 � 0 The denominator cannot equal 0.

(x � 3)(x � 3) � 0 Factor.

x � 3 � 0 or x � 3 � 0 Zero Product Property

� �3 � 3Therefore, x cannot equal �3 or 3.

x2 � 1�x2 � 9


ExercisesExercises


1. 8 2. �32

3. �4 4. �4

5. �2, 2 6. �4, 4

7. �2 8. �3, �2

9. �3, �1 10. � , 1

11. �1, 5 12. 2, 8

13. �5, 1 14. �2, 2

15. 4, 16 16. �3, 1�4

x2 � 4x � 4��4x2 � 11x � 3

k2 � 2k � 3��k2 � 20k � 64

y2 � y � 2��3y2 � 12

n2 � 2n � 3��n2 � 4n � 5

2x2 � 5x � 1��x2 � 10x � 16

25 � n2��n2 � 4n � 5

1�2

m2 � 1��2m2 � m � 1

k2 � 2k � 1��k2 � 4k � 3

a � 1��a2 � 5a � 6

x2 � 4��x2 � 4x � 4

2x � 18�x2 � 16

2n � 12��n2 � 4

m2 � 4�2m � 8

x2 � 2�x � 4

12 � a�32 � a

2b�b � 8



1. �7, 7 2. 4, 9 3. �5, �3


4. ; 0 5. ; 0, 0, 0 6. ; 0, 0, 0

7. ; c: 0, �2, d: 0 8. p � 6; 2

9. m � 2; 6 10. ; �3, 3

11. ; 2, 7 12. ; �3, 5

13. ; 2 14. ; �7, 1

15. ; 3, 9 16. ; �6, 2

17. ; �3, 4 18. ; � ,

ENTERTAINMENT For Exercises 19 and 20, use the following information.Fairfield High spent d dollars for refreshments, decorations, and advertising for a dance. Inaddition, they hired a band for $550.

19. Write an expression that represents the cost of the band as a fraction of the total amount spent for the school dance.

20. If d is $1650, what percent of the budget did the band account for? 25%

PHYSICAL SCIENCE For Exercises 21–23, use the following information.Mr. Kaminksi plans to dislodge a tree stump in his yard byusing a 6-foot bar as a lever. He places the bar so that 0.5 footextends from the fulcrum to the end of the bar under the treestump. In the diagram, b represents the total length of the barand t represents the portion of the bar beyond the fulcrum.

21. Write an equation that can be used to calculate the MA �mechanical advantage.

22. What is the mechanical advantage? 11

23. If a force of 200 pounds is applied to the end of the lever, what is the force placed on thetree stump? 2200 lb

b � t�

t

b

fulcrum

tree stumpt

550�d � 550

3�2

1�2

y � 4�2y � 3

2y2 � 9y � 4��4y2 � 4y � 3

2(x � 6)��3(x � 4)

2x2 � 18x � 36��3x2 � 3x � 36

r � 3�r � 6

r2 � r � 6��r2 � 4r � 12

t � 9�t � 3

t2 � 81��t2 � 12t � 27

r � 6�r � 7

r2 � 7r � 6��r2 � 6r � 7

y � 8�y � 2

y2 � 6y � 16��y2 � 4y � 4

x � 2�x � 3

x2 � 7x � 10��x2 � 2x � 15

2�b � 2

2b � 14��b2 � 9b � 14

1�m � 3

m � 3�m2 � 9

m2 � 4m � 12��m � 6

p2 � 8p � 12��p � 2

c2d2�8 � c3

5c3d4��40cd2 � 5c4d2

9m�5p3

36m3np2��20m2np5

2z2�xy

6xyz3�3x2y2z

1�4a2

12a�48a3

a2 � 2a � 15��a2 � 8a � 15

p2 � 16��p2 � 13p � 36

4n � 28�n2 � 49

Practice (Average)


NAME ______________________________________________ DATE ____________ PERIOD _____




NAME ______________________________________________ DATE ____________ PERIOD _____

12-212-2

Pre-Activity How can a rational expression be used in a movie theater?


What happens to the image on the screen if the distance between theprojector and the screen increases?

The intensity of the image decreases.

Reading the Lesson

1. Write yes or no to tell whether each expression is or is not a rational expression. If anexpression is not a rational expression, explain why.

a. b. c.

yes yes No; the numerator isnot a polynomial.

Complete each sentence.

2. An excluded value for a rational expression that contains the variable

is a value of x that makes the of the rational

expression equal to .

3. To simplify a rational expression, you divide the numerator and denominator of the

expression by their .

4. If you simplify , you will find that � . Write the equation

you should solve to find the excluded values. Do not solve the equation.

x2 � 5x � 6 � 0

5. Tell whether each statement is true or false for every rational expression and itssimplified form.

a. If a number n is an excluded value for the simplified form of a rational expression,then it must also be an excluded value for the original rational expression. true

b. If a number n is an excluded value for a rational expression, then it must also be anexcluded value for the simplified form of the expression. false


6. Explain how you can use what you know about simplifying fractions for rationalnumbers to remember how to simplify rational expressions.

Sample answer: In both situations, factor the numerator anddenominator. Then divide the numerator and denominator by the GCF.

7�x � 3

7x � 14��x2 � 5x � 6

7x � 14��x2 � 5x � 6

GCF

0denominatorx

�3x � 4��5x

n2 � 15��2n3 � n � 4

x � 2��6x � 7


Rational ExponentsYou have developed the following properties of powers when a is apositive real number and m and n are integers.

am � an � am � n (ab)m � ambm a0 � 1

(am)n � amn � am � n a�m �

Exponents need not be restricted to integers. We can define rationalexponents so that operations involving them will be governed by theproperties for integer exponents.

�a�12

��2 � a�12

� � 2� a �a�

13

��3 � a�13

� � 3 �a�n1

��n � a�n1

� � n� a

a�12

�squared is a. a

�13

�cubed is a. a

�n1

�to the n power is a.

a�12

�is a square root of a. a

�13

�is a cube root of a. a

�n1

�is an nth root of a.

a�12

�� a� a

�13

��

3�a� a�n1

��

n�a�

Now let us investigate the meaning of a�mn

�.

a�mn

�� am � �

n1

�� (am)

�n1

��

n�am� a�mn

�� a

�n1

� � m� �a�

n1

��m � � n�a��m

Th f�m

� n� m� � n��m

1�am

am�an

Enrichment

NAME ______________________________________________ DATE ____________ PERIOD _____



COOKING For Exercises 42–45, use the following information.

The formula t � �40

5(205��

11.8.855aa)

� relates the time t in minutes that it takes to cook

an average-size potato in an oven that is at an altitude of a thousands of feet.

42. What is the value of a for an altitude of 4500 feet? 4.5

43. Calculate the time is takes to cook a potato at an altitude of 3500 feet.

44. About how long will it take to cook a potato at an altitude of 7000 feet?

45. The altitude in Exercise 44 is twice that of Exercise 43. How do your cookingtimes compare for those two altitudes? The times are not doubled; thedifference is 12 minutes.

PHYSICAL SCIENCE For Exercises 46–48, use the following information.To pry the lid off a paint can, a screwdriver that is 17.5 centimeters long is used as a lever. It is placed so that 0.4 centimeter of its length extends inward from the rim of the can.

46. Write an equation that can be used to calculate the mechanical advantage. MA � �

s �r

r�

47. What is the mechanical advantage? 42.75

48. If a force of 6 pounds is applied to the end of the screwdriver, what is the force placed on the lid? 256.5 lb

FIELD TRIPS For Exercises 49–52, use the following information.Mrs. Hoffman’s art class is taking a trip to the museum. A bus that can seat up to 56 people costs $450 for the day, and group rate tickets at the museum cost $4 each.

49. If there are no more than 56 students going on the field trip, write an expressionfor the total cost for n students to go to the museum. 450 � 4n

50. Write a rational expression that could be used to calculate the cost of the trip perstudent. �450

n� 4n�

51. How many students must attend in order to keep the cost under $15 perstudent? 41

52. How would you change the expression for cost per student if the school were to cover the cost of two adult chaperones?

FARMING For Exercises 53 and 54, use the following information.Some farmers use an irrigation system that waters a circular region in a field. Suppose a square field with sides of length 2x is irrigated from the center of the square. The irrigation system can reach a radius of x.

53. Write an expression that represents the fraction of the field that is irrigated. ��

4xx2

2� or �

�4�

54. Calculate the percent of the field that is irrigated to the nearest whole percent. 79%

55. CRITICAL THINKING Two students graphed the following equations on theircalculators. See margin.

y � �xx

2 ��

146

� y � x � 4

They were surprised to see that the graphs appeared to be identical.

a. Explain why the graphs appear to be the same.

b. Explain how and why the graphs are different.

x

�450 � 4

n(n � 2)�

screwdriver

lid

rim of can(fulcrum of

lever)

s

r

43. about 29 min

44. about 41 min

FarmingAlthough the amount offarmland in the UnitedStates is declining, cropproduction has increasedsteadily due in part tobetter irrigation practices.Source: U.S. Department of

Agriculture

�

�

You can use a rational

expression to determine

how an amusement

park can finance a new

roller coaster. Visit

www.algebra1.com/webquest to continue

work on your WebQuest

project.


ELL

http://www.algebra1.com/webquest


Writing Have students write aparagraph explaining how toidentify excluded values andgive an example.

Getting Ready for Lesson 12-3BASIC SKILL Students will learndimensional analysis in Lesson12-3. In order to solve dimen-sional analysis problems, studentsneed a basic understanding ofconversion factors betweendifferent units of measure. UseExercises 75–80 to determineyour students’ familiarity withcompleting open sentences byusing conversion factors.

Answers

55a. Sample answer: The graphappear to be identical becausethe second equation is thesimplified form of the firstequation.

55b. Sample answer: The first graphhas a hole at x � 4 because it isan excluded value of theequation.

56. Sample answer: Use the rationalexpression for light intensity tohelp determine the brightness ofthe picture on the screen for thedistance between the projectorand the screen. Answers shouldinclude the following.

• Find the solutions for theexpression in the denominator.

• Use the light intensityexpression to determine thebrightness of a search light.



56. Answer the question that was posed at the beginning of the lesson. See margin.

How can a rational expression be used in a movie theater?


• a description of how you determine the excluded values of a rationalexpression, and

• an example of another real-world situation that could be described using a rational expression.

57. Which expression is written in simplest form? C

�xx

2

2��

3xx

��

22

� �23xx2��

32

�

�x2

x�

2 �3x

7�x

4� �

2xx2

2

��

x5�x �

123

�

58. In which expression are 1 and 5 excluded values? B

�xx

2

2��

63xx

��

52

� �xx

2

2��

36xx

��

25

�

�xx

2

2��

63xx

��

52

� �xx

2

2��

36xx

��

25

�DC

BA

DC

BA

WRITING IN MATH

Write an inverse variation equation that relates x and y. Assume that y variesinversely as x. Then solve. (Lesson 12-1)

59. If y � 6 when x � 10, find y when x � �12. xy � 60; �5

60. If y � 16 when x � �12

�, find x when y � 32. xy � 8; �14

�

61. If y � �2.5 when x � 3, find y when x � �8. xy � �7.5; 0.9375

Use a calculator to find the measure of each angle to the nearest degree.(Lesson 11-7)

62. sin N � 0.2347 14° 63. cos B � 0.3218 71°

64. tan V � 0.0765 4° 65. sin A � 0.7011 45°

Solve each equation. Check your solution. (Lesson 11-3)

66. �a � 3� � 2 1 67. �2z � 2� � z � 3 7

68. �13 �4�p� � p � 8 �3 69. �3r2 ��61� � 2r � 1 6

Find the next three terms in each geometric sequence. (Lesson 10-7)

70. 1, 3, 9, 27, … 81, 243, 729 71. 6, 24, 96, 384, … 1536, 6144, 24,576

72. �14

�, ��12

�, 1, �2, … 4, �8, 16 73. 4, 3, �94

�, �2176�, …

74. GEOMETRY Find the area of a rectangle if the length is 2x � y units and thewidth is x � y units. (Lesson 8-7) 2x2 � 3xy � y2 units2

BASIC SKILL Complete.

75. 84 in. � ft 7 76. 4.5 m � cm 45077. 4 h 15 min � s 15,300 78. 18 mi � ft 95,04079. 3 days � h 72 80. 220 mL � L 0.22

�8614�, �22

4536

�, �1702294

�

Mixed Review




4 Assess4 Assess

AssessAssess

GraphingCalculatorInvestigation

TeachTeach

Getting StartedGetting StartedA Follow-Up of Lesson 12-2

Know Your Calculator Whenstudents enter two functions thatshould produce identical graphs,it is impossible to tell from thescreen whether there are actuallytwo graphs or one. To make surethere are two graphs, and thatthey overlap, have students

press . Then have themuse the up and down arrow keysto switch between the twographs. Each time they press thekeys, the equation in the upperleft-hand corner of the screenshould change.

• Make sure students enter theequations exactly as shown inthe keystrokes. If students failto put the numerator anddenominator in parentheses,the resulting graph may beincorrect.

Ask students when they woulduse a graphing calculator toconfirm the simplification of arational expression. Sampleanswer: when the simplification isvery complicated

Answers

1.

[�10, 10] scl: 1 by [�10, 10] scl: 1

TRACE


A Follow-Up of Lesson 12-2

Simplify �x2 �

x2

1�0x

2�5

25�.

Factor the numerator and denominator.

• �x2 �

x2

1

�0x

2

�5

25� � �

(

(

xx

��

5

5

)

)

(

(

xx

��

5

5

)

)�

� �(

(

xx

��

5

5

)

)�

When x � �5, x � 5 � 0. Therefore, x cannot equal �5

because you cannot divide by zero.

Graph the original expression. • Set the calculator to Dot mode.

• Enter �x2 �

x2

1

�0x

2

�5

25� as Y1 and graph.

KEYSTROKES:

25

10 25

6

Graph the simplified expression.

• Enter �(

(

xx

��

5

5

)

)� as Y2 and graph.

KEYSTROKES:5

5

Since the graphs overlap, the two expressions are equivalent.

GRAPH)

X,T,�( �)

X,T,�(

ZOOM

) X,T,�

X,T,�( �)

X,T,�( ENTER

MODE


www.algebra1.com/other_calculator_keystrokes

[�10, 10] scl: 1 by [�10, 10] scl: 1

[�10, 10] scl: 1 by [�10, 10] scl: 1

Exercises 4a. Sample answer: Examine the values and verify that they are identical.Simplify each expression. Then verify your answer graphically. Name the excluded values. 1–3. See margin for graphs.1. �

x2 �3x

7�x �

610

� �x �

35

�; �5, �2 2. �xx2

2

��

196xx

��

864

� �xx

��

18

�; 8 3. �53xx

2

2��

160xx

��

35

� �53

�; �1

4. Simplify the rational expression �4x

22

x��

1

9

8x� and answer the following questions using

the TABLE menu on your calculator.

a. How can you use the TABLE function to verify that the original expression and the simplified expression are equivalent?

b. How does the TABLE function show you that an x value is an excluded value?

When simplifying rational expressions, you can use a TI-83 Plus graphing calculator to support your answer. If the graphs of the original expression and the simplified expressioncoincide, they are equivalent. You can also use the graphs to see excluded values.

It displays ERROR.


2. 3.

[�5, 15] scl: 1 by [�10, 10] scl: 1[�10, 10] scl: 1 by [�10, 10] scl: 1

http://www.algebra1.com/other_calculator_keystrokes





Students were briefly introducedto dimensional analysis inChapter 3. In this lesson, studentswill learn that dimensionalanalysis is a process to convertbetween different units ofmeasure through multiplying byrational expressions.

can you multiplyrational expressions to

determine the cost of electricity?Ask students:• What is the simplification of the

expression shown? 0.15h dollar

• In what way are the units inthese expressions similar tovariables? The common units inthe numerator and denominatorcan be factored out.

• How do you know that youended up with the correct unitsat the end of the simplification?The problem states that you wantto know the cost of the electricity.The final unit left is dollars, whichis the correct unit for cost.

MULTIPLY RATIONAL EXPRESSIONS The multiplication expression aboveis similar to the multiplication of rational expressions. Recall that to multiplyrational numbers expressed as fractions, you multiply numerators and multiplydenominators. You can use this same method to multiply rational expressions.

Multiplying Rational Expressions

Lesson 12-3 Multiplying Rational Expressions 655

From this point on, you

may assume that no

denominator of a rational

expression has a value

of zero.

Expressions Involving Monomialsa. Find .

Method 1 Divide by the greatest common factor after multiplying.

�5

8

acb2

3� � �

1

1

5

6

ac2

3

b� � �

1

8

2

0

0

aab2

3

bcc

3

2� Multiply the numerators.

Multiply the denominators.

� The GCF is 40abc2.

� �23ba

2c� Simplify.

Method 2 Divide by the common factors before multiplying.

�5

8

acb2

3� � �

1

1

5

6

ac2

3

b� � � Divide by common factors 5, 8, a, b, and c2.

� �23ba

2c� Multiply.

b. Find �4152mxy

p

2

2� � �2470mx3

3

yp

�.

�4

1

5

2

mxy

p

2

2� � �2

4

7

0

mx3

3

yp

� � � Divide by common factors 4, 9, x, y, m, and p.

� �5

9

0

mx

2

2

yp

� Multiply.

3 m2 127m3p�40x3y10 x2 1

3 1 y12xy2

�45mp2

5 1 p

2 c16c3�15a2b3 a 1

1 1 b2

5ab3�8c2

1 1

140abc2(2b2c)��

40abc2(3a)1

�58acb2

3� � �

1156ac2

3

b�

Example 1Example 1

can you multiply rational expressions to determine the cost of electricity?

can you multiply rational expressions to determine the cost of electricity?

←←


• Multiply rational expressions.

• Use dimensional analysis with multiplication.

There are 25 lights around a patio. Each light is 40 watts, and the cost of electricity is 15 cents per kilowatt-hour. You can use the expression below to calculate the cost of using the lights for h hours.

25 lights � h hours � �40

liwgh

attts

� � �110k0i0lo

ww

aatttts

� ��1 kilo

15w

caettn

�ts

hour��

1100

docellnatrs

�

Lesson x-x Lesson Title 655

Chapter 12 Resource Masters• Study Guide and Intervention, pp. 717–718• Skills Practice, p. 719• Practice, p. 720• Reading to Learn Mathematics, p. 721• Enrichment, p. 722• Assessment, p. 773






Transparencies

LessonNotes

1 Focus1 Focus


33


11

22


MULTIPLY RATIONALEXPRESSIONS

Teaching Tip The text statesthat from this point forward,students can assume that nodenominator of a rationalexpression has a value of zero.Ask students to discuss why thisis an important assumption.

a. Find � .

b. Find � .

a. Find � .

b. Find �

.

DIMENSIONAL ANALYSIS

SPACE The velocity that aspacecraft must have in orderto escape Earth’s gravitationalpull is called the escapevelocity. The escape velocityfor a spacecraft leaving Earthis about 40,320 kilometers perhour. What is this speed inmeters per second? 11,200 m/s

b � 1�4b � 40

b2 � 4b � 3��b2 � 7b � 30

b � 3�4b � 12

x � 8�

x2

x2 � 4x � 32��

x3x

�x � 4

d2q2r 3�

366q5r4

�60c2d2

5c2d4

�18q3r

x�6y3z2

14z�49xy4

7x2y�12z3

Sometimes you must factor a quadratic expression before you can simplify aproduct of rational expressions.


Expressions Involving Polynomialsa. Find �x �

x5

� � �x2 � 2

xx

2

� 15�.

�x �

x5

� � �x2 � 2

xx

2

� 15� � �

x �x

5� � �

(x � 5x)(

2

x � 3)� Factor the denominator.

� The GCF is x(x � 5).

� �x �

x3

� Simplify.

b. Find �a2 �

a7�a

1� 10

� � �3aa

��

23

�.

�a2 �

a7�a �

110

� � �3aa��

23

� � �(a �

a5�)(a

1� 2)

� � �3(

aa��

21)

� Factor the numerators.

� The GCF is (a � 1)(a � 2).

� �3(a

1� 5)� Multiply.

� 3a � 15 Simplify.

1 13(a � 5)(a � 2)(a � 1)��

(a � 1)(a � 2)1 1

x 1x2(x � 5)

��x(x � 5)(x � 3)1 1

Example 2Example 2

DIMENSIONAL ANALYSIS When you multiply fractions that involve units ofmeasure, you can divide by the units in the same way that you divide by variables.Recall that this process is called dimensional analysis.

Dimensional AnalysisOLYMPICS In the 2000 Summer Olympics in Sydney, Australia, Maurice Greenof the United States won the gold medal for the 100-meter sprint. His winningtime was 9.87 seconds. What was his speed in kilometers per hour? Round to thenearest hundredth.

�91.8070

smec

eotenrdss

� � �110k0i0lo

mm

eetteerrs

� � �610

mse

icnountdes

� � �60

1mhionuurtes

�

� �91.8070

smec

eotenrdss

� � �110k0i0lo

mm

eetteerrs

� � �610

mse

icnountdes

� � �60

1mhionuurtes

�

�

� Simplify.

��360

908.

k7ilhoomu

ertsers

� Multiply.

��36.47

1khilooumreters

� Divide numerator and denominator by 98.7.

His speed was 36.47 kilometers per hour.

60 � 60 kilometers��

9.87 � 10 hours

1100 � 1 � 60 � 60 kilometers��

9.87 � 1000 � 1 � 1 hours10

Example 3Example 3

OlympicsAmerican sprinter ThomasBurke won the 100-meterdash at the first modernOlympics in Athens,Greece, in 1896 in 12.0 seconds.Source: www.olympics.org

Look BackTo review dimensionalanalysis, see Lesson 3-8.

Study Tip


2 Teach2 Teach

Logical Challenge students to write at least one rational expressionconsisting of two quadratic expressions, one divided by the other.

The expression should simplify to .x � 1�x � 2


PowerPoint®

PowerPoint®

http://www.olympics.org


Application

26.�(b �

b3�

)(b1� 3)

�

Guided Practice



Find each product.

12. �x82� � �

4xx

4� 2x 13. �

160nr3

3� � �

4325nr3

2� �

2n

�

14. �1

6

0

wy3

xz3

2

� � �54xyzw

2� 15. �32ag

2

hb

� � �2

1

4

5

gab

2h2� �

125

ab

g�

16. �(x �

(x8�)(x

8�)

3)� � �

(x �(x

4�)(x

8�)

3)� �

xx

��

48

� 17. �(n �(n

1�)(n

1�)

1)� � �

(n �(n

1�)(n

4�)

4)�

18. �((zz

��

46))((zz

��

61))

� � �((zz

��

13))((zz

��

54))

� 19. �((xx

��

17))((xx

��

74))

� ��((xx

��

41))((xx

��

1100))

�

20. �x2 �

925

� � �xx

��

55

� 21. �yy

2

2

�

�

4

1� � �

yy

�

�

1

2� �

yy

�

�

21

�

22. �x2 � x

1� 12� � �

xx

��

35

� �(x � 4)

1(x � 5)� 23. �

x2 �x

4�x

6� 32� � �

xx

��

42

� �(x �

x8�)(x

6� 2)

�

24. �xx

��

34

� � �x2 � 7

xx � 12� �

(x �x

4)2� 25. �n2 � 8

nn � 15� � �

2nn�

210

� �n(n

2� 3)�

26. �b2 �

b212

�b

9� 11

� � �b2 �

b2�0b

9� 99

� 27. �a2

a�2 �

a1�6

6� � �

aa

2

2��

74aa��

142

� �((aa

��

34

))((aa

��

32

))

�

�(x �

95)2

�

�1

2

2

5

wy2

2

zx4

2�

1. OPEN ENDED Write two rational expressions whose product is �2x

�.

2. Explain why ��xx

��

65

� is not equivalent to ��xx��

56

�.

3. FIND THE ERROR Amiri and Hoshi multiplied �xx

��

33

� and .

Who is correct? Explain your reasoning.

Find each product.

4. �6

5

4

yy2

� � �5

8

yy� 8y 5. �

1152ss

2tt

3� � �

2st�

6. �m

3�m

4� � �

3(m4m

� 5)� 7. �

x2

2� 4� � �

x �4

2� 2(x � 2)

8. �nn

2 ��

146

� � �nn

��

24

� 9. � �x2 �

5x � 6� �

x �5

3�

10. Find �12s4ec

feoentd

� � �610

mse

icnountdes

� � �60

1mhionuurtes

� � �52

18m0

ifleeet

�. 16.36 mph

11. SPACE The moon is about 240,000 miles from Earth. How many days would it take a spacecraft to reach the moon if it travels at an average of 100 miles per minute? 1�

23

� days

�x2 �

x7

�

x5

� 10��

n2 �

n�

�

8n2

� 16�

�(m � 4

4

)

m(m

2

� 5)�

�1

1

0

6

ss3

tt

2

3�

�x2 �

4

4

xx � 3�

Concept Check


4–9 1–210, 11 3

1. Sample answer:�21

�, �1x

�

2. Sample answer:When the negativesign in front of thefirst expression is distributed, thenumerator is �x � 6.3. Amiri; sampleanswer: Amiri correctlydivided by the GCF.

Hoshi

�xx

+- 3

3� � �

x2 -44xx + 3�

= �xx

+- 3

3� � �

x2 -44xx + 3�

= �x2

1+ 3�

Amiri

�xx

-+ 3

3� � �

x2 -44xx + 3�

= �(x + 3

()x(x

--33)4)(

xx - 1)

�

= �(x + 3

4)(

xx - 1)�


Exercises Examples12–15 116–27 228–37 3


17. �nn

��

44

�

18. �((zz

�

�

66))((zz

�

�

53))

�

19. �((xx

�

�

17))((xx

�

�

71))

�



Study NotebookStudy NotebookHave students—• include examples of how to

multiply rational expressions, andhow to use dimensional analysis.


FIND THE ERRORRemind students

that only commonfactors can be eliminated from thenumerator and denominator.

About the Exercises …Organization by Objective• Multiply Rational

Expressions: 12–27• Dimensional Analysis:

28–31


Alert! Exercise 33 requires theInternet or other researchmaterials.

Assignment GuideBasic: 13–31 odd, 35–61



All: Practice Quiz 1 (1–10)

Diane Stilwell South M.S., Morgantown, WV

“I remind my students that only common factors, not common terms, can becancelled when simplifying rational expressions. I use Exercise 3 to determinewhether my students understand the difference between the factors and theterms of a polynomial.”

Teacher to TeacherTeacher to Teacher




NAME ______________________________________________ DATE ____________ PERIOD _____

12-312-3

Less

on

12-

3

Multiply Rational Expressions To multiply rational expressions, you multiply thenumerators and multiply the denominators. Then simplify.

Find � .

� � Multiply.

� Simplify.

� Simplify.

Find � .

� � � Factor.

� � Simplify.

� Multiply.

Find each product.

1. � 2. �

3. � 4. � 2

5. � 4 6. �

7. � 8. � (a � 5)(a � 2)

9. � 10. �

11. � 12. �

13. � 14. �v � 7��3v � 6

v2 � 8v��v2 � 11v � 24

v2 � 4v � 21��

3v2 � 6v

(a � 3)(a � 5)��(a � 2)(a � 4)

a2 � 3a � 10��a2 � 2a � 8

a2 � 7a � 12��a2 � 2a � 8

6pq�p � q

20p2q2�p2 � q2

3p � 3q�10pq

n � 1�n � 2

n2 � 25��n2 � 6n � 5

n2 � 1��n2 � 7n � 10

m � 1��(m � 1)2

2m � 1��m2 � 2m � 1

m2 � 1��2m2 � m � 1

x � 2�x � 2

2x2 � x � 1��x2 � 3x � 2

x2 � 6x � 8��2x2 � 9x � 4

a2 � 4�a � 5

a2 � 25�a � 2

4�x � 1

x � 1�2x � 2

8x � 8��x2 � 2x � 1

x � 4��2x � 8

x � 4��x2 � 8x � 16

x2 � 16�2x � 8

2n � 4�n � 4

2n � 8�n � 2

16�m � 5

m � 5�8

x � 2�x � 1

x � 4�x � 1

x � 2�x � 4

4n�3

4�mn

mn2�3

6a�b3

a2�b2

6ab�a2b2

x � 4�2x � 8

x � 4��(x � 4)(x � 4)

(x � 4)(x � 4)��2(x � 4)

x � 4��(x � 4)(x � 4)

(x � 4)(x � 4)��2(x � 4)

x � 4��x2 � 8x � 16

x2 � 16�2x � 8

x � 4��x2 � 8x � 16

x2 � 16�2x � 8

2ac�15b

(abcd)(2ac)��(abcd)(15b)

2a2bc2d�15ab2cd

a2b�3cd

2c2d�5ab2

a2b�3cd

2c2d�5ab2

ExercisesExercises

Example 1Example 1

Example 2Example 2

1

1

1 1

1 1


Find each product.

1. � 2. �

3. � 4. �

5. � 6. �

7. � 8. �

9. � 10. �

11. � 12. �

13. � 14. �

Find each product.

15. � � � 16 oz/s

16. � � � 56.25 m/min

17. ANIMAL SPEEDS The maximum speed of a coyote is 43 miles per hour over a distanceof approximately a quarter mile. What is a coyote’s maximum speed in feet per second?Round to the nearest tenth. 63.1 ft/s

18. BIOLOGY The heart of an average person pumps about 9000 liters of blood per day.How many quarts of blood does the heart pump per hour? (Hint: One quart is equal to0.946 liter.) Round to the nearest whole number. 396 qt/h

1 hour��60 minutes

1 day��24 hours

1000 meters��1 kilometer

81 kilometers��1 day

1 minute��60 seconds

1 hour��60 minutes

128 ounces��1 gallon

450 gallons��1 hour

t � 3�t � 5

t2 � t � 20��t2 � 7t � 12

t2 � 6t � 9��t2 � 10t � 25

(b � 1)(b � 1)��(b � 6)(b � 2)

b2 � 5b � 6��b2 � 2b � 8

b2 � 5b � 4��

b2 � 36

3(y � 4)��

y � 5y2 � 8y � 16��y � 3

3y � 9��y2 � 9y � 20

n � 2��5(n � 1)

n � 2��n2 � 9n � 8

n2 � 10n � 16��5n � 10

4��x(x � 7)

x��x2 � 5x � 14

4x � 8�

x2

1�a � 6

a � 3�a � 6

a � 4��a2 � a � 12

x � 4�x � 2

x � 2�x � 4

x2 � 16�x2 � 4

(c � 1)(c � 3)��

6c2 � 9�3c � 3

c2 � 1�2c � 6

m � 4�m � 2

(m � 6)(m � 4)��(m � 7)

m � 7��(m � 6)(m � 2)

9(x � 2)��

x � 272

��(x � 2)(x � 2)(x � 2)(x � 2)��8

3(a � 2b)��

5b24(a � 2b)��

20a2b312a2b�4

8m2ny�

3x36m4n2�

7x2y14xy2�27m2n

1�s2

12s3t2�36s2t

24st2�8s4t3

9xy�

815y3�24x

18x2�10y2

Practice (Average)


NAME ______________________________________________ DATE ____________ PERIOD _____




NAME ______________________________________________ DATE ____________ PERIOD _____

12-312-3

Pre-Activity How can you multiply rational expressions to determine the cost ofelectricity?


• Why are units of measure crossed out in the expression?

to eliminate common units in the numerator and thedenominator

• What is the expression after you multiply the numerators and multiplythe denominators?

or 0.15h dollars

Reading the Lesson

1. Complete the sentence. The product of two rational expressions can always be found by

multiplying the numerators and multiplying the .

2. When you multiply rational expressions, why do you eliminate common factors from theexpression(s) above and below the fraction bar(s)?

so that the final answer will be in simplest form

3. Complete the sentence. If the numerators or denominators of two rational expressions

involve quadratic expressions with two or three terms, try to theseexpressions before you multiply the rational expressions.

4. A student thinks that Example 2b on page 656 shows that you can multiply two rationalexpressions and get an answer that is not a rational expression. Do you agree? Explain.

Sample answer: No; 3a � 15 is not in fraction form, but it is a rational

expression, because it can be written as the fraction .


5. Suppose that a friend was absent when the class worked on this lesson. Tell how you canexplain to your friend the procedure for multiplying rational expressions.

Sample answer: Factor the numerators and denominators of the rationalexpressions. Without using the Distributive Property, write a singlefraction showing the product of the numerators divided by the product of the denominators. Divide numerator and denominator by the GCF.Then simplify.

3a � 15�

1

factor

denominators

15,000h�100,000


Continued FractionsThe following is an example of a continued fraction. By starting at the bottom you can simplify the expression to a rational number.

3 � � 3 �

� 3 � or

Express as a continued fraction.

� 2 � Notice that the numerator of the last fraction must be equal to 1 before the process stops.

� 2 �

� 2 � 1�1 � �

9�

1��1190�

10�19

48�19

48�19

67�13

28�13

4��173�

4�1 � �

67�

Enrichment

NAME ______________________________________________ DATE ____________ PERIOD _____

12-312-3

ExampleExample

Enrichment, p. 722


38. c and e; Sample answer: The expressions each have a GCF that canbe used to simplify the expressions.

Exchange RatesA system of floatingexchange rates amonginternational currencieswas established in 1976. It was needed because theold system of basing acurrency’s value on goldhad become obsolete.Source: www.infoplease.com

Find each product.

28. �2.54 c1en

inticmh

eters��

121ifnocohtes

� � �13

yfeaerdt

� 91.44 cm/yd

29. �60 k1ilhoomuerters

� � �110k0i0lo

mm

eetteerrs

� � �60

1mhionuurtes

� � �610mse

inco

untedss

� 16.67 m/s

30. �13s2ec

feoentd

� � �610

mse

icnountdes

� � �60

1mhionuurtes

� � �52

18m0

ifleeet

� 21.8 mi/h

31. 10 feet � 18 feet � 3 feet � �2

1

7

y

f

a

e

r

e

d

t

3

3� 20 yd3

32. DECORATING Alani’s bedroom is 12 feet wide and 14 feet long. What will itcost to carpet her room if the carpet costs $18 per square yard? $336

33. EXCHANGE RATES While traveling in Canada, Johanna bought some gifts tobring home. She bought 2 T-shirts that cost $21.95 (Canadian). If the exchangerate at the time was 1 U.S. dollar for 1.37 Canadian dollars, how much didJohanna spend in U.S. dollars? about $16.02

Online Research Data Update Visit www.algebra1.com/data_update tofind the most recent exchange rate between the United States and Canadiancurrency. How much does a $21.95 (Canadian) purchase cost in U.S. dollars?

34. CITY MAINTENANCE Street sweepers can clean 3 miles of streets per hour. A city owns 2 street sweepers, and each sweeper can be used for three hoursbefore it comes in for an hour to refuel. How many miles of streets can becleaned in 18 hours on the road? 108 mi

TRAFFIC For Exercises 35–37, use the following information.During rush hour one evening, traffic was backed up for 13 miles along a particularstretch of freeway. Assume that each vehicle occupied an average of 30 feet of spacein a lane and that the freeway has three lanes.

35. Write an expression that could be used to determine the number of vehiclesinvolved in the backup.

36. How many vehicles are involved in the backup? 6864

37. Suppose that there are 8 exits along this stretch of freeway, and if it takes eachvehicle an average of 24 seconds to exit the freeway. Approximately how manyhours will it take for all the vehicles in the backup to exit? 5.72 h

38. CRITICAL THINKING Identify the expressions that are equivalent to �xy

�. Explain why the expressions are equivalent.

a. �xy

��

33

� b. �33

��

xy

� c. �33xy� d. �

xy

3

3� e. �nn

3

3xy

�


How can you multiply rational expressions to determine the cost ofelectricity?


• an expression that you could use to determine the cost of using 60-watt lightbulbs instead of 40-watt bulbs, and

• an example of a real-world situation in which you must multiply rationalexpressions.

WRITING IN MATH

3 lanes � �113

lman

ilees

� � �52

18m0

ifleeet

� � �13v0eh

feicelte

�


ELL

http://www.algebra1.com/data_update



Modeling Use note cards orscrap pieces of paper to write amultiplication problem involvingrational expressions. Tape thepieces of paper on the board sothat each numerator and denom-inator is on a separate piece ofpaper. Ask volunteers to comeup and find common factors. Foreach common factor found, havestudents rewrite the expressionon a piece of paper and tape itover the existing expression,until the product is found.

Getting Ready for Lesson 12-4PREREQUISITE SKILL Students willlearn to divide rational expres-sions in Lesson 12-4, whichinvolves factoring polynomials.Use Exercises 56–61 to determineyour students’ familiarity withfactoring polynomials.

Assessment Options

Practice Quiz 1 The quizprovides students with a briefreview of the concepts and skillsin Lessons 12-1 through 12-3.Lesson numbers are given to theright of exercises or instructionlines so students can reviewconcepts not yet mastered.

Quiz (Lessons 12-1 through 12-3) is available on p. 773 of theChapter 12 Resource Masters.




40. Which expression is the product of and ? D

�13

zx3

y3

� �13

yx3z2

� �13

zx3

yz� �

13yx3z3

�

41. Identify the product of and . A

4a2�3(a � 1)

D4a�3(a � 1)

C4a�3

B4a�3(a � 1)

A

a2�3a � 3

�4

aa2 �

�

a4

�

DCBA

8x2z2�

2y3

13xyz�4x2y


Practice Quiz 1Practice Quiz 1

Graph each variation if y varies inversely as x. (Lesson 12-1) 1–2. See margin.1. y � 28 when x � 7 2. y � �6 when x � 9

Simplify each expression. (Lesson 12-2)

3. �4298aab

2� �

47ab� 4. �

y3y

�

�

3y1

2

� y 5. �b2b2

��

133bb

��

436

� �bb

��

19

� 6. �33nn2

2

��

153nn

��

24

� �nn

��

24

�

Find each product. (Lesson 12-3) 8. �a

2�x

9� 9. �

5(n4� 5)�

7. �32mm

2� � �

198mm2� 3m2 8. �

5a1�0x2

10� � �

a2 � 141xa

3

� 18� 9. �

n42n��

285

� � �5nn

��

510

� 10. �x2

x�2 �

x �9

6� � �

xx

2

2��

74xx��

142

�

Lessons 12-1 through 12-3

Mixed Review State the excluded values for each rational expression. (Lesson 12-2)

42. �s2s

��

366

� �6, 6 43. �a2 �

a2

3�a

2�5

10� �5, 3 44. �

x2 �x �

6x3� 9

� �3

Write an inverse variation equation that relates x and y. Assume that y variesinversely as x. Then solve. (Lesson 12-1)

45. If y � 9 when x � 8, find x when y � 6. xy � 72; 1246. If y � 2.4 when x � 8.1, find y when x � 3.6. xy � 19.44; 5.447. If y � 24 when x � �8, find y when x � 4. xy � �192; �4848. If y � 6.4 when x � 4.4, find x when y � 3.2. xy � 28.16; 8.8

Simplify. Assume that no denominator is equal to zero. (Lesson 8-2)

49. ��

779

12� �73 or �343 50. �

2

8

0

pp8

6

� �25p2� 51. �

246aa

3

6bc

4

2c7

� �4b

a

4

3c5�

Solve each inequality. Then check your solution. (Lesson 6-2)

52. �g8

� � �72

� g � 28 53. 3.5r 7.35 r 2.1 54. �94k� �

35

� k �145�

55. FINANCE The total amount of money Antonio earns mowing lawns and doingyard work varies directly with the number of days he works. At one point, heearned $340 in 4 days. At this rate, how long will it take him to earn $935?(Lesson 5-2) 11 days

PREREQUISITE SKILL Factor each polynomial.(To review factoring polynomials, see Lessons 9-3 through 9-6.)

56. x2 � 3x � 40 57. n2 � 64 58. x2 � 12x � 36

59. a2 � 2a � 35 60. 2x2 � 5x � 3 61. 3x3 � 24x2 � 36x

58. (x � 6)2 59. (a � 7)(a � 5) 60. (2x � 1)(x � 3) 61. 3x(x � 2)(x � 6)

10. �xx

��

42

�

56. (x � 5)(x � 8)57. (n � 8)(n � 8)


4 Assess4 Assess

Answers

39. Sample answer: Multiply rational expressions to perform dimensionalanalysis. Answers should include the following.

• 25 lights � h hours � � � �

• Sample answer: converting units of measure

1 dollar��100 cents

15 cents��1 kilowatt � hour

1 kilowatt��1000 watts

60 watts�

light

1. 2.

xO 2 4 6 8�2�4�6�8

�80�60�40

4020

8060 xy � 54

y

xO 2 4 6 8�2�4�6�8

�80�120�160

4080

120160

xy � 196

y





In Lesson 2-4, students learnedthat to divide rational numbers,you multiply by the reciprocal.In this lesson, students will learnthat the same procedure is usedfor dividing rational expressionsinvolving monomials, binomials,and polynomials.

can you determine thenumber of aluminum

soft drink cans made each year?Ask students:• Use the information in the

paragraph to write an equationthat represents c, the numberof aluminum soft drink cansproduced each year.

c � 63.9 billion

• How do you solve the equa-tion that you just wrote for c?Multiply both sides of the equation

by the reciprocal of , which is .

• About how many aluminumsoft drink cans are producedeach year? 102.24 billion

8�5

5�8

5�8

DIVIDE RATIONAL EXPRESSIONS Recall that to divide rational numbersexpressed as fractions you multiply by the reciprocal of the divisor. You can use thissame method to divide rational expressions.

Dividing Rational Expressions


can you determine the number of aluminum soft drink cans made each year?can you determine the number of aluminum soft drink cans made each year?

Expression Involving MonomialsFind �5

7x2� � �

1201x3�.

�57x2� � �

1201x3� � �

57x2� � �

1201x3� Multiply by �1

201x3�, the reciprocal of �12

01x3�.

� �57x2� � �

1201x3� Divide by common factors 5, 7, and x2.

� �23x� Simplify.

Example 1Example 1

• Divide rational expressions.

• Use dimensional analysis with division.

Most soft drinks come in aluminum cans. Although more cans are usedtoday than in the 1970s, the demandfor new aluminum has declined. This is due in large part to the greatnumber of cans that are recycled. In recent years, approximately63.9 billion cans were recycled

annually. This represents �58

� of all cans produced.

Year

Num

ber o

f Can

s Co

llect

ed (b

illio

ns)

70

60

50

40

30

20

10

0

1975

1980

1985

1990

1995

2000

Expression Involving BinomialsFind �n

n��

13

� � �2nn

��

42

�.

�nn

��

13

� � �2nn

��

42

� � �nn

��

13

� � �2nn

��

42

� Multiply by �2nn

��

42

�, the reciprocal of �2nn

��

42

�.

� �nn

��

13

� � �2(

nn

��

41)

� Factor 2n � 2.

� �nn

��

13

� � �2(

nn

��

41)

� The GCF is n � 1.

� �2(

nn

��

43)

� or �2nn

��

46

� Simplify.

Example 2Example 2

1 2x

1 3

1

1

LessonNotes

1 Focus1 Focus


Graphing Calculator and Spreadsheet Masters, p. 46






Transparencies

Lesson 12-4 Dividing Rational Expressions 661

Often the quotient of rational expressions involves a divisor that is a binomial.


Divide by a BinomialFind �5a

a��

150

� � (a � 2).

�5aa �

�150

� � (a � 2) � �5aa

��

150

� � �(a �

12)

� Multiply by �(a +1

2)�, the reciprocal of (a � 2).

� �5(

aa��

52)

� � �(a �

12)

� Factor 5a � 10.

� �5(

aa��

52)

� � �(a �

12)

� The GCF is a � 2.

� �a �

55

� Simplify.

Example 3Example 3

Expression Involving PolynomialsFind �m

2 � 34m � 2� � �

mm

��

21

�.

�m2 � 3

4m � 2� � �

mm

��

21

� � �m2 � 3

4m � 2� � �

mm

��

12

� Multiply by the reciprocal, �mm��

12

�.

� �(m � 1)

4(m � 2)� � �

mm

��

12

� Factor m2 � 3m � 2.

� �(m � 1)

4(m � 2)� � �

mm

��

12

� The GCF is m � 2.

� �(m �

41)2

� Simplify.

Example 4Example 4

DIMENSIONAL ANALYSIS You can divide rational expressions that involveunits of measure by using dimensional analysis.

MultiplicativeInverseAs with rationalnumbers, dividingrational expressionsinvolves multiplying bythe inverse. Rememberthat the inverse of

a � 2 is �a �

12

�.

Study Tip

Sometimes you must factor a quadratic expression before you can simplify thequotient of rational expressions.

1

1

1

1

Dimensional AnalysisSPACE In November, 1996, NASA launched the Mars Global Surveyor. It took309 days for the orbiter to travel 466,000,000 miles from Earth to Mars. What wasthe speed of the spacecraft in miles per hour? Round to the nearest hundredth.

Use the formula for rate, time, and distance.

rt � d rate � time � distance

r � 309 days � 466,000,000 mi t � 309 days, d � 466,000,000

r ��466

3,00090d,0a0y0s

mi� Divide each side by 309 days.

��466,0

30009,0d0a0y

ms

iles��

241

hdoauyrs

� Convert days to hours.

��466

7,040106,0h0o0umrs

iles� or about �

62,8317h.1o1umr

iles�

Thus, the spacecraft traveled at a rate of about 62,837.11 miles per hour.

Example 5Example 5

SpaceThe first successful Marsprobe was the Mariner 4,which arrived at Mars onJuly 14, 1965.Source: NASA


11

22

33

44


55


DIVIDE RATIONALEXPRESSIONS

Teaching Tip Tell students torewrite division problemsinvolving rational expressions asmultiplications of the reciprocal.Trying to do the division withoutrewriting may lead to confusionand errors.

Find � .

Find � .

Find � (x � 3).

Find � .

DIMENSIONAL ANALYSIS

AVIATION In 1986, anexperimental aircraft namedVoyager was piloted by JennaYeager and Dick Rutanaround the world non-stop,without refueling. The triptook exactly 9 days andcovered a distance of 25,012miles. What was the speed ofthe aircraft in miles per hour?Round to the nearest mile perhour. The speed of the aircraftwas about 116 miles per hour.

q2 � 9q � 14��

7

q � 13�q � 7

q2 � 11q � 26��

7

12�x � 7

12x � 36��

x � 7

3m � 6�m � 5

m � 4�m � 2

3m � 12�

m � 5

15x3�

424x�75

6x4

�5

Visual/Spatial Have students write division problems involving rationalexpressions on note cards, pieces of scrap paper, or any other item thatthey can manipulate. Then have students physically “flip” the fractions tomultiply by the reciprocal. The act of “flipping” the fractions will helpcement the concept in students’ minds.


2 Teach2 Teach

PowerPoint®

PowerPoint®



Study NotebookStudy NotebookHave students—• include examples of how to divide

rational expressions• include any other item(s) that they

find helpful in mastering the skillsin this lesson.

1. OPEN ENDED Write two rational expressions whose quotient is �x5yz�.

2. Tell whether the following statement is always, sometimes, or never true. Explainyour reasoning. Sometimes; sample answer: 0 has no reciprocal.Every real number has a reciprocal.

3. Explain how to calculate the mass in kilograms of one cubic meter of asubstance whose density is 2.16 grams per cubic centimeter. Sample answer:Divide the density by the given volume, then perform dimensional analysis.

Find each quotient.

4. �10

7n3� � �

52n1

2� 6n 5. �

a2�a

3� � �

aa

��

73

� �a

2�a

7�

6. �3mm

��

145

� � �6mm

��

524

� 18 7. �2xx

��

56

� � (x � 3) �x �

25

�

8. �k2 �

k �4k

3� 4

� � �2kk��

26

� �2(k

1� 2)� 9. �

x2 �2x

11�x �

418

� � �x2 �

x �5x

1� 6

�

10. Express 85 kilometers per hour in meters per second. 23.61 m/s

11. Express 32 pounds per square foot in square inches. �29

� lb/in2

12. COOKING Latisha was making candy using a two-quart pan. As she stirred

the mixture, she noticed that the pan was about �23

� full. If each piece of candy

has a volume of about �34

� ounce, approximately how many pieces of candy will Latisha make? about 57 pieces


Concept Check1. Sample answer:

�145y2z

� � �34xy�

Guided Practice

9. �2((xx��

12))((xx��

93))

�

Application


4–6 1, 27 3

8, 9 410–12 5


Find each quotient.

13. �ba2

2� � �ba3� ab 14. �

np2

4� � �

np3

2� n2p

15. �4yx4

3� � �

8yx2

2� �

2xy2� 16. �

170nm

2

2� � �

2154mn3

4� �

54mn

2�

17. �xs

2

2

yt

3

2

z� � �

xs

2

3

ytz2

3

� �szy2

2� 18. �

ag

4

2bhc3

3� � �

agb3

2

hc3

2� �

a3

bcg�

19. �b2

4�b

9� � (b � 3) �

b4�b

3� 20. �

m2

5�m

16� � (m � 4) �

m5�m

4�

21. �k

3�k

1� � (k � 2) �

(k � 13)(kk � 2)� 22. �

d5�d

3� � (d � 1) �

(d � 35)d(d � 1)�

23. �34xx

��

1128

� � �2xx

��

48

� �43((2xx��

49))

� 24. �42aa

��

86

� � �2aa��

44

� �aa

��

43

�

Complete.

25. 24 yd3 � ft3 648 26. 0.35 m3 � cm3 350,00027. 330 ft/s � mi/h 225 28. 1730 plants/km2 � plants/m2

0.00173

29. What is the quotient when �2xx��

56

� is divided by �x �

25

�? x � 3

30. Find the quotient when �mm

��

87

� is divided by m2 � 7m � 8. �(m � 7)

1(m � 1)�


Exercises Examples13–18 119–22 323, 24 229–36 425–28, 537–41



About the Exercises …Organization by Objective• Divide Rational

Expressions: 13–24, 29, 30,31–34

• Dimensional Analysis:25–28


Assignment GuideBasic: 13–37 odd, 38, 39, 42,45–75

Average: 13–37 odd, 40–42,45–75

Advanced: 14–36 even, 42,45–69 (optional: 70–75)



NAME ______________________________________________ DATE ____________ PERIOD _____

12-412-4

Less

on

12-

4

Divide Rational Expressions To divide rational expressions, multiply by thereciprocal of the divisor. Then simplify.

Find � .

� �

�

�

Find � .

� �

�

�

� x � 1

Find each quotient.

1. � 2. �

3. � 6xy 4. � 2

5. � 6. �

7. � 3(x � 2) 8. �

9. � 1 10. �

11. � 12. �

13. � 14. �(a � 3)(2a � 1)��

a � 7a � 3�4a2 � 1

a2 � 9��2a2 � 13a � 7

(a � 4)(a � 5)��(a � 2)(a � 3)

a2 � 9�a2 � 25

a2 � 7a � 12��a2 � 3a � 10

p �q�p � q

p2 � q2�p � q

p2 � 2pq � q2��p � q

�4(n � 2)��

n3 � n�4n � 12

n2 � 5n � 6��

n2 � 3n

3m(m � 7)��

m � 6m2 � 13m � 42��

3m2m2 � 49��m

x � 4�x � 2

x2 � 6x � 8��x2 � 4x � 4

4b2tu�

9s6a2bc�8st2u

a2b3c�3s2t

x � 3�15

x2 � 5x � 6��5

y � 6�y � 7

y � 6�y � 7

y2 � 36�y2 � 49

1�n � 2

n2 � 4�n

2n � 4�2n

m � 5�16

m � 5�8

y�16

3xy2�8

m�4

n�m

n�4

12�b2

b�a

12ab�a2b2

(x � 2)(x � 1)��x � 3

(x � 9)(x � 3)��(x � 9)(x � 2)

(x � 2)(x � 1)��x � 3

(x � 9)(x � 3)��(x � 9)(x � 2)

x2 � x � 2��x � 3

x2 � 6x � 27��x2 � 11x � 18

x � 3��x2 � x � 2

x2 � 6x � 27��x2 � 11x � 18

x � 3��x2 � x � 2

x2 � 6x � 27��x2 � 11x � 18

24�abd

10ab�c2d2

12c2d�5a2b2

10ab�c2d2

12c2d�5a2b2

c2d2�10ab

12c2d�5a2b2

c2d2�10ab

12c2d�5a2b2

Example 1Example 1

Example 2Example 2

ExercisesExercises

1

a b 11

1 1 1

1 1 1

d

1 1 12


Find each quotient.

1. � 2. �

3. � (a � 1) 4. � (z � 4)

5. � 8 6. �

Complete.

7. 1.75 m2 � cm2 17,500 8. 0.54 tons/yd3 � lb/ft3 40

Find each quotient.

9. � 10. �

11. � 12. �

13. � 14. �

15. � 16. �

TRAFFIC For Exercises 17 and 18, use the following information.On Saturday, it took Ms. Torres 24 minutes to drive 20 miles from her home to her office.During Friday’s rush hour, it took 75 minutes to drive the same distance.

17. What was Ms. Torres’s speed in miles per hour on Saturday? 50 mi/h

18. What was her speed in miles per hour on Friday? 16 mi/h

SHOPPING For Exercises 19 and 20, use the following information.

Ashley wants to buy some treats for her dog Foo. She can purchase a 1 -pound box of dog

treats for $2.99. She can purchase the same treats in a 2-pound package on sale for $4.19.

19. What is the cost of each in cents per ounce? Round to the nearest tenth.

1 lb: 15.0 cents/oz, 2 lb: 13.1 cents/oz

20. If a box of treats costs $3.49 at a rate of 14.5 cents per ounce, how much does the boxweigh in ounces and in pounds?

about 24.1 oz, or just over 1 lb1�2

1�4

1�4

y � 2�y � 2

(a � 6)(a � 5)��(a � 6)(a � 5)

y2 � 9y � 14��y2 � 7y � 18

y2 � 6y � 7��y2 � 8y � 9

a2 � 4a � 12��a2 � 3a � 10

a2 � 8a � 12��a2 � 7a � 10

x � 2��2(x � 3)

6x � 6��x2 � 5x � 6

3x � 3��x2 � 6x � 9

b � 4�b � 4

2b � 8�2b � 18

b2 � 2b � 8��b2 � 11b � 18

4��(n � 5)(n � 5)

n2 � 6n � 5��4n � 12

n � 1��n2 � 2n � 15

y � 5��2(y � 8)

2y � 4�y � 1

y2 � 3y � 10��y2 � 9y � 8

3(n � 8)��

n � 8n � 8�27

n2 � 9n � 8��9n � 9

(s � 2)(s � 10)��

7s � 2�s � 2

s2 � 8s � 20��7

x � 3��3(x � 4)

2x � 6�x � 3

4x � 12�6x � 24

y � 5�2y � 6

4y � 20�y � 3

z � 4�

3zz2 � 16�3z

2a��(a � 1)(a � 1)

2a�a � 1

np�xy

mnp2�

x3ymn2p3�

x4y220�3ab

21a3�35b

28a2�7b2

Practice (Average)


NAME ______________________________________________ DATE ____________ PERIOD _____




NAME ______________________________________________ DATE ____________ PERIOD _____

12-412-4

Less

on

12-

4

Pre-Activity How can you determine the number of aluminum soft drink cansmade each year?


Write an equation that you could use to determine the number ofaluminum cans, in billions, produced each year.

Sample answer: x � 63.9

Reading the Lesson

1. Why is it important to know the reciprocal of the divisor when you divide two rationalexpressions?

Sample answer: To divide two rational expressions, you multiply by thereciprocal of the divisor.

2. State the reciprocal of the divisor in each of the following.

a. � (b � 2) b. �

3. Supply the reason for each step below.

� Original expression

� � Multiply by the reciprocal of the divisor.

� � Factor y2 � 5y � 6.

� � Divide by the GCF.

� Multiply the fractions.


4. One way to remember something is to see how it is similar to something you alreadyknow. How is dividing rational expressions similar to dividing rational numbers that arein fraction form?

Sample answer: When you divide fractions for rational numbers, youinvert and multiply. This is the same method as multiplying by thereciprocal when you divide rational expressions.

y � 1�y � 2

y � 3�1

y � 1��( y � 2)( y � 3)

y � 3�1

y � 1��( y � 2)( y � 3)

y � 3�1

y � 1��y2 � 5y � 6

1�y � 3

y � 1��y2 � 5y � 6

3d�c

c�3d

2c2�d

1�b � 2

3b � 15�b � 1

5�8


Division by Zero?You may remember being told, “division by zero is not possible” or“division by zero is undefined” or “we never divide by zero.” Have youwondered why this is so? Consider the two equations below.

� n � m

Because multiplication is the inverse of division, these equations leadto the following.

0 � n � 5 0 � m � 0

There is no number that will make the first equation true. Anynumber at all will satisfy the second equation.

For each expression, give the values that must be excluded from the replacementset in order to prevent division by zero.

1. x � 1 2. x � 3. x � �2or x � 2

(x � y)2x2 � y2 � z2x � y � 3

(x � 1)(x � 1)��(x � 2)(x � 2)

1�2

2(x � 1)�2x � 1

x � 1�x � 1

0�0

5�0

Enrichment

NAME ______________________________________________ DATE ____________ PERIOD _____



Find each quotient.

31. �x2 � 2

2x � 1� � �

xx

��

11

� �(x � 1)

2(x � 1)� 32. �n

2 � 34n � 2� � �

nn

��

12

� �(n �

42)2

�

33. �aa

2

2��

86aa��

196

� � �23aa

��

89

� �32((aa

��

43))

� 34. �b2 �

b �4b

2� 4

� � �2bb��

44

� �2(

bb

��

42)2�

35. �xx2

2

��

5xx

��

26

� � �xx2

2

��

72xx

��

132

� �((xx

��

43

))

� 36. �xx

2

2��

2xx

��

3105

� � �xx

2

2��

32xx

��

1284

�

37. TRIATHLONS Irena is training for an upcoming triathlon and plans to run12 miles today. Jorge offered to ride his bicycle to help her maintain her pace. If Irena wants to keep a steady pace of 6.5 minutes per mile, how fast shouldJorge ride in miles per hour? about 9.2 mph

CONSTRUCTION For Exercises 38 and 39, use the following information.A construction supervisor needs to determinehow many truckloads of earth must be removedfrom a site before a foundation can be poured.The bed of the truck has the shape shown atthe right.

38. Use the formula V � �d(a

2� b)� � w to write an equation involving units that

represents the volume of the truck bed in cubic yards if a � 18 feet, b � 15 feet,w � 9 feet, and d � 5 feet. V ��

5 ft(18 ft2� 15 ft)� � 9 ft � �

217ydft

3

3�

39. There are 20,000 cubic yards of earth that must be removed from the excavationsite. Write an equation involving units that represents the number of truckloadsthat will be required to remove all of the earth. Then solve the equation.

TRUCKS For Exercises 40 and 41, use the following information.The speedometer of John’s truck uses the revolutions of his tires to calculate the speed of the truck.

40. How many revolutions per minute do the tires make when the truck is traveling at 55 miles per hour?

41. Suppose John buys tires with a diameter of 30 inches. Whenthe speedometer reads 55 miles per hour, the tires would still revolve at the same rate as before. However, with the new tires, the truck travels a different distance in each revolution. Calculate the actual speed when the speedometer reads 55 miles per hour. 68.7 mph

42. CRITICAL THINKING Which expression is not equivalent to the reciprocal of

�xx

2 �

�

4

2

yy

2

�? Justify your answer.

a. b. �2y

��1

x� c. �

x �1

2y� d. �

1x

� � �21y�

SCULPTURE For Exercises 43 and 44, use the following information. A sculptor had a block of marble in the shape of a cube with sides x feet long. A

piece that was �12

� foot thick was chiseled from the bottom of the block. Later, the

sculptor removed a piece �34

� foot wide from the side of the marble block.

43. Write a rational expression that represents the volume of the block of marblethat remained.

44. If the remaining marble was cut into ten pieces weighing 85 pounds each, writean expression that represents the weight of the original block of marble.

1�

�xx

2 �

� 2

4

yy2

�

26 in.

a

bd

w


TriathlonsThe IronmanChampionship Triathlonheld in Hawaii consists of a 2.4-mile swim, a 112-mile bicycle ride, and a 26.2-mile run.Source: www.infoplease.com

42. d; Sample answer:

�1x

� � �21y� � �

2y2x

�

yx

�

which is not equivalent to �

xx2

�

�

24yy2�

43.

44. (10 � 85 pounds) �

�x � �12

��x � �34

��(x)��x3

�x � �12

��x � �34

��(x)��

x3

36. �((xx

��

36))((xx

��

43))

�

39. n � 20,000 yd3 �

�217ydft

3

3� � �7

142

tr.u5cfkt3�;

727.27

about 770 rpm


ELL




Writing Have students write ashort paragraph explaining whydividing rational expressions isthe same as multiplying by thereciprocal.

Getting Ready for Lesson 12-5PREREQUISITE SKILL Studentswill learn to divide polynomialsin Lesson 12-5. In order to dividepolynomials, students mustrecall how to divide monomials.Use Exercises 70–75 to determineyour students’ familiarity withdividing monomials.

Answer

45. Sample answer: Divide the

number of cans recycled by to

find the total number of cansproduced. Answers should includethe following.

• x � 63,900,000 cans � �1 pound�33 cans

5�8

5�8



How can you determine the number of aluminum soft drink cans made each year?


• a rational expression that will give the amount of new aluminum needed to

produce x aluminum cans today when �58

� of the cans are recycled and 33 cans

are produced from a pound of aluminum.

46. Which expression is the quotient of �35bc� and �

1185bc

�? B

�1185bc2

2� �

12

� �1185bc

� 2

47. Which expression could be used for the width of the rectangle? C

x � 2 (x � 2)(x � 2)2

x � 2 (x � 2)(x � 2)DC

BAA � x2 � 4

x2 � x � 2x � 1

DCBA

WRITING IN MATH


Mixed Review

51. �7(x �

x2�

y)(yx � 5)�

65. {xx �0.7}



Find each product. (Lesson 12-3)

48. �x2 �

x7�x

5� 10� � �

x �1

2� 1 49. �x

x

2

2��

38xx

��

11

05

� � �xx

2

2��

54

xx

��

64

� �xx

��

22

�

50. �x

4�y

4� � �

x2 �

1

7

6

xy� 12� �

x �4

3� 51. �x

2 �x

8�x

y� 15� � �

7xx

�

�

1

3

4y�


52. �c2 �

c1�2c

6� 36

� �c �

16

� 53. �x2

2�5 �

x �x2

30� ��

xx

��

56

�

54. �a2 �

a �4a

3� 3

� �a �

11

� 55. �n2 �

n2

8�n

1�6

16� �

nn

��

44

�

Solve each equation. Check your solutions. (Lesson 9-6)

56. 3y2 � 147 {�7} 57. 9x2 � 24x � �16 ��43

��58. a2 � 225 � 30a {15} 59. (n � 6)2 � 14 ��6 � �14� �

Find the degree of each polynomial. (Lesson 8-4)

60. 13 � �18

� 0 61. z3 � 2z2 � 3z � 4 3 62. a5b2c3 � 6a3b3c2 10

Solve each inequality. Then check your solution. (Lesson 6-2)

63. 6 � 0.8g {gg 7.5} 64. �15b � �28 �bb �21

85�� 65. �0.049 � 0.07x

66. �37

�h � �439� �hh � �

17

�� 67. �1�24r

� �230� �rr � ��

210�� 68. �

y6

� �12

� {yy 3}

69. MANUFACTURING Tanisha’s Sporting Equipment manufactures tennis racketcovers at the rate of 3250 each month. How many tennis racket covers will thecompany manufacture by the end of the year? (Lesson 5-3) 39,000 covers

PREREQUISITE SKILL Simplify. (To review dividing monomials, see Lesson 8-2.)

70. �6xx4

2� �

x62� 71. �

255mm

4� �

m5

3� 72. �

14

85

aa

3

5� �52a2�

73. �bb

6

3cc

3

6� �bc3

3� 74. �

1

2

2

8

xx

3

4

yy

2

� �37

yx� 75. �

7xz

4

3z2� �

7zx4�


4 Assess4 Assess

ReadingMathematics

Getting StartedGetting Started

TeachTeach

AssessAssess

Study NotebookStudy Notebook

Have students describe what theythink of when they see a fraction.Some may think of fractions, nomatter the value of the numera-tor or denominator, as numbersthat are less than one. Others maythink of fractions as divisionproblems. Discuss with studentswhat the different parts of afraction mean.

• The concept that a singledenominator divides each termof the numerator will be veryimportant to students whendividing polynomials. Givestudents additional problems ofthis type so they can practicerewriting them as separatefractions.

Ask students to summarize whatthey have learned about writing andsimplifying rational expressions.

Investigating Slope-Intercept Form 665Reading Mathematics Rational Expressions 665

Several concepts need to be applied when reading rational expressions.

• A fraction bar acts as a grouping symbol, where the entire numerator is divided by the entire denominator.

Example 1 �6x

1�0

4�

It is to read the expression as the quantity six x plus four divided by ten.

It is to read the expression as six x divided by ten plus four, orsix x plus four divided by ten.

• If a fraction consists of two or more terms divided by a one-term denominator, the denominator divides each term.

Example 2 �6x

1�0

4�

It is to write �6x

1�0

4� � �

61x0� � �

140�.

� �35x� � �

25

� or �3x

5� 2�

It is also to write �6x

1�0

4� � �

2(32x

��5

2)�.

� �2(3

2x

��5

2)� or �

3x5� 2�

It is to write �6x

1�0

4� � �

6x1�0

4� � �

3x5� 4�.

Reading to LearnWrite the verbal translation of each rational expression. 1–6. See margin.

1. �m �

4

2� 2. �

x3

�

x1

� 3. �aa2��

2

8�

4. �xx2 �

�2

5

5� 5. �

x2 �x

3

�

x2

� 18� 6. �

xx2

2��

2

xx

��

2

3

0

5�

Simplify each expression.

7. �3x

9� 6� �

x �3

2� 8. �

4n �8

12� �

n �4

3� 9. �

5x2

1�0x

25x� �

x �2

5�

10. �x2 �

x7

�x

3

� 12� �

x �1

4� 11. �

x2 �

x2

�

xyy� y2� �

x �1

y� 12. �

x2 �

x2

8

�x

1

�6

16� �

xx

��

44

�

incorrect

correct

correct

incorrect

correct


3x

5

Reading Mathematics Rational Expressions 665

Answers

1. Sample answer: the quantity m plus two, divided by 42. Sample answer: three x divided by the quantity x minus 13. Sample answer: the quantity a plus 2 divided by the quantity a squared plus 84. Sample answer: the quantity x squared minus 25 divided by the quantity x plus 55. Sample answer: the quantity x squared minus 3x plus 18 divided by the quantity x minus 26. Sample answer: the quantity x squared plus 2x minus 35 divided by the quantity x squared

minus x minus 20

English LanguageLearners may benefit fromwriting key concepts from thisactivity in their Study Notebooksin their native language and thenin English.

ELL



Mathematical Background notesare available for this lesson on p. 640D.


In Lesson 8-2, students learnedto divide by monomials. In thislesson, students will first learnhow to divide binomials andpolynomials by monomials, thenthey will learn how to dividepolynomials by binomials.

is division used insewing?

Ask students:• Think back to the Reading

Mathematics activity on theprevious page. How might youread the expression given in thisexample? The quantity 36 yardsminus seven and one-half yards,divided by one and one-half yards.

• How can the fraction be writtenso that the denominator divideseach term?

�

• Use a calculator to find thenumber of flags that can bemade using the roll of fabric.19 flags

7�12

� yards�

1�12

� yards

36 yards�1�

12

� yards

DIVIDE POLYNOMIALS BY MONOMIALS To divide a polynomial by a monomial, divide each term of the polynomial by the monomial.

Dividing Polynomials


• Divide a polynomial by a monomial.

• Divide a polynomial by a binomial.

Marching bands often use intricate marchingroutines and colorful flags to add interest to their shows. Suppose a partial roll of fabric is used to make flags. The original roll was

36 yards long, and 7�12

� yards of the fabric were

used to make a banner for the band. Each flag

requires 1�12

� yards of fabric. The expression

can be used to represent

the number of flags that can be made using the roll of fabric.

36 yards � 7�1

2� yards

��1�

1

2� yards

is division used in sewing?is division used in sewing?

Divide a Binomial by a MonomialFind (3r2 � 15r) � 3r.

(3r2 � 15r) � 3r � �3r2 �

3r15r

� Write as a rational expression.

� �33rr

2� � �

135rr

� Divide each term by 3r.

� �33rr

2� � �

135rr

� Simplify each term.

� r � 5 Simplify.

Example 1Example 1

Divide a Polynomial by a MonomialFind (n2 � 10n � 12) � 5n.

(n2 � 10n � 12) � 5n � �n2 � 1

50nn � 12� Write as a rational expression.

� �5nn

2� � �

150nn� � �

51n2� Divide each term by 5n.

� �5nn

2� � �

150nn� � �

51n2� Simplify each term.

� �n5� � 2 � �

51n2� Simplify.

Example 2Example 2

1 1

r 5

1 1

n 2

LessonNotes

1 Focus1 Focus

Chapter 12 Resource Masters• Study Guide and Intervention, pp. 729–730• Skills Practice, p. 731• Practice, p. 732• Reading to Learn Mathematics, p. 733• Enrichment, p. 734• Assessment, pp. 773, 775


Teaching Algebra With ManipulativesMasters, pp. 10, 11, 17, 197





Transparencies


Use algebra tiles to find (x2 � 3x � 2) � (x � 1).Step 1 Model the polynomial x2 � 3x � 2.

Step 2 Place the x2 tile at the corner of the product mat. Place one of the 1 tiles as shown to make a length of x � 1.

Step 3 Use the remaining tiles to make a rectangular array.

The width of the array, x � 2, is the quotient.

Model and Analyze

Use algebra tiles to find each quotient.1. (x2 � 3x � 4) � (x � 1) (x � 4) 2. (x2 � 5x � 6) � (x � 2) (x � 3)3. (x2 � 16) � (x � 4) (x � 4) 4. (2x2 � 4x � 6) � (x � 3) (2x � 2)

5. Describe what happens when you try to model (3x2 � 4x � 3) � (x � 2).What do you think the result means? You cannot do it. There is a remainder.

x � 2x 2

xx

x � 1

x

1

1

x 2

x � 1

1

1

1

x 2 x x x

Lesson 12-5 Dividing Polynomials 667

DIVIDE POLYNOMIALS BY BINOMIALS You can use algebra tiles tomodel some quotients of polynomials.


Divide a Polynomial by a BinomialFind (s2 � 6s � 7) � (s � 7).

(s2 � 6s � 7) � (s � 7) � �s2

(�s �

6s7�)

7� Write as a rational expression.

� �(s �

(s7�)(s

7�)

1)� Factor the numerator.

� �(s �

(s7�)(s

7�)

1)� Divide by the GCF.

� s � 1 Simplify.

Example 3Example 3

Recall from Lesson 12-4 that when you factor, some divisions can be performedeasily.

1

1


2 Teach2 Teach

11

22


33


DIVIDE POLYNOMIALS BYMONOMIALS

Teaching Tip As an alternativeto dividing each term of thepolynomial by the monomial,students can factor and theneliminate the GCF.

Find (4x2 � 18x) � 2x.2x � 9

Find (2y2 � 3y � 9) � 3y.

� 1 �

DIVIDE POLYNOMIALS BYBINOMIALS

Find (2r2 � 5r �3) � (r � 3).2r � 1

3�y

2y�3

Algebra Activity

Materials algebra tiles and product mat

Point out to students that the divisor in this example is found along thehorizontal axis, and the quotient is found along the vertical axis. Suggest thatstudents arrange their problems in this way as they use algebra tiles in thisactivity.

PowerPoint®

PowerPoint®


44

55


Teaching Tip Remind studentsto pay close attention to thesigns of the binomials as theyperform the long division. Sinceeach binomial is being sub-tracted, the sign of the secondterm in the binomial changes.

Find (x2 � 7x �15) � (x � 2).The quotient is x � 9 with aremainder of 3.

Teaching Tip Ask students toexplain why a term with a zerocoefficient is added to thepolynomial. A coefficient with azero term has a value of zero, soit does not affect the quotient. Itis simply a placeholder.

Find (x3 � 34x � 45) � (x � 5).The quotient is x2 � 5x � 9.

Concept CheckExplain which method youwould use to find the quotient inthe following problems.

a. (x2 � 9x � 2) � (x � 1)Since the polynomial cannot befactored, this quotient must befound through long division.

b. (2x2 � 13x � 15) � (2x � 3)Since the polynomial can befactored, divide by the GCF.

Answers

2. Sample answer: A remainder ofzero means that the divisor is afactor of the dividend.

11. � 3 �

12. � 1 � 4�a

a�7

7�3x

x�3

Polynomial with Missing TermsFind (a3 � 8a � 24) � (a � 2).

Rename the a2 term using a coefficient of 0.

(a3 � 8a � 24) � (a � 2) � (a3 � 0a2 � 8a � 24) � (a � 2)

a2 � 2a � 12

a � 2�a3� �� 0�a2� �� 8�a�� 2�4�

Multiply a2 and a � 2.

2a2 � 8a Subtract and bring down 8a.

Multiply 2a and a � 2.

12a � 24 Subtract and bring down 24.

Multiply 12 and a � 2.

0 Subtract.

Therefore, (a3 � 8a � 24) � (a � 2) � a2 � 2a � 12.

(�) 12a � 24

(�) 2a2 � 4a

(�) a3 � 2a2

Example 5Example 5

In Example 3 the division could be performed easily by dividing by commonfactors. However, when you cannot factor, you can use a long division processsimilar to the one you use in arithmetic.


Long DivisionFind (x2 � 3x � 24) � (x � 4).

The expression x2 � 3x � 24 cannot be factored, so use long division.

Step 1 Divide the first term of the dividend, x2, by the first term of the divisor, x.

x x2 � x � x

x � 4�x�2�� 3�x� �� 2�4�

Multiply x and x � 4.

7x Subtract.

Step 2 Divide the first term of the partial dividend, 7x � 24, by the first term ofthe divisor, x.

x � 7 7x � x � 7

x � 4�x�2�� 3�x� �� 2�4�

7x � 24 Subtract and bring down the 24.

Multiply 7 and x � 4.

4 Subtract.

The quotient of (x2 � 3x � 24) � (x � 4) is x � 7 with a remainder of 4, which can

be written as x � 7 � �x �

44

�. Since there is a nonzero remainder, x � 4 is not a

factor of x2 � 3x � 24.

(�) 7x � 28

(�) x2 � 4x

(�) x2 � 4x

Example 4Example 4

FactorsWhen the remainder ina division problem is 0,the divisor is a factor ofthe dividend.

Study Tip

When the dividend is an expression like a3 � 8a � 21, there is no a2 term. In such situations, you must rename the dividend using 0 as the coefficient of themissing terms.


Intrapersonal Ask students to write a journal entry about what theylearned in this lesson. Tell them to include the aspects of the lessonthat they liked, and that they did not like. They should also explainwhich concepts they feel like they mastered, and which ones they arestill not comfortable with. Have students share their journal entries withyou privately.


PowerPoint®


1. Choose the divisors of 2x2 � 9x � 9 that result in a remainder of 0. b and ca. x � 3 b. x � 3 c. 2x � 3 d. 2x � 3

2. Explain the meaning of a remainder of zero in a long division of a polynomialby a binomial. See margin.

3. OPEN ENDED Write a third-degree polynomial that includes a zero term.Rewrite the polynomial so that it can be divided by x � 5 using long division.Sample answer: x3 � 2x2 � 8; x3 � 2x2 � 0x � 8

Find each quotient.

4. (4x3 � 2x2 � 5) � 2x 2x2 � x � �25x� 5. 2 � �

5a

� � �72b2�

6. (n2 � 7n � 12) � (n � 3) n � 4 7. (r2 � 12r � 36) � (r � 9)

8. �4m3

2�m

5�m

3� 21

� 2m2 � 3m � 7 9. (2b2 � 3b � 5) � (2b � 1)

10. ENVIRONMENT The equation C � �12

1

0

�

,00

p0p

� models the cost C in dollars for a

manufacturer to reduce the pollutants by a given percent, written as p in decimalform. How much will the company have to pay to remove 75% of the pollutantsit emits? $360,000

14a2b2 � 35ab2 � 2a2��

7a2b2

Concept Check

Guided Practice

7. r � 3 � �r �

99

�

9. b � 2 � �2b

3� 1�

Application




Find each quotient. 11–22. See margin.11. (x2 � 9x � 7) � 3x 12. (a2 � 7a � 28) � 7a

13. 14.

15. (x2 � 9x � 20) � (x � 5) 16. (x2 � 6x � 16) � (x � 2)

17. (n2 � 2n � 35) � (n � 5) 18. (s2 � 11s � 18) � (s � 9)

19. (z2 � 2z � 30) � (z � 7) 20. (a2 � 4a � 22) � (a � 3)

21. (2r2 � 3r � 35) � (r � 5) 22. (3p2 � 20p � 11) � (p � 6)

23. �3t2 �3t

1�4t

4� 24

� t � 6 24. �12n2

2�n

3�6n

5� 15

� 6n � 3

25. 26.

27. �6x3

2�x

9�x2

3� 6

� 3x2 � �2x

6� 3� 28. �

9g3

3

�

g �

5g2

� 8� 3g2 � 2g � 3 � �

3g2� 2�

29. Determine the quotient when 6n3 � 5n2 � 12 is divided by 2n � 3.

30. What is the quotient when 4t3 � 17t2 � 1 is divided by 4t � 1? t2 � 4t � 1

LANDSCAPING For Exercises 31 and 32, use the following information.A heavy object can be lifted more easily using a lever and fulcrum. The amount thatcan be lifted depends upon the length of the lever, the placement of the fulcrum,

and the force applied. The expression �W(L

x� x)� represents the weight of an object

that can be lifted if W pounds of force are applied to a lever L inches long with the fulcrum placed x inches from the object.

31. Suppose Leyati, who weighs 150 pounds, uses all of his weight to lift a rockusing a 60-inch lever. Write an expression that could be used to determine theheaviest rock he could lift if the fulcrum is x inches from the rock. �150(6

x0 � x)�

32. Use the expression to find the weight of a rock that could be lifted by a 210-pound man using a six-foot lever placed 20 inches from the rock. 546 lb

20b3 � 27b2 � 13b � 3��

4b � 33x3 � 8x2 � x � 7��

x � 2

12a3b � 16ab3 � 8ab��

4ab9s3t2 � 15s2t � 24t3��

3s2t2


Exercises Examples11–14 1, 2

15–18, 23, 24 319–22, 25, 26 4

27–30 5


25. 3x2 � 2x � 3 ��x �

12

�

26. 5b2 � 3b � 1 29. 3n2 � 2n � 3 ��2n

3� 3�


4, 5 1, 26, 7 3

9, 10 48 5



Study NotebookStudy NotebookHave students—• include examples of how to divide

polynomials.• include any other item(s) that they


About the Exercises …Organization by Objective• Divide Polynomials by

Monomials: 11–14• Divide Polynomials by

Binomials: 15–28, 29, 30



Average: 11–29 odd, 33–35,39–60


Answers

13. 3s � �

14. 3a2 � 4b2 � 2

15. x � 4

16. x � 8

17. n � 7

18. s � 2

19. z � 9 �

20. a � 7 �

21. 2r � 7

22. 3p � 2 � 1�p � 6

1�a � 3

33�z � 7

8t�s2

5�t




NAME ______________________________________________ DATE ____________ PERIOD _____

12-512-5

Less

on

12-

5

Divide Polynomials by Monomials To divide a polynomial by a monomial, divideeach term of the polynomial by the monomial.

Find (4r2 � 12r) � 2r.

(4r2 � 12r) � 2r �

� � Divide each term.

� � Simplify.

� 2r � 6 Simplify.

12r�2r

4r2�2r

12r�2r

4r2�2r

4r2 � 12r��2r

Find (3x2 � 8x � 4) � 4x.

(3x2 � 8x � 4) � 4x �

� � �

� � �

� � 2 � 1�x

3x�4

4�4x

8x�4x

3x2�4x

4�4x

8x�4x

3x2�4x

3x2 � 8x � 4��4x


ExercisesExercises

Find each quotient.

1. (x3 � 2x2 � x) � x x2 � 2x � 1 2. (2x3 � 12x2 � 8x) � 2x x2 � 6x � 4

3. (x2 � 3x � 4) � x x2 � 3 � 4. (4m2 � 6m � 8) � 2m2 2 � �

5. (3x3 � 15x2 � 21x) � 3x x2 � 5x � 7 6. (8m2n2 � 4mn � 8n) � n 8m2n � 4m � 8

7. (8y4 � 16y2 � 4) � 4y2 2y2 � 4 � 8. (16x4y2� 24xy � 5) � xy 16x3y � 24 �

9. 3x2 � 5x � 6 10. 5ab � 6b �

11. 2x2 � 3x � 12. � �

13. 1 � � 14. � 4 �

15. 3b2 � � 16. xy2 � 2xy � 4

17. 18.

9xyz � 2z � ab � 4 � �6

�a2b2

5�ab

12�y

2a3b3 � 8a2b2 � 10ab � 12��

2a2b29x2y2z � 2xyz � 12x��xy

2x2y3 � 4x2y2 � 8xy��2xy

6�a2

4b�a

6a2b2 � 8ab � 12��

2a2

6q�p

p�q

p2 � 4pq � 6q2��pq

6�m2n2

5�mn

m2n2 � 5mn � 6��

m2n2

14�m2

4�m

1�3

m2 � 12m � 42��

3m23�x

6x3 � 9x2 � 9��3x

4b�a

10a2b � 12ab � 8b��2a

15x2 � 25x � 30��5

5�xy

1�y2

4�m2

3�m

4�x

1 1 4 1 x

3x 2 12r 6


Find each quotient.

1. (6q2 � 18q � 9) � 9q 2. ( y2 � 6y � 2) � 3y 3.

� 2 � � 2 � 2 � �

4. 5. (x2 � 3x � 40) � (x � 5) 6. (3m2 � 20m � 12) � (m � 6)

� � x � 8 3m � 2

7. (a2 � 5a � 20) � (a � 3) 8. (x2 � 3x � 2) � (x � 7) 9. (t2 � 9t � 28) � (t � 3)

a � 8 � x � 10 � t � 6 �

10. (s2 � 9s � 25) � (s � 4) 11. 12.

s � 5 � 2r � 7 4w � 3

13. (x3 � 2x2 � 16) � (x � 2) 14. (s3 � 11s � 6) � (s � 3)

x2 � 4x � 8 s2 � 3s � 2

15. 16.

x2 � 8x � 19 � 3d2 � 4d � 5 �

17. 18.

k2 � 2k � 3 � 3y2 � 2y � 1 �

LANDSCAPING For Exercises 19 and 20, use the following information.Jocelyn is designing a bed for cactus specimens at a botanical garden. The total area can bemodeled by the expression 2x2 � 7x � 3, where x is in feet.

19. Suppose in one design the length of the cactus bed is 4x, and in another, the length is 2x � 1. What are the widths of the two designs?

� � ; x � 3

20. If x � 3 feet, what will be the dimensions of the cactus bed in each of the designs?12 ft by 3.5 ft; 7 ft by 6 ft

21. FURNITURE Teri is upholstering the seats of four chairs and a bench. She needs

square yard of fabric for each chair, and square yard for the bench. If the fabric at

the store is 45 inches wide, how many yards of fabric will Teri need to cover the chairsand the bench if there is no waste? 1 yd1

�5

1�2

1�4

3�4x

7�4

x�2

3�3y � 2

2�2k � 3

9y3 � y � 1��3y � 2

2k3 � 7k2 � 7��2k � 3

2�2d � 3

39�x � 2

6d3 � d2 � 2d � 17��2d � 3

x3 � 6x2 � 3x � 1��x � 2

5�s � 4

20w2 � 39w � 18��5w � 6

6r2 � 5r � 56��3r � 8

10�t � 3

68�x � 7

44�a � 3

n2�2m

7�m2

n�4

2m3n2 � 56mn � 4m2n3��

8m3n

7�a

b�2a

2�3y

y�3

1�q

2q�3

12a2b � 3ab2 � 42ab��

6a2b

Practice (Average)


NAME ______________________________________________ DATE ____________ PERIOD _____




NAME ______________________________________________ DATE ____________ PERIOD _____

12-512-5

Pre-Activity How is division used in sewing?


• One way to find the number of flags is to the terms in

the numerator, and then divide by the .

• Another way to find the number of flags is to each term

of the numerator by the and then .

Reading the LessonComplete each sentence.

1. To divide a polynomial by a monomial, you can divide each of thepolynomial by the monomial.

2. You can use factoring to divide a polynomial by a binomial if a of thepolynomial is equal to the binomial divisor.

3. If you cannot see a way to factor a polynomial, then you can divide it by a binomial by

using .

4. In Example 4, the polynomial that is being divided cannot be factored. In such cases, thequotient can be written as the sum of a polynomial and a fraction whose numerator is a

number and whose denominator is equal to the .

5. Tell whether the following statement is true or false. If you say that it is false, give anexample that supports your answer.

To divide a polynomial by a binomial of the form x � a, the polynomial must have atleast two terms.

False; sample answer: The monomial x5 can be divided by x � 3.

6. If you are dividing a polynomial by a binomial, what number should you use to representa missing term of the polynomial? 0


7. If you want to remember one method that you can always use to divide a polynomial bya binomial, which method should you select? long division

binomial divisor

long division

factor

term

subtractdenominator

divide

denominator

subtract


Synthetic DivisionYou can divide a polynomial such as 3x3 � 4x2 � 3x � 2 by a binomialsuch as x � 3 by a process called synthetic division. Compare theprocess with long division in the following explanation.

Divide (3x3 � 4x2 � 3x � 2) by (x � 3) using synthetic division.

1. Show the coefficients of the terms in descending order.

2. The divisor is x � 3. Since 3 is to besubtracted, write 3 in the corner .

3. Bring down the first coefficient, 3.4. Multiply. 3 � 3 � 95. Add. �4 � 9 � 56. Multiply. 3 � 5 � 157. Add. �3 � 15 � 128. Multiply. 3 � 12 � 369. Add. �2 � 36 � 34

Check Use long division

Enrichment

NAME ______________________________________________ DATE ____________ PERIOD _____

12-512-5

3 �4 �3 �29 15 36

3 3 5 12 34

3x2 � 5x � 12, remainder 34

ExampleExample

Enrichment, p. 734

33. DECORATING Anoki wants to put a decorative border 3 feet above the floor around his bedroom walls. If the border comes in 5-yard rolls, how many rolls of border should Anoki buy? 3 rolls

PIZZA For Exercises 34 and 35, use the following information.

The expression ��6d4

2� can be used to determine the number of slices of a round

pizza with diameter d.

34. Write a formula to calculate the cost per slice of a pizza s that costs C dollars.

35. Copy and complete the table below. Which size pizza offers the best price per slice? 18-inch

SCIENCE For Exercises 36–38, use the following information.The density of a material is its mass per unit volume. 36–37. See margin.36. Determine the densities for the

materials listed in the table.

37. Make a line plot of the densities computed in Exercise 36. Use densities rounded to the nearest whole number.

38. Interpret the line plot made in Exercise 37.The densities are clustered around 9.

39. GEOMETRY The volume of a prism with a triangular base is 10w3 � 23w2 � 5w � 2. The height of the prism is 2w � 1, and the height of thetriangle is 5w � 1. What is the measure of the base

of the triangle? �Hint: V � �12

�Bh� 2w � 4

CRITICAL THINKING Find the value of k in each situation.

40. k is an integer and there is no remainder when x2 � 7x � 12 is divided by x � k. 3, 4

41. When x2 � 7x � k is divided by x � 2, there is a remainder of 2. 12

42. x � 7 is a factor of x2 � 2x � k. 63

2w � 1

5w � 1

14 ft

12 ft

42 in.42 in.

34.5 in.

34.5 in.


34. s � ��64

dC2�

ScienceWhen air is heated it isless dense than the airsurrounding it, and theheated air rises. This iswhy a hot air balloon isable to fly.Source: www.howstuffworks.com

Material Mass (g) Volume (cm3)

aluminum 4.15 1.54

gold 2.32 0.12

silver 6.30 0.60

steel 7.80 1.00

iron 15.20 1.95

copper 2.48 0.28

blood 4.35 4.10

lead 11.30 1.00

brass 17.90 2.08

concrete 40.00 20.00

�

�

�

10-inch 14-inch 18-inch

$4.99 $8.99 $12.99

5 10 16$1.02 $0.93 $0.82

Size

Price

Number of slices

Cost per slice


ELL

Answer

36. aluminum: 2.69 g/cm3, gold: 19.33 g/cm3,silver: 10.5 g/cm3, steel: 7.8 g/cm3,iron: 7.79 g/cm3, copper: 8.86 g/cm3,blood: 1.06 g/cm3, lead: 11.3 g/cm3,brass: 8.61 g/cm3, concrete: 2 g/cm3

http://www.howstuffworks.com


Speaking Place students in pairs.Assign each student two polyno-mial division problems. Tell thestudents to study their problemsfor a few minutes to decide howthe quotients should be found.Then ask the students to explaintheir decisions to each other, anddiscuss whether the methodschosen are correct.

Getting Ready for Lesson 12-6PREREQUISITE SKILL Studentswill learn to add and subtractrational expressions with likedenominators in Lesson 12-6. Toprepare students for these con-cepts, make sure they understandhow to add polynomials. UseExercises 57–60 to determineyour students’ familiarity withadding polynomials.

Assessment Options

Quiz (Lessons 12-4 and 12-5) isavailable on p. 773 of the Chapter12 Resource Masters.

Mid-Chapter Test (Lessons 12-1through 12-5) is available on p. 775 of the Chapter 12 ResourceMasters.

Answers

37.

Den

sity

(g

/cm

3 )

2

0

46

8

10

12

14

16

18

20

Alu

min

um

Go

ld

Silv

er

Stee

l

Iro

n

Co

pp

er

Blo

od

Lead

Bra

ss

Co

ncr

ete




How is division used in sewing?


• a description showing that and �

result in the same answer, and

• a convincing explanation to show that �a �

cb

� � �ac

� � �bc

�.

44. Which expression represents the length of the rectangle? D

m � 7 m � 8

m � 7 m � 8

45. What is the quotient of x3 � 5x � 20 divided by x � 3? B

x2 � 3x � 14 � �x

2�2

3� x2 � 3x � 14 � �

x2�2

3�

x2 � 8x � �x �

43

� x2 � 3x � 14 � �x

2�2

3�DC

BA

DC

BA A � m2 � 4m � 32 m � 4

7�1

2� yards

��1�

1

2� yards

36 yards��1�

1

2� yards

36 yards � 7�1

2� yards

��1�

1

2� yards

WRITING IN MATH

Mixed Review



Find each quotient. (Lesson 12-4)

46. �xx

2

2��

5xx��

162

� � �x2 �

x �x �

220

� x � 5 47. �mm2

2

��

8mm

��

615

� � �mm2

2

��

9mm

��

220

� �mm

��

41

�

Find each product. (Lesson 12-3)

48. �b2 �

b1�9b

3� 84

� � �b2 �

b2

15�b

9� 36

� b � 7 49. �zz

2

2��

196zz��

1389

� � �z2 �

z1�8z

5� 65

� �z �

16

�

Simplify. Then use a calculator to verify your answer. (Lesson 11-2)

50. 3�7� � �7� 51. �72� � �32� 52. �12� � �18� � �48�2�7� 10�2� 6�3� � 3�2�

Factor each polynomial, if possible. If the polynomial cannot be factored, write prime. (Lesson 9-6)

53. d2 � 3d � 40 54. x2 � 8x � 16 55. t2 � t � 1(d � 5)(d � 8) (x � 4)2 prime

56. BUSINESS Jorge Martinez has budgeted $150 to have business cards printed. A card printer charges $11 to set up each job and an additional $6 per box of100 cards printed. What is the greatest number of cards Mr. Martinez can haveprinted? (Lesson 6-3) 2300 cards

PREREQUISITE SKILL Find each sum. 57. 4m3 � 6n2 � n 58. 4x2 � 13xy � 2y2

(To review addition of polynomials, see Lesson 8-5.)

57. (6n2 � 6n � 10m3) � (5n � 6m3) 58. (3x2 � 4xy � 2y2) � (x2 � 9xy � 4y2)

59. (a3 � b3) � (�3a3 � 2a2b � b2 � 2b3) 60. (2g3 � 6h) � (�4g2 � 8h)�2a3 � 2a2b � b2 � 3b3 2g3 � 4g2 � 2h


4 Assess4 Assess

43. Sample answer: Division can be used to find the number of pieces of fabric available whenyou divide a large piece of fabric into smaller pieces. Answers should include thefollowing.

• The two expressions are equivalent. If you use the Distributive Property, you canseparate the numerator into two expressions with the same denominator.

• When you simplify the right side of the equation, the numerator is a � b and thedenominator is c. This is the same as the expression on the left.





In Lesson 2-2, students firstlearned to add and subtractrational numbers. In this lesson,students will learn how to addand subtract rational expressionswith binomials in thedenominator

can you use rationalexpressions to interpret

graphics?Ask students:• What conditions must be met

to add fractions? They musthave like denominators.

• Do you really need to changepercents into fractions beforeadding them? Explain. No.Percents are understood to befractions of 100, so they can beadded without first converting themto fraction form.

ADD RATIONAL EXPRESSIONS Recall that to add fractions with likedenominators you add the numerators and then write the sum over the commondenominator. You can add rational expressions with like denominators in the same way.

Sometimes the denominators of rational expressions are binomials. As long aseach rational expression has exactly the same binomial as its denominator, theprocess of adding is the same.

Rational Expressions with Like Denominators


• Add rational expressions with

like denominators.

• Subtract rational expressions

with like denominators.

can you use rationalexpressions to interpret graphics?

can you use rational expressions to interpret graphics?

Numbers in DenominatorFind �

31n2� � �

71n2�.

�31n2� � �

71n2� � �

3n1�2

7n� The common denominator is 12.

� �1102n� Add the numerators.

� Divide by the common factor, 2.

� �56n� Simplify.

510n�126

Example 1Example 1

The graphic at the right shows the number of credit cards Americans have. To determine what fraction of those surveyed have no more than two credit cards, you can use addition. Remember that percents can be written as fractions with denominators of 100.

No credit one or two no more thancards plus credit cards equals two credit cards.

�12020

� � �13030

� � �15050

�

Thus, �15050

� or 55% of those surveyed have no more than two credit cards.

��

JOHN M. DOE

JOHN M. DOE

JOHN M. DOE

USA TODAY Snapshots®

By Marcy E. Mullins, USA TODAY

Most Americans haveone or two credit cardsOne in five Americans say theyhave no credit cards. The numberof cards among those who havethem:

Source: Gallup Poll of 1,025 adults April 6-8.Margin of error: ±3 percentage points.

One or two

Three or four

None

Five or six

Seven or more

33%

23%

22%

11%

9%

LessonNotes

1 Focus1 Focus







Transparencies

Lesson 12-6 Rational Expressions with Like Denominators 673

SUBTRACT RATIONAL EXPRESSIONS To subtract rational expressionswith like denominators, subtract the numerators and write the difference over the common denominator. Recall that to subtract an expression, you add its additive inverse.


Binomials in DenominatorFind �

x2�x

1� � �

x �2

1�.

�x

2�x

1� � �

x �2

1� � �

2xx��

12

� The common denominator is x � 1.

� �2(

xx�

�

1

1)� Factor the numerator.

� Divide by the common factor, x � 1.

� �21

� or 2 Simplify.

12(x � 1)�

x � 11

Example 2Example 2

CommonMisconceptionAdding the additiveinverse will help you avoidthe following error in thenumerator.(3x � 4) � (x � 1) �

3x � 4 � x � 1.

Study Tip

Find a PerimeterGEOMETRY Find an expression for the perimeter of rectangle PQRS.

P � 2� � 2w Perimeter formula

� 2��43aa

��

57bb

�� 2��23aa

��

37bb

�� 43aa

��

57bb

�, w � �23aa

��

37bb

�

�The common denominator is 3a � 7b.

� Distributive Property

� �123aa

��

176bb

� Combine like terms.

� �4(

3

3

aa�

�

7

4

bb)

� Factor.

The perimeter can be represented by the expression �4(

3

3

aa�

�

7

4

bb)

�.

8a � 10b � 4a � 6b��

3a � 7b

2(4a � 5b) � 2(2a � 3b)��

3a � 7b

4a � 5b3a � 7b

2a � 3b3a � 7b

P Q

S R

Example 3Example 3

Subtract Rational Expressions Find �3

xx��

24

� � �xx

��

12

�.

�3xx��

24

� � �xx

��

12

� � The common denominator is x � 2.

� The additive inverse of (x � 1) is �(x � 1).

��3x �

x4��

2x � 1


� �2xx��

25

� Simplify.

(3x � 4) � [�(x � 1)]��

x � 2

(3x � 4) � (x � 1)��

x � 2

Example 4Example 4


2 Teach2 Teach

11

22

33


44


ADD RATIONALEXPRESSIONS

Teaching Tip Remind studentsto avoid the temptation to

simplify before adding. The

denominators must be thesame in order to add, andsimplifying the fraction willchange the denominator.

Find � .

Find � . 6

GEOMETRY Find anexpression for the perimeterof rectangle WXYZ.

SUBTRACT RATIONALEXPRESSIONS

Find � .

2(3x � 7)��

x � 3

x � 5�x � 3

7x � 9�x � 3

4(4x � 3y)��

2x � y

3x � 2y2x � y

XW

YZ

5x � 4y2x � y

12�c � 2

6c�c � 2

4b�3

16b�15

4b�15

3n�12

Online Lesson Plans

USA TODAY Education’s Online site offers resources andinteractive features connected to each day’s newspaper.Experience TODAY, USA TODAY’s daily lesson plan, isavailable on the site and delivered daily to subscribers.This plan provides instruction for integrating USA TODAYgraphics and key editorial features into your mathematicsclassroom. Log on to www.education.usatoday.com.

PowerPoint®

PowerPoint®

http://www.education.usatoday.com




55


Teaching Tip Alert students tothe fact that they must rewritethe numerator as well as thedenominator when rewritinginverse denominators. In thisexample, the numerator changesfrom positive to negative.

Find � .

Have students—• include examples of how to add

and subtract rational expressions.• include any other item(s) that they


FIND THE ERRORTell students to

pay close attention tosigns any time that they are deal-ing with inverses. It is easy toforget to change a sign whenfinding the inverse of anexpression.

Answer

2. Sample answer: When you addrational expressions with likedenominators, you combine thenumerators and keep the commondenominator. This is the sameprocess as adding fractions withlike denominators.

8s�11 � s

�5s�s � 11

3s�11 � s

Sometimes you must express a denominator as its additive inverse to have likedenominators.


Inverse DenominatorsFind �

m2�m

9� � �

94�m

m�.

The denominator 9 � m is the same as �(�9 � m) or �(m � 9). Rewrite the secondexpression so that it has the same denominator as the first.

�m

2�m

9� � �

94�m

m� � �

m2�m

9� � �

�(m4m

� 9)� 9 � m � �(m � 9)

� �m

2�m

9� � �

m4�m

9� Rewrite using like denominators.

� �2m

m��

49m

� The common denominator is m � 9.

� �m�

�2m

9� Subtract.

Example 5Example 5

Guided Practice

1. Sample answer: �xx

��

62

� � �xx

��

42

� � 1

3. Sample answer:Two rational expressions whosesum is 0 are additiveinverses, while tworational expressionswhose difference is 0 are equivalent expressions.4. Russell; sampleanswer: Ginger factored incorrectly inthe next to last step ofher work.


5–8 1–39, 10, 12, 13 4

11 5

1. OPEN ENDED Write two rational expressions with a denominator of x � 2 that have a sum of 1.

2. Describe how adding rational expressions with like denominators is similar to adding fractions with like denominators. See margin.

3. Compare and contrast two rational expressions whose sum is 0 with tworational expressions whose difference is 0.

4. FIND THE ERROR Russell and Ginger are finding the difference of �74

xx

��

23

�

and �3x��

48x

�.

Who is correct? Explain your reasoning.

Find each sum.

5. �a �

42

� � �a �

42

� �a2

� 6. �x

3�x

1� � �

x �3

1� 3

7. �2n

��

n1

� � �n �

11

� �3n

��

n1

� 8. �41t��

41t

� � �21

t��

43t

� �61

t��

42t

�

Find each difference.

9. �152a� � �

172a� ��

a6

� 10. �n �

73

� � �n �

43

� �n �

33

�

11. �m

3�m

2� � �

2 �6

m� �

3mm

��

26

� 12. �x

x�

2

y� � �

xy�

2

y� x � y

Ginger

�47xx

+- 3

2� - �

3x

--48x

� = �3-2

--47xx

� - �3x

--48x

�

= �-2 +

38

--47xx - x

�

= �-36

--48xx

�

= �-2

3(3

--44xx)

�

= -2

Russell

�47

xx

-+ 2

3� - �

3x

--

48x

� = �47

xx

-+ 2

3� + �

4xx--83

�

= �7x +

4xx

+-

23

- 8�

= �84

xx

--

63

�

= �2(

44xx

--

33)

�

= 2

Concept Check


Verbal/Linguistic Place students in small groups. Have students taketurns reading problems involving addition and subtraction of rationalexpressions to the group. As the student reads, have the other groupmembers record the problem. Then have all members discuss theprocedures for adding or subtracting the expressions.

Differentiated Instruction ELL

PowerPoint®


13. SCHOOL Most schools create daily attendance reports to keep track of theirstudents. Suppose that one day, out of 960 students, 45 were absent due toillness, 29 were participating in a wrestling tournament, 10 were excused to go to their doctors, and 12 were at a music competition. What fraction of thestudents were absent from school on this day? �

110�

Application



Find each sum.

14. �m3� � �

23m� m 15. �

172z� � �

�75z� z 16. �

x �5

3� � �

x �5

2� �

2x5� 5�

17. �n �

27

� � �n �

25

� n � 1 18. �y

2

�

y3

� � �y �

63

� 2 19. �r �

3r5

� � �r

1�5

5� 3

20. �kk

��

51

� � �k �

41

� 1 21. �nn

��

23

� � �n

��

13

� �nn

��

33

� 22. �4xx��

25

� � �xx

��

32

� �5xx��

22

�

23. �2aa��

43

� � �aa

��

24

� �3aa��

41

� 24. �52ss

��

11

� � �32ss

��

21

� �82

ss

��

11

� 25. �92bb

��

36

� � �52bb

��

46

� �124bb��

67

�

26. What is the sum of �132xx��

27

� and �92x��

35x

�? 1

27. Find the sum of �121xx��

55

� and �121x

x��

152

�. �222xx��

57

�


28. �57x� � �

37x� �

27x� 29. �

43n� � �

23n� �

23n� 30. �

x �5

4� � �

x �5

2� �

25

�

31. �a �

65

� � �a �

63

� �13

� 32. �x �

27

� � �x��

57

� �x �

77

� 33. �z �

42

� � �z��

62

� �z

1�0

2�

34. �3x

5� 5� � �

3x3�x

5� �1 35. �

7m4� 2� � �

7m7m

� 2� 36. �

x2�x

2� � �

22�x

x� �

x4�x

2�

37. �y

5

�

y3

� � �3

5

�

yy

� �y1�

0y3

� 38. �3t

8� 4� � �

3t6�t

4� �2 39. �

5x15

�x

1� � �

5x��3

1� �3

40. Find the difference of �102aa

��

162

� and �6 �

6a2a

�. �8aa

��

36

�

41. What is the difference of �2bb��

1152

� and ��2b

3b�

�12

8�? �

42

bb

��

21

32

�

42. POPULATION The United States populationin 1998 is described in the table. Use this information to write the fraction of the population that is 80 years or older.

43. CONSERVATION The freshman class chose to plant spruce and pine trees at a wildlife sanctuary for a service project. Some students can plant 140 trees onSaturday, and others can plant 20 trees after school on Monday and again onTuesday. Write an expression for the fraction of the trees that could be plantedon these days if n represents the number of spruce trees and there are twice asmany pine trees. �6

n0�

�286,96,9851,60,00000

�

35. �47m

��7m

2�

Age Number of People

0–19 77,525,000

20–39 79,112,000

40–59 68,699,000

60–79 35,786,000

80–99 8,634,000

100� 61,000

Source: Statistical Abstract of the United States


Exercises Examples14–17 1

18–25, 27, 2, 342, 43, 45,

4628–35, 438, 39,

44, 47, 4826, 36, 37 5



About the Exercises …Organization by Objective• Add Rational Expressions:

14–25, 26, 27• Subtract Rational

Expressions: 28–39, 40, 41


Assignment GuideBasic: 15–43 odd, 45, 46, 49–71



All: Practice Quiz 2 (1–10)



Rational Expressions With Like Denominators

NAME ______________________________________________ DATE ____________ PERIOD _____

12-612-6

Less

on

12-

6Add Rational Expressions To add rational expressions with like denominators, addthe numerators and then write the sum over the common denominator. If possible, simplifythe resulting rational expression.

Find � .

� � Add the numerators.

� Simplify.

� Divide by 3.

� Simplify.4n�5

12n�15

12n�15

5n � 7n��15

7n�15

5n�15

7n�15

5n�15

Find � .

� �

�

�

� or 33�1

3(x � 2)�x � 2

3(x � 2)�x � 2

3x � 6�x � 2

6�x � 2

3x�x � 2

6�x � 2

3x�x � 2


ExercisesExercises

Find each sum.

1. � 2. �

3. � 4. � m � 2

5. � 6. �

7. � 1 8. � x � 3

9. � 1 10. � 2

11. � 2 12. �

13. � 4 14. � 2

15. � 3 16. � 9a � 4

17. � x 18. �17a � 2�2a � 1

8a � 16�2a � 1

9a � 14�2a � 1

4x � x2�x � 4

�8x�x � 4

6a2�a

3a2 � 4a��a

x � 6�x � 3

2x � 3�x � 3

a � 6�a � 1

a � 4�a � 1

x � 6�x � 2

3x � 2�x � 2

5a�b2

10a�3b2

5a�3b2

x � 5�x � 2

x � 1�x � 2

3m � 3�2m � 1

m � 1�2m � 1

�a�a � 4

2a � 4�a � 4

2x � 10�5

3x � 5�5

1�y � 6

y � 5�y � 6

2m � 8�m � 1

m � 4�m � 1

m � 4�m � 1

5x�x � 5

3x�x � 5

2x�x � 5

m � 4�2

m � 8�2

2x � 1�

6x � 2�6

x � 3�6

x2 � x�

8x�8

x2�8

7�a

4�a

3�a

4n

5

1

1


Find each sum.

1. � 2. �

3. � 4. �

5. � 4 6. � 1

7. � 8. �

9. � 6 10. � 2

11. � 12. �


13. � 14. � n 15. � �1

16. � 17. � 18. �

19. � 20. � 1 21. �

22. � 23. � 1 24. �

25. GEOMETRY Find an expression for the perimeter of rectangle ABCD. Use the formula P � 2� � 2w.

26. MUSIC Kerrie is burning an 80-minute CD-R containing her favorite dance songs.Suppose she has burned 41 minutes of songs and has five more songs in the queue thattotal x minutes. When she is done, write an expression for the fraction of the CD thathas been filled with music. 41 � x

�80

4(4a � 3b)��

2a � b

5a � 4b2a � b

BA

CD

3a � 2b2a � b

30t � 5��6t � 1

5�1 � 6t

30t�6t � 1

4a � 6�2a � 2

6a � 4�2a � 2

9y�y � 2

7y�2 � y

2y�y � 2

8p�p � 5

4p�5 � p

4p�p � 5

3�3 � 2y

2y�2y � 3

1�d � 6

6�d � 6

7�d � 6

8�c � 1

�2�c � 1

6�c � 1

28�5

s � 14�5

s � 14�5

1�2

x � 7�2

x � 6�2

r � 5�3

r � 2�3

4n�5

9n�5

y�4

y�8

3y�8

10t � 2�3t � 1

4t � 3�3t � 1

6t � 5�3t � 1

7a � 2�2a � 2

2a � 4�2a � 2

5a � 2�2a � 2

4y � 5�3y � 2

2y � 1�3y � 2

2p � 10�p � 4

4p � 14�p � 4

3r � 4�r � 5

2r � 1�r � 5

r � 5�r � 5

x � 7�x � 2

�2�x � 2

x � 5�x � 2

�8�n � 2

n � 6�n � 2

4�c � 1

4c�c � 1

s � 6�

2s � 4�4

s � 8�4

2w � 13��

9w � 4�9

w � 9�9

3u�4

5u�16

7u�16

n�2

3n�8

n�8

Practice (Average)


NAME ______________________________________________ DATE ____________ PERIOD _____




NAME ______________________________________________ DATE ____________ PERIOD _____

12-612-6

Pre-Activity How can you use rational expressions to interpret graphics?


Write a subtraction expression that you can evaluate to find what percentof the people surveyed have three or more credit cards.

Sample answer: �

Reading the Lesson1. To add or subtract rational expressions with like denominators, add or subtract the

and then write the sum or difference over the

.

2. For each addition or subtraction problem, write the needed expression in each box on theright side of the equation.

a. � �

b. � �

c. � �

d. � �

e. � �

f. � �


3. How can you use what you know about addition and subtraction of rational numbersthat have like denominators to remember how to add and subtract rational expressionsthat have like denominators?

To add or subtract rational numbers that have like denominators, youadd or subtract their numerators and keep the same denominator. You do the same to add or subtract rational expressions that have likedenominators.

8 � ( �9 )��6x � 1

9�1 � 6x

8�6x � 1

7 � 5 �3x � 4

5�4 � 3x

7�3x � 4

d � c � (c � d)��c � 2d

c � d�c � 2d

d � c�c � 2d

3 � ( 6m � 1 )��

2m � 56m � 1�2m � 5

3�2m � 5

7x � (x � 3)��

x � 1x � 3�x � 1

7x�x � 1

5n � 8 �7

8�7

5n�7

common denominatornumerators

55�100

100�100


Sum and Difference of Any Two Like PowersThe sum of any two like powers can be written an � bn, where n is a positive integer. The difference of like powers is an � bn. Underwhat conditions are these expressions exactly divisible by (a � b) or(a � b)? The answer depends on whether n is an odd or even number.

Use long division to find the following quotients. (Hint: Writea3 � b3 as a3 � 0a2 � 0a � b3.) Is the numerator exactlydivisible by the denominator? Write yes or no.

1. 2. 3. 4.

yes no no yes

5. 6. 7. 8.

no no yes yes

a4 � b4�a � b

a4 � b4�a � b

a4 � b4�a � b

a4 � b4�a � b

a3 � b3�a � b

a3 � b3�a � b

a3 � b3�a � b

a3 � b3�a � b

Enrichment

NAME ______________________________________________ DATE ____________ PERIOD _____


44. GEOMETRIC DESIGN A student center is a square room that is 25 feet wide and 25 feet long. The walls are 10 feet high and each wall is painted white with a red diagonal stripe as shown. What fraction of the walls are painted red? �1

5�

HIKING For Exercises 45 and 46, use the following information.A tour guide recommends that hikers carry a gallon of water on hikes to the bottomof the Grand Canyon. Water weighs 62.4 pounds per cubic foot, and one cubic footof water contains 7.48 gallons.

45. Tanika plans to carry two 1-quart bottles and four 1-pint bottles for her hike.Write a rational expression for this amount of water written as a fraction of a cubic foot. �

7.148� ft3

46. How much does this amount of water weigh? 8.3 lb

GEOMETRY For Exercises 47 and 48, use the following information.Each figure has a perimeter of x units.

a. b. c.

47. Find the ratio of the area of each figure to its perimeter. �1x6�; �

1x8�; �

2x4�

48. Which figure has the greatest ratio? a

49. CRITICAL THINKING Which of the following rational numbers is notequivalent to the others? ca. �

2 �3

x� b. �

x��

32

� c. ��2 �

3x

� d. ��x �

32

�


How can you use rational expressions to interpret graphics?


• an explanation of how the numbers in the graphic relate to rationalexpressions, and

• a description of how to add two rational expressions whose denominators are3x � 4y and 4y � 3x.

51. Find �kk

��

27

� � �k��

37

�. A

�kk

��

17

� �kk

��

57

� �kk

��

17

� �kk

��

57

�

52. Which is an expression for the perimeter of rectangle ABCD? B

�2r

1�4r

6s� �

r1�4r

3s�

�r

1�4r

6s� �

r2�8r

3s�DC

BA

9r2r � 6s

5r2r � 6s

A B

D C

DCBA

WRITING IN MATH

3x12

4x12

5x12x

6x6

x3

x3

x4

x4

x4

x4

15 ft

20 ft

5 ft

5 ft

5 ft


HikingDue to its popularity, the Grand Canyon is one of the most threatenednatural areas in the United States.Source: The Wildlife Foundation



ELL


Modeling Have students useconstruction paper, scrap paper,etc. to create models of rationalexpressions. The models need tobe keyed so that they fit togetherlike pieces of a puzzle if they havelike denominators, and do not fittogether if the denominators arenot alike. Display students’models on a bulletin board toremind them of the propertiesthat rational expressions musthave in order to be added orsubtracted.

Getting Ready for Lesson 12-7BASIC SKILL Students will learn toadd and subtract rational expres-sions with unlike denominators inLesson 12-7. In order to add andsubtract rational expressions withunlike denominators, studentsmust find the least commondenominator. Use Exercises 63–71to determine your students’familiarity with finding leastcommon multiples.

Assessment Options

Practice Quiz 2 The quizprovides students with a briefreview of the concepts and skillsin Lessons 12-4 through 12-6.Lesson numbers are given to theright of exercises or instructionlines so students can reviewconcepts not yet mastered.



Practice Quiz 2Practice Quiz 2 Lessons 12-4 through 12-6


Find each quotient. (Lessons 12-4 and 12-5)

53. �x3 �

x �7x

2� 6

� x2 � 2x � 3 54. 8x2 � 9

55. �b2

4�b

9� � (b � 3) �

b4�b

3� 56. �

x �x

2� � �

x2 �x5

2

x � 6� �

x �x

3�

Factor each trinomial. (Lesson 9-3)

57. a2 � 9a � 14 58. p2 � p � 30 59. y2 � 11yz � 28z2

(a � 7)(a � 2) (p � 6)(p � 5) (y � 4z)(y � 7z)

Find each sum or difference. (Lesson 8-5)

60. (3x2 � 4x) � (7 � 9x) 61. (5x2 � 6x � 14) � (2x2 � 3x � 8)3x2 � 5x � 7 7x2 � 3x � 22

62. CARPENTRY When building a stairway, a carpenter considers the ratio of riser to tread. If each stair being built is to have a width of 1 foot and a height of 8 inches, what will be the slope of the stairway?

BASIC SKILL Find the least common multiple for each set of numbers.

63. 4, 9, 12 36 64. 7, 21, 5 105 65. 6, 12, 24 2466. 45, 10, 6 90 67. 5, 6, 15 60 68. 8, 9, 12 7269. 16, 20, 25 400 70. 36, 48, 60 720 71. 9, 16, 24 144

�23

�

56x3 � 32x2 � 63x � 36��

7x � 4

Mixed Review

Find each quotient. (Lessons 12-4 and 12-5)

1. �a �

a3

� � �aa��

131

� �a �

a11

� 2. �4zz��

38

� � (z � 2) �z �

43

� 3. �((2xx��

21))((xx��

32))

��((2xx��

31))((xx��

15))

�

4. (9xy2 � 15xy � 3) � 3xy 5. (2x2 � 7x � 16) � (2x � 3) 6. �y2 �

y1

�

9y4

� 9� y � 15 � �

y5�1

4�

Find each sum or difference. (Lesson 12-6)

7. �x �

27

� � �x �

57

� �x �

77

� 8. �m

2�m

3� � �

m��6

3� 2 9. �

53xx

��

12

� � �23xx

��

12

� �3x

3�x

2�

10. MUSIC Suppose the record shown played for 16.5 minutes on one side

and the average of the radii of the grooves on the record was 3�34

� inches.

Write an expression involving units that represents how many inches the needle passed through the grooves while the record was being played. Then evaluate the expression.

33 revolutions per minute

tread

riser

3. �xx

��

15

� 4. 3y � 5 � �x1y� 5. x � 5 � �

2x1� 3�

� � ; 4125� or about 12,959 in.16.5 minutes��

1

33�13

� revolutions��

1 minute

2� �3�34

�in.��1 revolution


4 Assess4 Assess

Answer

50. Sample answer: Since any rational number can be expressed asa fraction, values on graph can be written as rational expressionsfor clarification. Answers should include the following.

• The numbers in the graphic are percents that can be written asrational expressions with a denominator of 100.

• To add the rational expressions, factor �1 out of eitherdenominator so that it is like the other.




can rational expressionsbe used to describe

elections?Ask students:• In what year will the senator

face his or her first reelection?The second reelection? Thesenator will face reelection in2010, and 2016.

• When will the next threepresidential elections takeplace? 2008, 2012, 2016

• Based on this information,when will the senator’sreelection fall in the same yearas a presidential election? 2016

Vocabulary• least common multiple

(LCM)• least common

denominator (LCD)

Rational Expressions withUnlike Denominators


LCM of PolynomialsFind the LCM of x2 � 8x � 15 and x2 � x � 6.

Express each polynomial in factored form.

x2 � 8x � 15 � (x � 3)(x � 5)

x2 � x � 6 � (x � 2)(x � 3)

Use each factor the greatest number of times it appears.

LCM � (x � 2)(x � 3)(x � 5)

Example 2Example 2

ADD RATIONAL EXPRESSIONS The number of years in which a specificsenator’s election coincides with a presidential election is related to the commonmultiples of 4 and 6. The least number of years that will pass until the next electionfor both a specific senator and the President is the least common multiple of thesenumbers. The is the least number that is a commonmultiple of two or more numbers.

least common multiple (LCM)

• Add rational expressions with unlike denominators.

• Subtract rational expressions with unlike denominators.

The President of the United States is elected every four years, and senators are elected every six years. A certain senator is elected in 2004, the same year as a presidential election, and is reelected in subsequent elections. In what year is the senator’s reelection the same year as a presidential election?

LCM of MonomialsFind the LCM of 15m2b3 and 18mb2.

Find the prime factors of each coefficient and variable expression.

15m2b3 � 3 � 5 � m � m � b � b � b

18mb2 � 2 � 3 � 3 � m � b � b

Use each prime factor the greatest number of times it appears in any of thefactorizations.

15m2b3 � 3 � 5 � m � m � b � b � b

18mb2 � 2 � 3 � 3 � m � b � b

LCM � 2 � 3 � 3 � 5 � m � m � b � b � b or 90m2b3

Example 1Example 1

can rational expressions be used to describe elections?can rational expressions be used to describe elections?

LessonNotes

1 Focus1 Focus



Prerequisite Skills Workbook, pp. 17–18

School-to-Career Masters, p. 23





Transparencies

Add Rational ExpressionsUse the following steps to add rational expressions with unlike denominators.

Step 1 Find the LCD.

Step 2 Change each rational expression into an equivalent expression with theLCD as the denominator.

Step 3 Add just as with rational expressions with like denominators.

Step 4 Simplify if necessary.

Lesson 12-7 Rational Expressions with Unlike Denominators 679www.algebra1.com/extra_examples

Monomial DenominatorsFind �a �

a1

� � �a

3�a

3�.

Factor each denominator and find the LCD.

a � a3a � 3 � a

LCD � 3a

Since the denominator of �a

3�a

3� is already 3a, only �

a �a

1� needs to be renamed.

�a �

a1

� � �a

3�a

3� � �

3(a3(

�a)

1)� � �

a3�a

3� Multiply �a �

a1

� by �33

�.

� �3a

3�a

3� � �

a3�a

3� Distributive Property

� �3a � 3

3�a

a � 3� Add the numerators.

� �43aa�

1

1

Divide out the common factor a.

� �43

� Simplify.

Example 3Example 3

Polynomial DenominatorsFind �

y2 �

y �

4y2� 4

� � �yy

�

�

22

�.

�y2 �

y �

4y2

� 4� � �

yy

�

�

2

2� � �

(yy

�

�

2

2

)2� � �yy

�

�

2

2� Factor the denominators.

� �(y

y�

�

2

2

)2� � �yy

�

�

2

2� � �

yy

�

�

2

2� The LCD is (y � 2)2.

� �(y

y�

�

2

2

)2� � �(

yy

2

�

�

2

4

)2� (y � 2)(y � 2) � y2 � 4

� �y �

(

2

y�

�

y2

2

)2

� 4� Add the numerators.

� �y2

(y�

�

y2

�

)2

6� or �

(y �

(y2

�

)(y2)

�2

3)� Simplify.

Example 4Example 4

Recall that to add fractions with unlike denominators, you need to rename thefractions using the least common multiple (LCM) of the denominators, known as the .least common denominator (LCD)

Lesson 12-7 Rational Expressions with Unlike Denominators 679

2 Teach2 Teach

11

22

33

44


ADD RATIONALEXPRESSIONS


Students first learned to find theprime factorization of monomialsin Lesson 9-1. Without thisknowledge, students would notbe able to find the LCM ofmonomials and polynomials. Ifstudents need a refresher onfinding prime factorizations,refer them to Lesson 9-1.

Teaching Tip Point out tostudents that it is possible fortwo monomials to have nocommon multiple other thantheir product.

Find the LCM of 12b4c5 and32bc2. 96b4c5

Find the LCM of x2 � 3x � 28 and x2 � 8x � 7.(x � 4)(x � 7)(x � 1)

Find � .

Find � .

(x � 2)(x � 1)��

(x � 3)2

x � 3�x � 3

x � 7��x2 � 6x � 9

2(3z � 14)��

5z

z � 6�

zz � 2�

5z

PowerPoint®


55

66


SUBTRACT RATIONALEXPRESSIONS

Teaching Tip Remind studentsthat whatever they multiply thedenominator by to produce theLCD, must also be multiplied bythe numerator.

Find � .

Teaching Tip Tell studentsthat with a test item of thiscomplexity, they need to workout the difference withoutlooking at the item choices,then compare their differencewith the item choices.

Find � . C

A

B

C

D 6x � 22��(x � 2)(x � 3)

6x � 22��(x � 2)2(x � 3)

6x � 22��(x � 2)2(x � 3)2

6x � 22��(x � 2)2(x � 3)

x � 5��x2 � x � 6

x � 4�(2 � x)2

�c2 � 29c � 120��

4(5 � c)(5 � c)

6�5 � c

c�20 � 4c


Binomials in DenominatorsFind �

3a4� 6� � �

a �a

2�.

�3a

4� 6� � �

a �a

2� � �

3(a4� 2)� � �

a �a

2� Factor.

� �3(a

4�(a

2�)(a

2�)

2)� � �

3(a3�a(a

2)�(a

2�)

2)� The LCD is 3(a � 2)(a � 2).

� Subtract the numerators.

� Multiply.

� ��3(

3aa�

2 �2)

1(a0a

��

2)8

� or ��33(aa

2

��

21)0(aa

��

28)

� Simplify.

4a � 8 � 3a2 � 6a��

3(a � 2)(a � 2)

4(a � 2) � 3a(a � 2)��

3(a � 2)(a � 2)

Example 5Example 5

Polynomials in DenominatorsMultiple-Choice Test Item

Read the Test Item

The expression �h2 �

h �4h

2� 4

� � �hh2��

44

� represents the difference of two rational

expressions with unlike denominators.

Solve the Test Item

Step 1 Factor each denominator and find the LCD.

The LCD is (h � 2)(h � 2)2.

Step 2 Change each rational expression into an equivalent expression with the LCD. Then subtract.

�(h

h��

22)2� � �

(h �h2�)(h

4� 2)

� � �((hh

��

22))2� � �

((hh

��

22))

� � �(h �

(h2�)(h

4�)

2)� � �

((hh

��

22

))

�

� �((hh

��

22))2((hh

��

22))

� � �((hh

��

24))2((hh

��

22))

�

� �(h

h�

2 �2)

42h(h

��

42)

� � �(h

h�

2 �2)

22h(h

��

82)

�

�

�

� �(h �

�22h)(

�h �

122)2� The correct answer is B.

h2 � h2 � 4h � 2h � 4 � 8��

(h � 2)2(h � 2)

(h2 � 4h � 4) � (h2 � 2h � 8)��

(h � 2)2(h � 2)

h2 � 4h � 4 � (h � 2)2

h2 � 4 � (h � 2)(h � 2)

Example 6Example 6StandardizedTest Practice

Test-Taking TipExamine all of the answerchoices carefully. Look fordifferences in operations,positive and negativesigns, and exponents.

SUBTRACT RATIONAL EXPRESSIONS As with addition, to subtractrational expressions with unlike denominators, you must first rename theexpressions using a common denominator.

Find �h2 �

h �4h

2� 4

� � �hh2��

44

�.

�(h �

2h2)

�(h

1�2

2)2� �(h �

�22h)(

�h �

122)2�

�(h �

2h2)

�2(h

12� 2)

� �(h

��

22h)�(h

1�

22)

�DC

BA


Example 6 If students areworking on a standardized testthey are allowed to write on,suggest that they highlight any

subtle differences that they see in the answer choices. With thedifferences highlighted, it may be easier for students to spotand eliminate the incorrect choices.

PowerPoint®



Concept Check1–2. See margin.3. Sample answer:�2x

x� 6�, �

x �5

3�

Guided PracticeGUIDED PRACTICE KEYExercises Examples

4 15, 6 2

7 38–10 4

11–14 515 6


Find the LCM for each pair of expressions. 18. (x � 4)(x � 2)16. a2b, ab3 a2b3 17. 7xy, 21x2y 21x2y 18. x � 4, x � 2

19. 2n � 5, n � 2 20. x2 � 5x � 14, (x � 2)2 21. p2 � 5p � 6, p � 1(2n � 5)(n � 2) (x � 7)(x � 2)2 (p � 1)(p � 6)

Find each sum.

22. �x32� � �

5x

� �3 �

x25x

� 23. �a23� � �

a72� �

2 �a3

7a�

24. �67a2� � �

35a� �

7 �6a

120a

� 25. �73m� � �

5m4

2� �15m

35m�

228

�

26. �x �

35

� � �x �

44

� �(x �

7x5)

�(x

8� 4)

� 27. �n �

n4

� � �n �

33

� �(n �

n2

4�)(n

1�2

3)�

28. �a

7�a

5� � �

a �a

2� �

(a �8a2

5)�(a

9�a

2)� 29. �

x6�x

3� � �

x �x

1� �

(x �7x2

3)�(x

3�x

1)�

30. �3x

5� 9� � �

x �3

3� �

3(x1�4

3)� 31. �

3mm� 2� � �

9m2� 6� �

13

�

32. �5��

3a

� � �a2 �

525

� �(a �

3a5�)(a

2�0

5)� 33. �

y21�8

9� � �

3��

7y

� �(y �

7y3�)(y

3�9

3)�

34. �x2 � 2

xx � 1� � �

x �1

1� �

(2xx��

11)2� 35. �

(2xx��

11)2� � �

x2 �x �

3x2� 4

� �(x3x

�

2 �4)(

6xx��

16)2�

36. �4x2

x�

2

9� � �

(2x �x

3)2� �(2

2xx3

��

3)5(x2

2

x��

33x)2� 37. �

a2 �a2

b2� � �(a �

ab)2�

a3 � a2b � a2 � ab��

(a � b)(a � b)2

1. Describe how to find the LCD of two rational expressions with unlikedenominators.

2. Explain how to rename rational expressions using their LCD.

3. OPEN ENDED Give an example of two rational expressions in which the LCDis equal to twice the denominator of one of the expressions.

Find the LCM for each pair of expressions.

4. 5a2, 7a 5. 2x � 4, 3x � 6 6. n2 � 3n � 4, (n � 1)2

35a2 6(x � 2) (n � 4)(n � 1)2

Find each sum.

7. �56x� � �

107x2� �

121x0�x2

7� 8. �

a �a

4� � �

a �4

4� �

(aa

2

��

48)a(a

��

146)

�

9. �y2

2

�

y25

� � �yy

�

�

5

5� �

y(y

2

�

�

51)2(yy

�

�

255)

� 10. �a2 �

a �4a

2� 3

� � �a �

63

� �(a �

7a3)

�(a

8� 1)

�


11. �63wz

2� � �4zw� �

2z4�w2

wz� 12. �

2a4�a

6� � �

a �3

3� �

2aa

��

33

�

13. �bb2 �

�186

� � �b �

14

� �(b � 4)

4(b � 4)� 14. �

x �x

2� � �

x2 � 33x � 10� �

(xx2

��

2)5(xx

��

35)

�

15. Find �y2 � 7

2

yy

� 12� � �

yy

�

�

2

4�. C

�(yy2

�

�

4)

5

(

yy

�

�

6

3)� �

(yy2

�

�

4)

2

(

yy

�

�

6

3)�

�(yy2

�

�

4)

7

(

yy

�

�

6

3)� �

(yy2

�

�

4)

5

(

yy

�

�

6

3)�DC

BA


Exercises Examples16, 17, 154–5718–21 222–25 326–37 438–49 550–53 6








• include examples of how to addand subtract rational expressionswith unlike denominators.


About the Exercises …Organization by Objective• Add Rational Expressions:

22–37• Subtract Rational

Expressions: 38–53


Assignment GuideBasic: 17–57 odd, 58–77



Answers

1. Sample answer: To find the LCD,determine the least commonmultiple of all of the factors of thedenominators.

2. Sample answer: Multiply both thenumerator and denominator byfactors necessary to form the LCD.

Visual/Spatial Have students work Example 5 by writing each fractionwith a different color. Once the two fractions have the samedenominator, then switch to a third color and combine the numerators.Using different colors may help students visualize how each fraction ischanged while helping students differentiate one fraction from another.




Rational Expressions with Unlike Denominators

NAME ______________________________________________ DATE ____________ PERIOD _____

12-712-7

Less

on

12-

7

Add Rational Expressions Adding rational expressions with unlike denominators issimilar to adding fractions with unlike denominators.

AddingStep 1 Find the LCD of the expressions.

RationalStep 2 Change each expression into an equivalent expression with the LCD as the denominator.

ExpressionsStep 3 Add just as with expressions with like denominators.Step 4 Simplify if necessary.

Find � .

Factor each denominator.n � n

4n � 4 � nLCD � 4n

Since the denominator of is already

4n, only needs to be renamed.

� � �

� �

�

�3n � 2�n

12n � 8�4n

8n � 4�4n

4n � 12�4n

8n � 4�4n

4(n � 3)��4n

8n � 4�4n

n � 3�n

n � 3�n

8n � 4�4n

8n � 4�4n

n � 3�n Find � .

� � �

� � � �

� �

�

�7x � 18

��2x2(x � 3)

x � 6x � 18��

2x2(x � 3)

6(x � 3)��2x2(x � 3)

x��2x2(x � 3)

2(x � 3)�2(x � 3)

3�x2

x�x

1��2x(x � 3)

3�x2

1��2x(x � 3)

3�x2

1��2x2 � 6x

3�x2

1��2x2 � 6x


ExercisesExercises

Find each sum.

1. � 2. �

3. � 4. �

5. � 6. �

7. � 8. �

9. � 10. �

11. � 12. �

13. � 0 14. �2q � 4

��(q � 4)(q � 4)

q � 1��q2 � 5q � 4

q�q2 � 16

2 � y��y2 � y � 6

y � 2��y2 � 5y � 6

a � 1�a � 2

a � 2�a � 2

a � 2�a2 � 4

2x � y�3x � y

3y�9x � 3y

4x�6x � 2y

8m � 4��3(m � 1)(m � 1)

2��3(m � 1)

6��3(m � 1)

a2 � 16��(a � 4)(a � 4)

4�a � 4

a�a � 4

3y � 4�(y � 2)2

2�y � 2

y��y2 � 4y � 4

5y � 14��(y � 6)(y � 2)

1�y � 2

4�y � 6

6h � 10��(h � 1)(h � 2)

2�h � 2

4�h � 1

2 � 2a�

a26

�3a8

�4a2

2x � 3�

x33

�x3

2�x2

4x � 45�

9x25

�x2

4�9x

4 � 9x�

24x3�8

1�6x

10�3a

7�3a

1�a


Find the LCM for each pair of expressions.

1. 3a3b2, 18ab3 2. w � 4, w � 2 3. 5d � 20, d � 418a3b3 (w � 4)(w � 2) 5(d � 4)

4. 6p � 1, p � 1 5. x2 � 5x � 4, (x � 1)2 6. s2 � 3s � 10, s2 � 4(6p � 1)(p � 1) (x � 4)(x � 1)2 (s � 5)(s � 2)(s � 2)

Find each sum.

7. � 8. �

9. � 10. �

11. � 12. �

13. � 14. �


15. � 16. �

17. � 18. �

19. � 20. �

21. SERVICE Members of the ninth grade class at Pine Ridge High School are organizinginto service groups. What is the minimum number of students who must participate forall students to be divided into groups of 4, 6, or 9 students with no one left out? 36

22. SAFETY When the Cooper family goes on vacation, they set the house lights on timersfrom 5 P.M. until 11 P.M. The lights come on at different times in each of three rooms:every 40 minutes, every 50 minutes, and every 100 minutes, respectively. The timerturns each of them off after 30 minutes. After 5 P.M., how many times do all the lightscome on at the same time in one evening? at what time(s)? once; 8:20 P.M.

y2 � 2y � 9��(y � 3)(y � 2)(y � 2)

�3t2 � 2t � 1��(t � 2)(t � 5)2

3y � 3�y2 � 4

4y��y2 � y � 6

4t � 8��t2 � 10t � 25

t � 3��t2 � 3t � 10

4b2 � 12b � 36��(b � 3)( b � 3)2

�3�b � 3

b � 3��b2 � 6b � 9

�2s2 � 7s � 13��4(s � 3)(s � 3)

2s � 3�4s � 12

s � 1�s2 � 9

m2 � 4m � 18��(m � 3)(m � 6)

2�m � 6

m � 4�m � 3

18p � 10xp��

15x22p�3x

6p�5x2

h2 � 2h � 9��

(h � 3)2h � 2�h � 3

h � 3��h2 � 6h � 9

2a2 � 2a � 6��

(a � 5)26a � 24

��a2 � 10a � 25

2a � 6�a � 5

p2 � 1��(p � 4)(p � 1)

p�p � 4

p � 1��p2 � 3p � 4

4y2 � 7y � 20��(y � 4)2(y � 4)

3y � 2��y2 � 8y � 16

y � 3�y2 � 16

n � 5��(n � 3)(n � 3)

2�2n � 6

8�n2 � 9

n2 � n � 14��(n � 6)(n � 2)

7�n � 6

n�n � 2

5b � 3�

4bb � 2�b

b � 5�4b

7y � 20x��

6x2y210�3xy2

7�6x2y

Practice (Average)


NAME ______________________________________________ DATE ____________ PERIOD _____




NAME ______________________________________________ DATE ____________ PERIOD _____

12-712-7

Pre-Activity How can rational expressions be used to describe elections?


• How can you find the years after 2004 when an election for senator willoccur? Sample answer: Add multiples of 6 to 2004.

• How can you find the years after 2004 when an election for President ofthe United States will occur? Sample answer: Add multiples of 4 to 2004.

Reading the Lesson

1. Answer each question about the monomials 49k2n3 and 21kn5.

a. What prime numbers are factors of these monomials? 3 and 7

b. How many times are these prime factors used in each monomial? In 49k2n3,3 is not used, but 7 is used twice. In 21kn5, 3 is used once and 7 isused once.

c. How many times should you use 3 as a factor in the LCM of the two monomials? Howmany times should you use 7 as a factor in the LCM? one time; two times

d. How many times should you use k as a factor in the LCM? How many times shouldyou use n as a factor in the LCM? two times; five times

2. How is the LCD for two rational expressions related to the LCM of the denominators?They are equal.

3. How does the LCD of two rational expressions help you add or subtract the expressions?It helps you rename the expressions as rational expressions with likedenominators.


4. Making a short list of the steps in a procedure can help you remember the procedure.Make a short list of the main steps you can use to add or subtract rational expressionswith unlike denominators. Sample answer: 1. Write the prime factorization ofthe denominator of each expression. 2. Use the prime factorizations towrite the LCM of the denominators. 3. Using the LCM as the LCD of theexpressions, write each of the rational expressions to have the LCD asits denominator. 4. Add or subtract numerators and keep the samedenominator. 5. Simplify if possible.


Graphing Circles by Completing SquaresOne use for completing the square is to graph circles. The general equation for a circle with center at the origin and radius r is x2 � y2 � r2. An equation represents a circle if it can be transformed into the sum of two squares.

x2 � 6x � y2 � 4y � 3 � 0

(x2 � 6x � �) � ( y2 � 4y � �) � 3 � (�6)2 and � (4)2

(x2 � 6x � 9) � ( y2 � 4y � 4) � 3 � 9 � 4 Add to both sides.

(x � 3)2 � ( y � 2)2 � 42 Factor trinomials.

Notice that the center of the circle is at the point (3, �2).

Transform each equation into the sum of two squares. Thengraph the circle represented by the equation. Use thecoordinate plane provided at the bottom of the page.

1. x2 � 14x � y2 � 6y � 49 � 0 2. x2 � y2 � 8y � 9 � 0

(x � 7)2 � (y � 3)2 � 32 x2 � (y � 4)2 � 52

1�2

1�2

x

y

O

Enrichment

NAME ______________________________________________ DATE ____________ PERIOD _____



38. �37x� � �

63x2� �

14x6x

�2

3� 39. �

154x2� � �

35x� �

41�5x

225x

�

40. �131yx2� � �

76xy� �

22x6�y2

7xy� 41. �

75xa� � �

231ax2� �

5ax7x

�2

a�

42. �xx

2

��

11

� � �xx

2

��

11

� �x�

�2x

1� 43. �

k �k

5� � �

k �3

3� �

(kk

2

��

56)k(k

��

135)

�

44. �2k

k� 1� � �

k �2

2� �

(2kk

2

��

12)k(k

��

22)

� 45. �mm

��

11

� � �2m

4� 5� �

(m2m

�

2

1�)(2

mm

��

95)

�

46. �x2

2�x

5x� � �

x��3x

5� �

(2x

��

35x)

� 47. �a��

36

� � �a2

��

66a

� ��a(

3aa��

66)

�

48. �5 �

nn

� � �n2 �

325

� �(�n

n�

2 �5)(

5nn

��

53)

� 49. �36a��

32a

� � �aa2��

24

� ��

33a(a

��

52)

�

50. �x2 �

33xx � 2� � �

x23�x

4�x

6� 4

� 51. �a2 �

53aa � 4� � �

aa2��

11

�

52. �xx

2

2��

42xx

��

53

� � �x �

21

� �xx

��

13

� 53. �m2 �

m8�m

4� 16

� � �mm

��

44

�

54. MUSIC A music director wants to form a group of students to sing and dance at community events. The music they will sing is 2-part, 3-part, or 4-part harmony. The director would like to have the same number of voices on each part. What is the least number of students that would allow for an even distribution on all these parts? 12

55. CHARITY Maya, Makalla, and Monya can walk one mile in 12, 15, and20 minutes respectively. They plan to participate in a walk-a-thon to raise money for a local charity. Sponsors have agreed to pay $2.50 for each mile that is walked. What is the total number of miles the girls would walk in one hour and how much money would they raise? 12 mi; $30

56. PET CARE Kendra takes care of pets while their owners are out of town. One week she has three dogs that all eat the same kind of dog food. The firstdog eats a bag of food every 12 days, the second dog eats a bag every 15 days,and the third dog eats a bag every 16 days. How many bags of food shouldKendra buy for one week? 2 bags

57. AUTOMOBILESCar owners need to follow a regularmaintenanceschedule to keeptheir cars runningsafely and efficiently.The table showsseveral items thatshould be performedon a regular basis. Ifall of these items areperformed when acar’s odometer reads36,000 miles, whatwould be the car’smileage reading thenext time all of theitems should beperformed?66,000 mi

Inspection or Service

engine oil and oil filter change

transmission fluid level check

brake system inspection

chassis lubrication

power steering pump fluid level check

tire and wheel rotation and inspection

every oil change

every oil change

every 6000 miles

every 6000 miles

every 15,000 miles

every 3000 miles(about 3 months)

Frequency


Pet CareKell, an English Mastiffowned by Tom Scott of theUnited Kingdom, is theheaviest dog in the world.Weighing in at 286 pounds,Kell eats a high protein diet of eggs, goat’s milk,and beef.Source: The Guinness Book

of Records

50. �(x �

91x)�(x �

62)2�

51.

53. �m3 � 11m2 � 56m � 48��

(m � 4)(m � 4)2

4a2 � 2a � 4��(a � 4)(a � 1)(a � 1)


ELL


Speaking Using a problem fromthe exercise set, have studentswork the problem with you,explaining their steps.

Getting Ready for Lesson 12-8PREREQUISITE SKILL Studentswill learn about mixed expres-sions and complex fractions inLesson 12-8. This will includesimplifying rational expressions.Use Exercises 72–77 to determineyour students’ familiarity withsimplifying rational expressions.

Assessment Options


Answers

58. Sample answer: This method willalways work.

� � � � �

� �

�

59. Sample answer: You can userational expressions and theirleast common denominators todetermine when elections willcoincide. Answers should includethe following.

• Use each factor of thedenominators the greatestnumber of times it appears.

• 2012

ay � bx�

xy

bx�yx

ay�xy

x�x

b�y

b�y

a�x

b�y

a�x



Find each sum. (Lesson 12-6)

62. �2m

3m� 1� � �

2m3� 1� 63. �

2x4�x

3� � �

2x5� 3� 64. �

y2

�

y3

� � �3 �

5y

�

Find each quotient. (Lesson 12-5)

65. �b2 �

b8�b

2� 20� b � 10 66. �t

3 �t

1�9t

4� 9

� 67. �4m2

2�m

8�m

7� 19

�

Factor each trinomial, if possible. If the trinomial cannot be factored usingintegers, write prime. (Lesson 9-4)

68. 2x2 � 10x � 8 69. 5r2 � 7r � 6 70. 16p2 � 4pq � 30q2

2(x � 4)(x � 1) (5r � 3)(r � 2) 2(2p � 3q)(4p � 5q)71. BUDGETING JoAnne Paulsen’s take-home pay is $1782 per month. She

spends $525 on rent, $120 on groceries, and $40 on gas. She allows herself 5% of the remaining amount for entertainment. How much can she spend onentertainment each month? (Lesson 3-9) $54.85

PREREQUISITE SKILL Find each quotient. (To review dividing rational expressions, see Lesson 12-4.)

72. �x2

� � �35x� �

56

� 73. �5ab

2� � �

140ab2� �

a2b� 74. �

x �x

7� � �

xx

��

73

� �x �

x3

�

75. �2n

3�n

5� � �

2n12

�n2

5� �

41n� 76. �

x3�x

2� � (x � 1) 77. �x

2 �x

7�x

6� 12� � (x � 3)

Mixed Review62. �3

2mm

��

31

�

63. �42xx

��

53

�

64. �2yy��

35

�

66. t2 � 4t � 3 �

�t �

34

�

67. 2m � 3 � �2m

2� 7�


76. �(x � 23)(xx � 1)�

77. �xx��

46

�


58. CRITICAL THINKING Janelle says that a shortcut for adding fractions with unlike denominators is to add the cross products for the numerator and write the denominator as the product of the denominators. She gives thefollowing example.

�27

� � �58

� � �2 � 8

7�� 8

5 � 7� � �

5516�

Explain why Janelle’s method will always work or provide a counterexample to show that it does not always work. See margin.


How can rational expressions be used to describe elections?


• an explanation of how to determine the least common multiple of two ormore rational expressions, and

• if a certain senator is elected in 2006, when is the next election in which thesenator and a President will be elected?

60. What is the least common denominator of �a2 � 2

6ab � b2� and �

a2 �6

b2�? D

(a � b)2 (a � b)(a � b)

(a � b)2 (a � b)2(a � b)

61. Find �(2

x��

x4)2� � �

x2x�

�x

5� 6

�. C

�(x �

8x3)

�(x

2�2

2)2� �(xx

2

��

22)x(x

��

137)

�

�(x �

6x3)

�(x

2�2

2)2� �(x �

223�)(x

6�x

2)�DC

BA

DC

BA

WRITING IN MATH


4 Assess4 Assess




are rational expressionsused in baking?

Ask students:• Why can’t you find the number

of cookies that Katelyn canmake by just simplifying thegiven expression? The units aredifferent. The amount of dough isgiven in pounds, and the amount ofdough per cookie is given in ounces.

• What is a method that youmight be able to use to find outhow many cookies Katelyn canmake. You could use dimensionalanalysis to convert pounds toounces.

SIMPLIFY MIXED EXPRESSIONS Recall that a number like 2�12

� is a mixed

number because it contains the sum of an integer, 2, and a fraction, �12

�. An expression

like 3 � �xx

��

23

� is called a because it contains the sum of a

monomial, 3, and a rational expression, �xx

��

23

�. Changing mixed expressions to

rational expressions is similar to changing mixed numbers to improper fractions.

mixed expression

Vocabulary• mixed expression• complex fraction

Mixed Expressions and Complex Fractions


• Simplify mixed expressions.

• Simplify complex fractions.

Katelyn bought 2�12

� pounds of chocolate chip

cookie dough. If the average cookie requires

1�12

� ounces of dough, the number of cookies

that Katelyn can bake can be found by

simplifying the expression .2�

1

2� pounds

��1�

1

2� ounces

Mixed Expression to Rational ExpressionSimplify 3 � �

x �6

3�.

3 � �x �

63

� � �3(

xx

��

33)

� � �x �

63

� The LCD is x � 3.

� �3(x

x�

�3)

3� 6

� Add the numerators.

� �3x

x�

�9

3� 6


� �3xx

��

135

� Simplify.

Example 1Example 1

SIMPLIFY COMPLEX FRACTIONS If a fraction has one or more fractions in the numerator or denominator, it is called a . You simplify an algebraic complex fraction in the same way that you simplify a numerical complex fraction.

numerical complex fraction algebraic complex fraction

� �83

� � �75

� � �ba

� � �dc

�

� �83

� � �57

� � �ba

� � �dc

�

� �4201� � �

abdc�

�ba

�

��dc

�

�8

3�

��7

5�

complex fraction

are rational expressions used in baking?are rational expressions used in baking?

LessonNotes

1 Focus1 Focus



School-to-Career Masters, p. 24





Transparencies

Simplifying a Complex Fraction

Any complex fraction , where b � 0, c � 0, and d � 0, can be expressed as �abdc�.

�ba

�

�

�dc

�

Lesson 12-8 Mixed Expressions and Complex Fractions 685www.algebra1.com/extra_examples

Complex Fraction Involving NumbersBAKING Refer to the application at the beginning of the lesson. How many

cookies can Katelyn make with 2�12

� pounds of chocolate chip cookie dough?

To find the total number of cookies, divide the amount of cookie dough by theamount of dough needed for each cookie.

� � �116

poouunncdes

�

� Simplify.

� Express each term as an improper fraction.

� Multiply in the numerator.

� �820��32

� � �abdc�

� �1660

� or 26�23

� Simplify.

Katelyn can make 27 cookies.

�ab

�

��dc

�

�820�

�

�3

2�

�116� � �5

2�

��3

2�

16 � 2�12

�

�1�

1

2�

Convert pounds to ounces.Divide by common units.

2�1

2� pounds

��1�

1

2� ounces

2�1

2� pounds

��1�

1

2� ounces

Example 2Example 2

Complex Fraction Involving Monomials

Simplify .

� �x2

ay2

� � �xa

2

3

y� Rewrite as a division sentence.

� �x2

ay2

� � �xa2

3

y� Rewrite as multiplication by the reciprocal.

� �x2

ay2

� � �xa2

3

y� Divide by common factors x2, y, and a.

� a2y Simplify.

�x2

ay2

�

�

�xa

2

3

y�

�x2

ay2�

�

�xa

2

3y�

Example 3Example 3

1 y a2

1 1 1

Fraction BarRecall that when applyingthe order of operations, a fraction bar serves as agrouping symbol. Simplifythe numerator anddenominator of acomplex fraction beforeproceeding with division.

Study Tip

Lesson 12-8 Mixed Expressions and Complex Fractions 685

2 Teach2 Teach

11


22

33


SIMPLIFY MIXEDEXPRESSIONS

Teaching Tip Point out thatsince one of the terms of amixed expression is amonomial, the LCD will be thesame as the denominator of therational expression.

Simplify 3 � .

SIMPLIFY COMPLEXFRACTIONS

BAKING Suppose Katelyndecided to make her cookieswith 2 pounds of dough. Howmany cookies would she beable to make? 21 cookies

Simplify .a4c2�

b3

�ac

5

2

b�

�

�acb4

4

�

3x � 1�x � 2

7�x � 2

Auditory/Musical Challenge students to come up with a musicalmnemonic they can use to remember how to simplify a complexfraction. For example, using the Key Concept box on p. 685, studentscould replace the variables a, b, c, and d with words such as Apple,Battle, Core, Door and have their tune include a phrase like “Apple and

Door over Battle and Core” to represent .ad�bc


PowerPoint®

PowerPoint®




44


Simplify .

Have students—• add the definitions/examples of


• include examples of how tosimplify mixed expressions andcomplex fractions.


FIND THE ERRORAsk students to

explain the mistakeLian made. Ask students howmany of them have made similarmistakes. Discuss ways studentsmight avoid errors in factoring.

Answers

1. Sample answer: Both mixednumbers and mixed expressionsare made up by the sum of aninteger and a fraction or rationalexpression.

2. Sample answer: � �

� �

�4�5

6�5

2�3

5�6

2�3

�23

��

�56

�

(b � 2)(b � 1)��(b � 3)(b � 4)

b � �b �

2

3�

��b � 4

1. Describe the similarities between mixed numbers and mixed rationalexpressions. 1–2. See margin.

2. OPEN ENDED Give an example of a complex fraction and show how to simplify it.

3. FIND THE ERROR Bolton and Lian found the LCD of �2x

4� 1� � �

x �5

1� � �

x �2

1�.

Who is correct? Explain your reasoning.Bolton; he used the factors correctly to determine the LCD.

Write each mixed expression as a rational expression.

4. 3 � �4x

� �3x

x� 4� 5. 7 � �

65y� �

42y6y

� 5� 6. �

a3�a

1� � 2a �

6a2 �3a

a � 1�


7. �11

49� 8. �

xy

4

5� 9. �ax

��

by

��xa �

�

by

�

�

�xa2

2

�

�

by

2

2

�

�xy

3

2�

�

�yx

3

�

3�12

�

�4�3

4�

Lian

�2x

4+ 1� – �

x5+ 1� + �

x2– 1�

LCD: 2 (x + 1 ) (x – 1 )

Bolton

�2x

4+ 1� – �

x5+ 1� + �

x –2

1�

LCD: (2x + 1)(x + 1)(x – 1)


Concept Check

Guided Practice

Complex Fraction Involving Polynomials

Simplify .

The numerator contains a mixed expression. Rewrite it as a rational expression first.

� The LCD of the fractions in the numerator is a � 2.

� Simplify the numerator.

� Factor.

� �(a �

a3�)(a

2� 5)

� � (a � 3) Rewrite as a division sentence.

� �(a �

a3�)(a

2� 5)

� � �a �

13

� Multiply by the reciprocal of a � 3.

� �(a �

a3�)(a

2� 5)

� � �a �

13

� Divide by the GCF, a � 3.

� �aa

��

52

� Simplify.

�(a �

a3�)(a

2� 5)

��

a � 3

�a2 �

a2�a �

215�

��a � 3

�a(

aa��

22)

� � �a1�5

2��

a � 3

a � �a

1�5

2�

��a � 3

a � �a

1�5

2�

��a � 3

Example 4Example 4


4–6 17, 10 2

8 39 4

1

1


PowerPoint®

Answers

14.

15.

16.

17.

18.

19.

20.

21.

22. p2 � p � 15��

p � 4

x2 � 7x � 17��

x � 3

3s3 � 3s2 � 1��

s � 1

5n3 � 15n2 � 1��

n � 3

r 3 � 3r2 � r � 4��

r � 3

b3 � ab2 � a � b��

a � b

6a2 � a � 1��

2a

2m2 � m � 4��

m

6wz � 2z��

w


10. ENTERTAINMENT The student talent committee isarranging the performances for their holiday pageant. Thefirst-act performances and theirlengths are shown in the table.What is the average length ofeach performance?

7�12090

� min

Holiday Pageant Line-Up

A

B

C

D

E

7

4

6

8

10

Length (min)Performance

12121415

Application




Write each mixed expression as a rational expression. 14–22. See margin.

11. 8 � �n3

� �8n

n� 3� 12. 4 � �

5a

� �4a

a� 5� 13. 2x � �

xy

� �2xy

y� x�

14. 6z � �2wz� 15. 2m � �

4 �m

m� 16. 3a � �

a2�a

1�

17. b2 � �aa

��

bb

� 18. r2 � �rr

��

43

� 19. 5n2 � �nn2��

39

�

20. 3s2 � �ss2��

11

� 21. (x � 5) � �xx

��

23

� 22. (p � 4) � �pp

�

�

1

4�


23. �34

� 24. �18445

� 25.

26. n 27. �yx

2

2((xy

�

�

42

))

� 28. �st2

3

((ss

��

tt))

�

29. �y �

14

� 30. �a �

11

� 31. �nn

��

23

�

32. 1 33. �((xx

�

�

32

))((xx

�

�

14

))

�34.

35. What is the quotient of b � �1b

� and a � �1a

�? �ab

((ba

2

2��

11

))

�

36. What is the product of �25bc

2� and the quotient of �

42bc

3� and �

78bc2

3�? �

3325b2�

37. PARTIES The student council is planning a party for the school volunteers.There are five 66-ounce bottles of soda left from a recent dance. When poured

over ice, 5�12

� ounces of soda fills a cup. How many servings of soda can they get

from the bottles they have? 60

n � �n �

3512

�

��n � �

n6�3

2�

x � �x

1�5

2�

��x � �

x2�0

1�

�xx

2

2��

49

xx

��

2118

�

��xx2

2

��

130xx

��

2284

�

�n2

n�

2

9�n

2�n

18�

��n2

n�

2 �n �

5n30

�

�a2 �

a22�a

1� 3�

��a � 3

�y2 �

y2

3

�

y1

� 4�

��y � 1

�st2

3�

��ss

��

tt�

�xy

��

4

2�

��xy

2

2�

�mn3

2�

�

�mn2

2�

1�ab2

�ba3�

�

�ab

2�

8�27

�

�4�4

5�

5�34

�

�7�2

3�

34. �((nn�

�

152))((nn�

�

29))

�


Exercises Examples11–22, 35 1

23–26, 237–40

27–32, 36 333, 34 4


�

�


About the Exercises …Organization by Objective• Simplify Mixed

Expressions: 11–22• Simplify Complex

Fractions: 23–34



Average: 11–37 odd, 38, 39,42–68





NAME ______________________________________________ DATE ____________ PERIOD _____

12-812-8

Less

on

12-

8

Simplify Mixed Expressions Algebraic expressions such as a � and 5 � are

called mixed expressions. Changing mixed expressions to rational expressions is similarto changing mixed numbers to improper fractions.

x � y�x � 3

b�c

Simplify 5 � .

5 � � � LCD is n.

� Add the numerators.

Therefore, 5 � � .5n � 2�n

2�n

5n � 2�n

2�n

5 � n�n

2�n

2�n Simplify 2 � .

2 � � �

� �

�

�

Therefore, 2 � � .2n � 9�n � 3

3�n � 3

2n � 9�n � 3

2n � 6 � 3��n � 3

3�n � 3

2n � 6�n � 3

3�n � 3

2(n � 3)��n � 3

3�n � 3

3�n � 3


ExercisesExercises


1. 4 � 2. � 3

3. 3x � 4. � 2

5. 10 � 6. � 2

7. � y2 8. 4 �

9. 1 � 10. � 2m

11. x2 � 12. a � 3 �

13. 4m � 14. 2q2 �

15. � 4y2 16. q2 �pq2 � q3 � p � q��

p � qp � q�p � q

2 � 4y4 � 4y2��

y2 � 12

�y2 � 1

q � 2pq2 � 2q3��

p � qq

�p � q8mt � 3n��

2t3n�2t

a2 � a � 11��

a � 3a � 2�a � 3

x3 � 3x2 � x � 2��

x � 3x � 2�x � 3

4 � 2m2 � 4m��

m � 24

�m � 2x � 1�

x1�x

8x�2x � 1

4�2x � 1

y3 � 2y2 � y��

y � 2y

�y � 2

3h � 8�h � 4

h�h � 4

10x � 110��

x � 560

�x � 5

4 � 2x2�

x24

�x2

3x3 � 1�

x21

�x2

1 � 27x�

9x1

�9x4a � 6�

a6�a



1. 14 � 2. 7d � 3. 3n �

4. 5b � 5. 3 � 6. 2s �

7. 2p � 8. 4n2 � 9. (t � 1) �


10. 11. 12.

13. 14. 15.

16. b 17. 18.

TRAVEL For Exercises 19 and 20, use the following information.

Ray and Jan are on a 12 -hour drive from Springfield, Missouri, to Chicago, Illinois. They

stop for a break every 3 hours.

19. Write an expression to model this situation.

20. How many stops will Ray and Jan make before arriving in Chicago? 3

21. CARPENTRY Tai needs several 2 -inch wooden rods to reinforce the frame on a futon.

She can cut the rods from a 24 -inch dowel purchased from a hardware store. How

many wooden rods can she cut from the dowel? 10

1�2

1�4

1�4

1�2

(y � 6)(y � 6)��(y � 7)(y � 7)

y � �y �

67

�

��y � �

y �7

6�

(g � 10)(g � 4)��(g � 9)(g � 5)

g � �g

1�0

9�

��g � �

g �5

4�

�bb

2

2��

b3b

��

124

�

��

�bb2

��

3b

�

k(k � 6)��

k � 5�k2

k�

2 �4k

6�k

5�

��k2 �

k �9k

8� 8

�

1�q � 4

�q2

q�

27�

q1�

612

�

��q � 31

��a(a � 4)

�a

a�

24

�

��

�a2 �

a16

�

3(x � y)��

x

�x2

x�

2y2

�

��x

3�x

y�

mn�18

�m6n

2�

�

�3nm2�

6�5

3�25

�

�2�

56

�

t2 � 6t � 9��

t � 54n3 � 4n2 � 1��

n � 12p2 � 5p � 1��

p � 3

4�t � 5

n � 1�n2 � 1

p � 1�p � 3

2s2 � 3s � 1��

s � 13t2 � t � 2��

t2 � 110b2 � b � 3��

2b

s � 1�s � 1

t � 5�t2 � 1

b � 3�2b

3n2 � n � 6��

n7dp � 4d��

p14u � 9�

u

6 � n�n

4d�p

9�u

Practice (Average)


NAME ______________________________________________ DATE ____________ PERIOD _____

12-812-8

12�12

�

�3�

14

�

Skills Practice, p. 749 and Practice, p. 750 (shown)



NAME ______________________________________________ DATE ____________ PERIOD _____

12-812-8

Pre-Activity How are rational expressions used in baking?


What is another way to write ? Sample answer: 2 � 1

Reading the Lesson

1. Tell whether each expression is a mixed expression or complex fraction. Write M formixed expression and C for complex fraction.

a. 7x � M b. C c. C d. (b � 6) � M

2. Complete each statement about mixed expressions and complex fractions.

a. A mixed expression is the sum of a monomial and a .

To change it to a rational expression, find the , rename the monomial as a rational expression using that denominator, and add.

b. A complex fraction is a fraction that has one or more in itsnumerator or denominator.

3. Complete each statement.

a. One method of simplifying a complex fraction is first to rewrite it as a

expression. Then by the of the divisor.

b. Another method is to express a fraction of the form as , and then

.


4. Describe an easy way to remember what a mixed expression is.

Sample answer: Think of a mixed expression as being similar to a mixednumber. A mixed number can be written as the sum of a whole numberand a fraction. A mixed expression is the sum (or difference) of amonomial and a fraction.

multiply

ad�bc

�ab

�

��dc

�

reciprocalmultiplydivision

fractions

LCD

rational expression

b � 3�b � 2

y � 12�14

�

��34

�

5 � �s �

21

�

��s2

x � 2�x � 5

1�2

1�2

2�12

�

�1�

12

�


Complex FractionsComplex fractions are really not complicated. Remember that a fraction can be interpreted as dividing the numerator by the denominator.

� � � � � �

Let a, b, c, and d be numbers, with b � 0, c � 0, and d � 0.

� � � � � Notice the pattern:denominator ofthe answer (bc)

←←

�ab

�

��dc

�

→→

numerator ofthe answer (ad )

ad�bc

d�c

a�b

c�d

a�b

�ab

�

��dc

�

14�15

2(7)�3(5)

7�5

2�3

5�7

2�3

�23

�

�

�57

�

Enrichment

NAME ______________________________________________ DATE ____________ PERIOD _____

12-812-8

Simplify .

�5x(3)�4(x + 2)

�54x�

�

�x �

32

�

�54x�

�

�x �

32

�

Simplify .

��x �

28

�

�

�3x

1� 2�

�2x

� � 4�3x – 2

�x2

� � 4�3x � 2


Enrichment, p. 752

ACOUSTICS For Exercises 38 and 39, use the following information.If a vehicle is moving toward you at v miles per hour and blowing its horn at afrequency of f, then you hear the horn as if it were blowing at a frequency of h.

This can be defined by the equation h � , where s is the speed of sound,

approximately 760 miles per hour.

38. Simplify the complex fraction in the formula. �s �

fsv

�

39. Suppose a truck horn blows at 370 cycles per second and is moving toward you at 65 miles per hour. Find the frequency of the horn as you hear it.404.60 cycles/s

40. POPULATION According to the 2000 Census, New Jersey was the most denselypopulated state, and Alaska was the leastdensely populated state. The population of New Jersey was 8,414,350, and thepopulation of Alaska was 626,932. The landarea of New Jersey is about 7419 squaremiles, and the land area of Alaska is about570,374 square miles. How many morepeople were there per square mile inNew Jersey than in Alaska? about 1133

41. BICYCLES When air is pumped into a bicycle tire, the pressure P requiredvaries inversely as the volume of the air V and is given by the equation P � �

Vk�.

If the pressure is 30 lb/in2 when the volume is 1�23

� cubic feet, find the pressure

when the volume is �34

� cubic feet. 66�23

� lb/in2

42. CRITICAL THINKING Which expressions are equivalent to 0? a and c

a. � b. � c. �


How are rational expressions used in baking?


• an example of a situation in which you would divide a measurement by afraction when cooking, and

• an explanation of the process used to simplify a complex fraction.

44. The perimeter of hexagon ABCDEF is 12. Which expression can be used to represent the measure

of B�C�? C

�6nn

�

�

9

8

6� �

9nn

�

�

9

8

6�

�6

4

nn

�

�

9

3

6

2� �

9

4

nn

�

�

9

3

6

2�

45. Express in simplest form. C

�mn

2� �n1

� �mn

2� �

3265np2

3�DCBA

�6

5

mpn

�

�

�2

2

0

4

mn2

p�

DC

BA

3nn � 8

F C

A B

E D

WRITING IN MATH

2a � �1

2�

�1 � b

�1

2� � 2a�b � 1

a � �13

�

�b

a � �13

�

�b

a��3

a� � 1

a�1 � �

3

a�

Alaska

NewJersey

f�1 � �v

s�



�


ELL


Modeling Have students usefraction bars to prove that thesimplification of complexfractions formula is correct. Forexample, students can modeland then use the formula to

confirm that � � 6.

Getting Ready for Lesson 12-9PREREQUISITE SKILL Studentswill learn to solve rational equa-tions in Lesson 12-9. In order tosolve complicated rationalequations, students should firstbe sure they can solve simplerequations. Use Exercises 63–68 todetermine your students’ famili-arity with solving linear andquadratic equations.

Answer

43. Sample answer: Mostmeasurements used in baking arefractions or mixed numbers,which are examples of rationalexpressions. Answers shouldinclude the following.

• You want to find the number ofbatches of cookies you canmake using the 7 cups of flouryou have on hand when a batchrequires 1 cups of flour.

• Divide the expression in thenumerator of a complex fractionby the expression in thedenominator.

1�2

1�9

2�3



Mixed Review

48. �(37aa2

��

23)(aa

��

142)2�

58. 1.6014 105

5.331 104

2.799 104

2.298 104

1.812 104

1.575 104

1.119 104

1.08 104


Find each sum. (Lesson 12-7)

46. �142yx2� � �

68y� �

9x3�y2

4y� 47. �

a �a

b� � �

2b �b

3a� �

(3aa�

2 �b)

3(2

abb

��

3ba

2

)�

48. �3a2 �

a �10

3a � 8� � �

a2 � 82aa

� 16� 49. �

(nn

��

24)2� � �

n2n�

�n

5� 6

�

Find each difference. (Lesson 12-6)

50. �x72� � �

x32� �

x42� 51. �

(x �x

3)2� � �(x �

33)2� �

x �1

3�

52. �t2 �

2t � 2� � �

t2 �tt � 2� ��

t �1

1� 53. �

n2 � 22nn

� 24� � �

n2 � 28n � 24� �

n �2

6�

54. BIOLOGY Ana is working on a biology project for her school’s science fair. For her experiment, she needs to have a certain type of bacteria that doubles its population every hour. Right now Ana has 1000 bacteria. If Ana does notinterfere with the bacteria, predict how many there will be in ten hours.(Lesson 10-6) 1,024,000

Solve each equation by factoring. Check your solutions. (Lesson 9-5)

55. s2 � 16 �4 56. 9p2 � 64 ��83

� 57. z3 � 9z � 45 � 5z2

{�5, �3, 3}FAMILIES For Exercises 58–60, refer to the graph. (Lesson 8-3)

58. Write each number in the graph using scientific notation.

59. How many times as great is the amount spent on food as the amount spent on clothing? Express your answer in scientific notation. 2.59 101

60. What percent of the total amount is spent on housing?33.3%

TELEPHONE RATES For Exercises 61 and 62, use the following information.(Lesson 5-4) 61. C � 0.16m � 0.99A 15-minute call to Mexico costs $3.39. A 24-minute call costs $4.83.

61. Write a linear equation to find the total cost C of an m-minute call.

62. Find the cost of a 9-minute call. $2.43

PREREQUISITE SKILL Solve each equation.(To review solving equations, see Lessons 3-2 through 3-4.)

63. �12 � �x4

� �48 64. 1.8 � g � 0.6 2.4 65. �34

�n � 3 � 9 16

66. 7x2 � 28 �2, 2 67. 3.2 � ��8

��7

n� �14.4 68. ��3n

��

6(�4)� � �9 ��

530�

Cost of parenthood risingAn average middle-income family will spend $160,140to raise a child born in 1999. Costs of raising a child frombirth through age 17:

USA TODAY Snapshots®

By Hilary Wasson, and Sam Ward, USA TODAY

Source: Agriculture Department

Housing

Food

Transportation

MiscellaneousChild care

and educationHealthcare

Clothing

$53,310

$27,990

$22,980

$18,120

$15,750

$11,190

$10,800

�(n2n

�

2 �2)2

8(nn

��

23)

�


4 Assess4 Assess

Online Lesson Plans

USA TODAY Education’s Online site offers resources andinteractive features connected to each day’s newspaper.Experience TODAY, USA TODAY’s daily lesson plan, isavailable on the site and delivered daily to subscribers.This plan provides instruction for integrating USA TODAYgraphics and key editorial features into your mathematicsclassroom. Log on to www.education.usatoday.com.

http://www.education.usatoday.com




are rational equationsimportant in the

operation of a subway system?Ask students:• Why is it important to know

where trains are? Sampleanswer: so that the trains do notrun into each other.

• If you’re given the position oftwo moving trains, is thatsufficient information todetermine whether they willcollide in the future? No. Youneed to know their position,direction of travel, and speed.

SOLVE RATIONAL EQUATIONS are equations thatcontain rational expressions. You can use cross products to solve rational equations,but only when both sides of the equation are single fractions.

Rational equations

Another method you can use to solve rational equations is to multiply each sideof the equation by the LCD to eliminate fractions.

Vocabulary• rational equations• work problems• rate problems• extraneous solutions

Solving Rational Equations


are rational equations important in the operation of a subway system?are rational equations important in the operation of a subway system?

Use Cross ProductsSolve �

x1�2

5� � �

(x �4

2)�.

�x

1�2

5� � �

(x �4

2)� Original equation

12(x � 2) � 4(x � 5) Cross multiply.

12x � 24 � 4x � 20 Distributive Property

8x � �4 Add �4x and �24 to each side.

x � ��48

� or ��12

� Divide each side by 8.

Example 1Example 1

Use the LCDSolve �n �

n2

� � �nn

��

36

� � �n1

�.

�n �

n2

� � �nn

��

36

� � �n1

� Original equation

n(n � 6)��n �n

2� � �

nn

��

36

�� n(n � 6)��n1

�� The LCD is n(n � 6).

1 1 1

��n(n1� 6)� � �

n �n

2�� n(n

1� 6)� � �

nn

��

36

�� n(n

1� 6)� � �

n1


1 1 1

(n � 6)(n � 2) � n(n � 3) � n � 6 Simplify.

(n2 � 8n � 12) � (n2 � 3n) � n � 6 Multiply.

n2 � 8n � 12 � n2 � 3n � n � 6 Subtract.

�5n � 12 � n � 6 Simplify.

�6n � �18 Subtract 12 from each side.

n � 3 Divide each side by �6.

Example 2Example 2

• Solve rational equations.

• Eliminate extraneous solutions.

The Washington, D.C., Metrorail is one of the safestsubway systems in the world, serving a populationof more than 3.5 million. It is vital that a rail systemof this size maintain a consistent schedule. Rationalequations can be used to determine the exactpositions of trains at any given time.

Washington MetropolitanArea Transit Authority

Red Line

Orange Line

Blue Line

Green Line

Yellow Line

19.4 mi

24.14 mi

19.95 mi

20.59 mi

9.46 mi

DistanceTrain

LessonNotes

1 Focus1 Focus


Graphing Calculator and Spreadsheet Masters, p. 45






Transparencies

Lesson 12-9 Solving Rational Equations 691

A rational equation may have more than one solution.

Rational equations can be used to solve .work problems


Multiple SolutionsSolve �

a��

41

� � �a3

� � 1.

�a��

41

� � �3a

� � 1 Original equation

a(a � 1)��a��

41

� � �3a

�� a(a � 1)(1) The LCD is a(a � 1).

1 1

��a(a1� 1)� � �

a��

41

�� a(a1� 1)� � �

3a

�� a(a � 1) Distributive Property

1 1

�4a � 3a � 3 � a2 � a Simplify.

�a � 3 � a2 � a Add like terms.

0 � a2 � 2a � 3 Set equal to 0.

0 � (a � 3)(a � 1) Factor.

a � 3 � 0 or a � 1 � 0

a � �3 a � 1

CHECK Check by substituting each value in the original equation.

�a��

41

� � �3a

� � 1 �a��

41

� � �3a

� � 1

��3

��4

1� � �

�33� � 1 a � �3 �

1��

41

� � �31

� � 1 a � 1

2 � (�1) � 1 �2 � 3 � 1

1 � 1 1 � 1

The solutions are 1 or �3.

Example 3Example 3

Work ProblemLAWN CARE Abbey has a lawn care service. One day she asked her friendJamal to work with her. Normally, it takes Abbey two hours to mow and trimMrs. Harris’ lawn. When Jamal worked with her, the job took only 1 hour and 20 minutes. How long would it have taken Jamal to do the job himself?

Explore Since it takes Abbey two hours to do the yard, she can finish �12

� the job

in one hour. The amount of work Jamal can do in one hour can be

represented by �1t�. To determine how long it takes Jamal to do the job,

use the formula Abbey’s work � Jamal’s work � 1 completed yard.

Plan The time that both of them worked was 1�13

� hours. Each rate multiplied

by this time results in the amount of work done by each person.

Solve Abbey’s work plus Jamal’s work equals total work.

�12

��43

�� 1t��

43

�� 1

�46

� � �34t� � 1 Multiply.

��

Example 4Example 4

Look BackTo review solvingquadratic equations byfactoring, see Lessons 9-4through 9-7.

Study Tip

Work ProblemsWhen solving workproblems, remember that each term shouldrepresent the portion of a job completed in oneunit of time.

Study Tip

(continued on the next page)

Lesson 12-9 Solving Rational Expressions 691

2 Teach2 Teach

11

22

33

44


SOLVE RATIONALEQUATIONS

Teaching Tip Remind studentsto check their solutions bysubstituting them back into theoriginal equation.

Solve � . 9

Solve � � .

Solve a � � .�1, 3

TV INSTALLATION On Satur-days, Lee helps her fatherinstall satellite TV systems.The jobs normally take Lee’s

father about 2 hours. But

when Lee helps, the jobs only

take them 1 hours. If Lee

were installing a satellite sys-tem herself, how long wouldthe job take? The job would

take Lee 3 hours by herself.3�4

1�2

1�2

a2 � a � 2��

a � 1

a2 � 5�a2 � 1

1�6

2�x2 � x

1�x

5�x � 1

2�x � 6

8�x � 3

PowerPoint®


55


TRANSPORTATION Refer tothe application at the begin-ning of the lesson. Supposetwo Red Line trains leavetheir stations at opposite endsof the line at exactly 2:00 P.M.One train travels between thetwo stations in 48 minutesand the other train takes 54minutes. At what time do thetwo trains pass each other?The trains pass each other atabout 25 minutes after they lefttheir stations, at 2:25 P.M.

Rational equations can also be used to solve .rate problems


6t��46

� � �34t�� 6t � 1 The LCD is 6t.

1 2

6t��46

�� 6t��34t�� 6t Distributive Property

1 1

4t � 8 � 6t Simplify.

8 � 2t Add �4t to each side.

4 � t Divide each side by 2.

Examine The time that it would take Jamal to do the yard by himself is fourhours. This seems reasonable because the combined efforts of the twotook longer than half of Abbey’s usual time.

Rate ProblemTRANSPORTATION Refer to the application at the beginning of the lesson. The Yellow Line runs between Huntington and Mt. Vernon Square. Supposeone train leaves Mt. Vernon Square at noon and arrives at Huntington 24 minutes later, and a second train leaves Huntington at noon and arrives at Mt. Vernon Square 28 minutes later. At what time do the two trains pass each other?

Determine the rates of both trains. The total distance is 9.46 miles.

Train 1 �92.446

mmin

i� Train 2 �

92.846

mmin

i�

Next, since both trains left at the same time, the time both have traveled when they pass will be the same. And since they started at opposite ends of the route, the sum of their distances is equal to the total route, 9.46 miles.

�9.

2446t� � �

9.2486t� � 9.46 The sum of the distances is 9.46.

168��9.2446t� � �

9.2486t�� 168 � 9.46 The LCD is 168.

7 6

�1618

� � �9.

2446t� � �

1618

� � �9.

2486t� � 1589.28 Distributive Property

1 1

66.22t � 56.76t � 1589.28 Simplify.

122.98t � 1589.28 Add.

t � 12.92 Divide each side by 122.98.

The trains passed at about 12.92 or about 13 minutes after leaving their stations,which is 12:13 P.M.

Example 5Example 5

Rate ProblemsYou can solve rateproblems, also calleduniform motion problems,more easily if you firstmake a drawing.

Study Tip

Train 2Train 1

9.46 mi

M H

Train 1

Train 2

r t d

�92.446

mmin

i� t min �

92.446t� mi

�92.846

mmin

i� t min �

92.486t� mi


Kinesthetic Have students recreate the classic train problem inExample 5 on their own by either walking or riding bikes toward eachother over a known distance. Have the two students start at the sametime, one traveling at a rate that is faster than the other, and record thetime that they meet. Then work out the problem to see if they meet atthe expected time.


PowerPoint®


EXTRANEOUS SOLUTIONS Multiplying each side of an equation by the LCDof two rational expressions can yield results that are not solutions to the originalequation. Recall that such solutions are called .extraneous solutions

No SolutionSolve �

x3�x

1� � �

6xx

��

19

� � 6.

�x

3�x

1� � �

6xx

��

19

� � 6 Original equation

(x � 1)��x3�x

1� � �

6xx

��

19

�� (x � 1)6 The LCD is x � 1.

1 1

(x � 1)��x3�x

1�� (x � 1)��6x

x��

19

�� (x � 1)6 Distributive Property

1 1

3x � 6x � 9 � 6x � 6 Simplify.

9x � 9 � 6x � 6 Add like terms.

3x � 3 Add 9 to each side.

x � 1 Divide each side by 3.

Since 1 is an excluded value for x, the number 1 is an extraneous solution. Thus,the equation has no solution.

Example 6Example 6

Extraneous SolutionSolve �

12�n

n� � �

nn2��

31

� � 1.

�1

2�n

n� � �

nn2��

31

� � 1

�1

2�n

n� � �

(n �n1�)(n

3� 1)

� � 1

��n

2�n

1� � �

(n �n1�)(n

3� 1)

� � 1

(n � 1)(n � 1)��n

2�n

1� � �

(n �n1�)(n

3� 1)

�� (n � 1)(n � 1)1

1 1 1

(n � 1)(n � 1)��n

2�n

1�� (n � 1)(n � 1)��(n �

n1�)(n

3� 1)

�� (n � 1)(n � 1)

1 1 1

�2n(n � 1) � (n � 3) � n2 � 1

�2n2 � 2n � n � 3 � n2 � 1

�3n2 � n � 4 � 0

3n2 � n � 4 � 0

(3n � 4)(n � 1) � 0

3n � 4 � 0 or n � 1 � 0

n � ��43

� n � 1

The number 1 is an extraneous solution, since 1 is an excluded value for n.

Thus, ��43

� is the solution of the equation.

Example 7Example 7

Rational equations can have both valid solutions and extraneous solutions.


66

77




EXTRANEOUS SOLUTIONS

Solve � � 1.

no solutions

Solve � �

7

Have students—• complete the definitions/examples

for the remaining terms on theirVocabulary Builder worksheets forChapter 12.

• include examples of how to solverational equations.


2�16 � x2

1�x � 4

3�x � 4

3�3x � 3

3�x � 1

Unlocking Misconceptions

Make sure students understand that the equation in Example 6 doeshave a solution; however, that solution is undefined. Therefore, we saythat it has no solutions.

PowerPoint®



NAME ______________________________________________ DATE ____________ PERIOD _____

12-912-9

Less

on

12-

9

Solve Rational Equations Rational equations are equations that contain rationalexpressions. To solve equations containing rational expressions, multiply each side of theequation by the least common denominator.

Rational equations can be used to solve work problems and rate problems.

Solve � � 4.

� � 4

6� � � � 6(4) The LCD is 6.

2(x � 3) � 3x � 24 Distributive Property

2x � 6 � 3x � 24 Distributive Property

5x � 30 Simplify.

x � 6 Divide each side by 5.

The solution is 6.

x�2

x � 3�3

x�2

x � 3�3

x�2

x � 3�3 WORK PROBLEM Marla can

paint Percy’s kitchen in 3 hours. Percy canpaint it in 2 hours. Working together, howlong will it take Marla and Percy to paintthe kitchen?In t hours, Marla completes t � of the job and

Percy completes t � of the job. So an equation

for completing the whole job is � � 1.

� � 1

2t � 3t � 6 Multiply each term by 6.

5t � 6 Add like terms.

t � Solve.

So it will take Marla and Percy 1 hours to paintthe room if they work together.

1�5

6�5

t�2

t�3

t�2

t�3

1�2

1�3


ExercisesExercises

Solve each equation.

1. � � 8 20 2. � 1 3. � 1

4. � 9 5. s � � s � 3 �4 6. � � �4

7. � � 2 � 8. � �

9. � � 1 �2 10. � x2 � 9 �3 or 2

11. GREETING CARDS It takes Kenesha 45 minutes to prepare 20 greeting cards. It takesPaula 30 minutes to prepare the same number of cards. Working together at this rate,how long will it take them to prepare the cards? 18 min

12. BOATING A motorboat went upstream at 15 miles per hour and returned downstream at20 miles per hour. How far did the boat travel one way if the round trip took 3.5 hours?30 mi

x2 � 9�x � 3

m�1 � m

m � 1�m � 1

10�17

7x � 2�6

4x � 3�6

5 � 2x�2

3�2

q�q � 1

q � 4�q � 1

m�3

m�3

m � 4�m

1�3

4�s � 3

10�n � 1

8�n � 1

2x � 2�15

x � 1�5

6�x � 1

3�x

x�4

x � 5�5


Solve each equation. State any extraneous solutions.

1. � 2. � 3. �

8 4 �3

4. � 5. � � 6. � � �1

� �1 �2

7. � � 8. � � 0 9. � � 1

�7 � �

10. � � 1 11. � � 12. � � �3

4 4; extraneous: 2

13. � � 14. � � � 15. � � 0

�1, � , �1 �3, 2

16. � � 1 17. � � 1 18. � � 1

�3; extraneous: �2 �2, 4 �5, �2

PUBLISHING For Exercises 19 and 20, use the following information.Tracey and Alan publish a 10-page independent newspaper once a month. At production,Alan usually spends 6 hours on the layout of the paper. When Tracey helps, layout takes 3 hours and 20 minutes.

19. Write an equation that could be used to determine how long it would take Tracey to dothe layout by herself.

Sample answer: � � � � � � 1

20. How long would it take Tracey to do the job alone? 7 h 30 min

TRAVEL For Exercises 21 and 22, use the following information.Emilio made arrangements to have Lynda pick him up from an auto repair shop after hedropped his car off. He called Lynda to tell her he would start walking and to look for him on the way. Emilio and Lynda live 10 miles from the auto shop. It takes Emilio 2 hours to walkthe distance and Lynda 15 minutes to drive the distance.

21. If Emilio and Lynda leave at the same time, when should Lynda expect to spot Emilio onthe road? in 13 min

22. How far will Emilio have walked when Lynda picks him up? 1 mi

1�2

1�4

10�3

1�t

10�3

1�6

n � 6�n2 � 16

2n�n � 4

x�x � 3

x � 7�x2 � 9

p � 2�p2 � 4

2p�p � 2

9�2

2�5

6 � z�6z

1�z � 1

1�n

n � 5�n � 3

n � 2�n

7�3

m � 2�m � 2

2�m � 2

3�2

y2�2 � y

3y � 2�y � 2

1�d

d � 4�d � 2

d � 3�d

2x�2x � 3

4x�2x � 1

1�2

13�2

1�9t � 3

3t�3t � 3

3�p � 2

5�p � 1

q � 4�18

q�3

2q � 1�6

1�5

y � 2�5

y � 2�4

5y�6

1�2

4y�3

2h � 1�h � 2

2h�h � 1

k � 1�k � 9

k � 5�k

x � 4�x � 6

x�x � 5

7�n � 6

5�n � 2

Practice (Average)


NAME ______________________________________________ DATE ____________ PERIOD _____




NAME ______________________________________________ DATE ____________ PERIOD _____

12-912-9

Pre-Activity How are rational equations important in the operation of a subwaysystem?


What is some information that would be important in establishing aschedule for a subway system? Sample answer: the number oftrains that operate on each subway line, the speeds at whichthe trains can travel, the distances between subway stops oneach line, the amount of time a train spends stopped to allowpassengers to exit and board the train

Reading the Lesson

1. Is � a rational equation? Explain. No; the expression on the left side of the equation is not a rational expression.

2. How can you tell by looking at a rational equation whether you can solve it by usingcross products? The equation should have a single fraction on each side.

3. How does multiplying both sides of a rational equation by the LCD help you solve theequation? It gets rid of the fractions.

4. For Example 4 in your textbook, look at the first equation of the Solve stage.

a. What does the expression � � represent in this situation?

the part of the total work that Abbey can do in hours

b. What does the expression � � represent?

the part of the total work that Jamal can do in hours

c. What does the number 1 on the right side represent? 1 whole job

5. When you solve a rational equation, in which equation should you substitute toeliminate possible extraneous solutions? the original equation


6. Think of a word that can help you remember that multiplying by the LCD is one methodyou can use to solve a rational equation. Sample answer: Use the word lucid. Ifyour thinking is LuCiD, you will be successful using the LCD.

4�3

4�3

1�t

4�3

4�3

1�2

3�x

�x � 3��4


Winning DistancesIn 1999, Hicham El Guerrouj set a world record for the mile run witha time of 3:43.13 (3 h 43 min 13 s). In 1954, Roger Bannister ran thefirst mile under 4 minutes at 3:59.4. Had they run those times in thesame race, how far in front of Bannister would El Guerrouj have beenat the finish?

Use � r. Since 3 min 43.13 s � 223.13 s, and 3 min 59.4 s � 239.4 s,

El Guerrouj’s rate was and Bannister’s rate was .

r t d

El Guerrouj 223.13 5280 feet

Bannister 223.13 � 223.13 or 4921.2 feet

Therefore, when El Guerrouj hit the tape, he would be 5280 � 4921.2,or 358.8 feet, ahead of Bannister. Let’s see whether we can develop aformula for this type of problem.

L t D th di t d

5280�239.4

5280�239.4

5280�223.13

5280 ft�239.4 s

5280 ft�223.13 s

d�t

Enrichment

NAME ______________________________________________ DATE ____________ PERIOD _____

12-912-9

r t d

Enrichment, p. 758


Guided Practice

Application



Concept Check


11. �4a

� � �a �

32

� 8 12. �3x

� � �x �

12

� 3 13. �x �

x3

� � �xx

��

36

� 3

14. �x �

x1

� � �xx

��

61

� ��32

� 15. �23n� � �

12

� � �2n

6� 3� �3 16. �

54

� � �3

2

y� � �

7

6

y� ��

145�

17. �aa

��

11

� � �a

2�a

1� � �1 0 18. �

x2 �7

5x� � �

5 �3

x� � �

4x

� �277� 19. �

2x4�x

3� � �

2x2�x

3� � 1 �

12

�

20. �5 �

5p

� � �p

p�

2

5� � �2 21. �

3aa� 6� � �

5a �a

10� � �

25

� 22. �c �

c4

� � �4 �

6c

� � c

23. �2bb��

25

� � 2 � �b �

32

� 1 24. �k �

73

� � �12

� � �k �

34

� 5, 10

25. �xx

2

��

24

� � x2 � 4 �2, 1 26. �n

2�n

1� � �

nn2��

51

� � 1 �4

27. �z2 �

35zz � 4� � �

z �2

4� � �

z �3

1� 7 28. �

m2 � 84m � 12� � �

mm� 2� � �

m �1

6�

�1; extraneous 629. QUIZZES Each week, Mandy’s algebra teacher gives a 10-point quiz. After

5 weeks, Mandy has earned a total of 36 points for an average of 7.2 points per quiz. She would like to raise her average to 9 points. On how many quizzes must she score 10 points in order to reach her goal? 9

BOATING For Exercises 30 and 31, use the following information.Jim and Mateo live across a lake from each other at a distance of about 3 miles. Jimcan row his boat to Mateo’s house in 1 hour and 20 minutes. Mateo can drive hismotorboat the same distance in a half hour.

30. If they leave their houses at the same time and head toward each other, howlong will it be before they meet? about 22 min

31. How far from the nearest shore will they be when they meet? about 0.82 mi

32. CAR WASH Ian and Nadya can each wash a car and clean its interior in about 2 hours, but Chris needs 3 hours to do the work. If the three work together, how long will it take to clean seven cars? 5 hr 15 min

�

1. OPEN ENDED Explain why the equation n � �n �

11

� � �n �

11

� � 1 has nosolution. See margin.

2. Write an expression to represent the amount of work Aminta can do in h hours if it normally takes her 3 hours to change the oil and tune up her car. �h

3� � 1

3. Find a counterexample for the following statement.Sample answer: �

5x� � �

x �7

4�

The solution of a rational equation can never be zero.


4. �2x

� � �x �

31

� 2 5. �a �

71

� � �a �

53

� �8 6. �35x� � �

32

� � �71

x0� 15

7. �x �

x1

� � �x �

x4

� � 6 �54

� 8. �k �

51

� � �7k

� � �k �

11

� ��73

� 9. �xx

��

22

� � �x �

22

� � ��37�

�1, �52�

10. BASEBALL Omar has 32 hits in 128 times at bat. He wants his batting average

to be .300. His current average is �13228

� or .250. How many at bats does Omar

need to reach his goal if he gets a hit in each of his next b at bats? 10


Exercises Examples11–14 1

15–19, 21, 223, 26, 2722, 24, 25 3

29–34 4, 520, 28 6, 7



4, 5 16–8 2

9 710 4, 5

�

20. 3; extraneous 521. �622. �6, 1


ELL


Writing Have students create arate problem and then solve it,showing their work for each step.

Assessment Options


Answers

1. Sample answer: When you solvethe equation, n � 1. But n ≠ 1, sothe equation has no solution.

36. Sample answer: Rationalequations are used in solving rateproblems, so they can be used todetermine traveling times,speeds, and distances related tosubways. Answers should includethe following.

• Sample answer: Since bothtrains leave at the same time,their traveling time is the same.The sum of the distances ofboth trains is equal to the totaldistance between the twostations. So, add the twoexpressions to represent thedistance each train travels andsolve for time.

Building the Best Roller Coaster

It is time to complete your project. Use the information and datayou have gathered about the building and financing of a rollercoaster to prepare a portfolio or Web page. Be sure to includegraphs and/or tables in the presentation.

www.algebra1.com/webquest


42. �2(

62m

��m

3)�

43. �y2 � 2

1y � 1�




SWIMMING POOLS For Exercises 33 and 34, use the following information.The pool in Kara’s backyard is cleaned and ready to be filled for the summer. It measures 15 feet long and 10 feet wide with an average depth of 4 feet.

33. What is the volume of the pool? 600 ft3

34. How many gallons of water will it take to fill the pool? (1 ft3 � 7.5 gal) 4500 gal

35. CRITICAL THINKING Solve � 2 � 0. ��134�


How are rational equations important in the operation of a subway system?


• an explanation of how rational equations can be used to approximate the timethat trains will pass each other if they leave distant stations and head towardeach other.

37. What is the value of a in the equation �a �

a2

� � �aa

��

36

� � �1a

�? A

3 2 6 0

38. Which value is an extraneous solution of �n

��

12

� � �3nn

2

2��

72nn

��

88

�? D

6 2 �1 �2DCBA

DCBA

WRITING IN MATH

�xx

��

3

2� � �

x2

x�

�x

5

� 2�

��x � 1


39. �xx

��

12

� 40. �aa

��

57

� 41. �xx

�� 5

1�

Find each difference. (Lesson 12-7)

42. �2m

3� 3� � �

6 �m

4m� 43. �

y2 � 2

yy � 1� � �

y �1

1� 44. �

aa2��

29

� � �6a2 �

21a7a � 3�

Factor each polynomial. (Lesson 9-2)

45. 20x � 8y 4(5x � 2y) 46. 14a2b � 21ab2 47. 10p2 � 12p � 25p � 307ab(2a � 3b) (2p � 5)(5p � 6)

48. CHEMISTRY One solution is 50% glycol, and another is 30% glycol. How much of each solution should be mixed to make a 100-gallon solution that is 45% glycol? (Lesson 7-2) 75 gal of 50%, 25 gal of 30%

x � 2 � �x �

2

5�

��x � 6 � �

x �6

1�

�a2

a2

��

1

6

3

aa

��

5

42�

��

�aa2

2

��

3

4

aa��

1

3

8�

�x2

x2�

�8x

x��

1

6

5�

��xx

2

2��

2

2

xx

��

15

3�

Mixed Review

4a2 � 7a � 2��(6a � 1)(a � 3)(a � 3)


About the Exercises …Organization by Objective• Solve Rational Equations:

11–19, 21–27• Extraneous Solutions: 20–28



Average: 11–29 odd, 30, 31,35–48

Advanced: 12–28 even, 32–48

4 Assess4 Assess

http://www.algebra1.com/webquest


Study Guide and Review

www.algebra1.com/vocabulary_review696 Chapter 12 Rational Expressions and Equations

See pages642–647.

12-112-1

ExampleExample

Vocabulary and Concept CheckVocabulary and Concept Check

complex fraction (p. 684)excluded values (p. 648)extraneous solutions (p. 693)inverse variation (p. 642)

least common multiple (p. 678)least common denominator (p. 679)mixed expression (p. 684)product rule (p. 643)

rate problem (p. 692)rational equation (p. 690)rational expression (p. 648)work problem (p. 691)

State whether each sentence is true or false. If false, replace the underlined expression to make a true sentence.

1. A expression is a fraction whose numerator and denominator are polynomials.

2. The complex fraction can be simplified as . true

3. The equation �x �

x1

� � �2xx

��

13

� � 2 has an extraneous solution of . true

4. The mixed expression 6 � �aa

��

23

� can be rewritten as . false, �5aa

��

23

0�

5. The least common multiple for (x2 � 144) and (x � 12) is . false, x2 � 144

6. The excluded values for �x2 �

4xx� 12� are . true�3 and 4

x � 12

�5aa

��

136

�

1

�65

�

�4

5�

��2

3�

false, rational

mixed

Inverse VariationConcept Summary

• The product rule for inverse variations states that if (x1, y1) and (x2, y2) are solutions of an inverse variation, then x1y1 � k and x2y2 � k.

• You can use �xx

1

2

� � �yy

2

1

� to solve problems involving inverse variation.

If y varies inversely as x and y � 24 when x � 30, find x when y � 10.

�xx

1

2

� � �yy

2

1

� Proportion for inverse variations

�3x0

2

� � �1204� x1 = 30, y1 = 24, and y2 = 10

720 � 10x2 Cross multiply.

72 � x2 Thus, x � 72 when y � 10.

Exercises Write an inverse variation equation that relates x and y. Assume thaty varies inversely as x. Then solve. See Examples 3 and 4 on page 644.

7. If y � 28 when x � 42, find y when x � 56. xy � 1176; 218. If y � 15 when x � 5, find y when x � 3. xy � 75; 259. If y � 18 when x � 8, find x when y � 3. xy � 144; 48

10. If y � 35 when x � 175, find y when x � 75. xy � 6125; 81.67


Have students flip back through their organizers to make sure thatthey have included information for every lesson page in theorganizer. Now is a good time to ask if students have anyquestions about the concepts that they recorded in theirorganizers.Encourage students to refer to their Foldables while completingthe Study Guide and Review and use them in preparing for theChapter Test.

TM

For more informationabout Foldables, seeTeaching Mathematicswith Foldables.

Lesson-by-LessonReviewLesson-by-LessonReview

Vocabulary and Concept CheckVocabulary and Concept Check

• This alphabetical list ofvocabulary terms in Chapter 12includes a page referencewhere each term wasintroduced.

• Assessment A vocabularytest/review for Chapter 12 isavailable on p. 772 of theChapter 12 Resource Masters.

For each lesson,

• the main ideas aresummarized,

• additional examples reviewconcepts, and

• practice exercises are provided.

The Vocabulary PuzzleMakersoftware improves students’ mathematicsvocabulary using four puzzle formats—crossword, scramble, word search using aword list, and word search using clues.Students can work on a computer screenor from a printed handout.

Vocabulary PuzzleMaker

ELL

MindJogger Videoquizzesprovide an alternative review of conceptspresented in this chapter. Students workin teams in a game show format to gainpoints for correct answers. The questionsare presented in three rounds.

Round 1 Concepts (5 questions)Round 2 Skills (4 questions)Round 3 Problem Solving (4 questions)

MindJogger Videoquizzes

ELL



Answers

11.

12. n

13.

14.

15.

16.

17.

18.

19.

20. b � 7

(x � 4)2�(x � 2)2

3�a2 � 3a

30�x � 10

3axy�

10

14a2b�

3

x � 3�x(x � 6)

a � 5�a � 2

x�4y2z

Chapter 12 Study Guide and Review 697

Chapter 12 Study Guide and ReviewChapter 12 Study Guide and Review

ExampleExample

Dividing Rational ExpressionsConcept Summary

• Divide rational expressions by multiplying by the reciprocal of the divisor.

Find �yy

2

2�

�

1664

� � �yy

�

�

48

�.

�yy

2

2

�

�

1

6

6

4� � �

yy

�

�

4

8� � �

yy

2

2

�

�

1

6

6

4� � �

yy

�

�

8

4� Multiply by the reciprocal of �

yy

�

�

48

�.

� �(

(

yy

�

�

4

8

)

)

(

(

yy

�

�

4

8

)

)�

1

� �yy

�

�

8

4�

1

1

or �yy

�

�

4

8� Simplify.

See pages648–653.

12-212-2

See pages655–659.

12-312-3

See pages660–664.

12-412-4

Rational ExpressionsConcept Summary

• Excluded values are values of a variable that result in a denominator of zero.

Simplify �x2 �

x1�2x

4� 32

�. State the excluded values of x.

�x2 �

x1�2x

4� 32

� �

1

�(x �

x4�)(x

4� 8)

�

1

Factor.

� �x �

18

� Simplify.

The expression is undefined when x � �4 and x � �8.

Exercises Simplify each expression. See Example 5 on page 650. 11–14. See margin.

11. �1

3

2

xx

2

yy3z

� 12. �nn

2 ��

33n

� 13. �a2 �

a2

3�a

2�5

10� 14. �x

x

2

3��

1x02x�

�42

2x1

�

Multiplying Rational ExpressionsConcept Summary

• Multiplying rational expressions is similar to multiplying rational numbers.

Find �x2 � x

1� 12� � �

xx

��

35

� .

�x2 � x

1� 12� � �

xx

��

35

� � �(x � 4)

1(x � 3)�

1

� �xx

��

35

� Factor.

� �(x � 4)

1(x � 5)� Simplify.

Exercises Find each product. See Examples 1–3 on pages 655 and 656. 15–20. See margin.

15. �79b2� � �

6ba2� 16. �

5

8

xa

2

by

� � �1225ax

2b� 17. (3x � 30) � �

x2 �10

100�

18. �3aa2 �

�96

� � �aa2 �

�23a

� 19. �x2 �

x �x �

212

� � �x2x�

�x

4� 6

� 20. �b2 �

b1�9b

3� 84

�� b2 �

b2

15�b

9� 36

�

ExampleExample

ExampleExample1

1



Answers

25. 2ac2 � 4a2c �

26. x2 � 4x � 2

27. x2 � 2x � 3

28. 4b � 1 � 8�12b � 1

3c2�b


ExampleExample

Rational Expressions with Like DenominatorsConcept Summary

• Add (or subtract) rational expressions with like denominators by adding(or subtracting) the numerators and writing the sum (or difference) overthe denominator.

Find �m

m�

2

4� � �

m1�6

4� .

�m

m�

2

4� � �

m1�6

4� � �

mm

2 ��

146

� Subtract the numerators.

� �(m �

m4)

�(m

4� 4)

�

1

or m � 4 Factor.

Exercises Find each sum or difference. See Examples 1–4 on pages 672 and 673.

29. �m �

54

� � �m �

51

� �2m

5� 3� 30. �

2n�

�5

5� � �

2n2�n

5� 1 31. �

a �a2

b� � �

a��b2

b� a � b

32. �7ba2� � �

5ba2� �

2ba2� 33. �

x2�x

3� � �

x �6

3� 2 34. �

mm�

2

n� � �

2mm

n��

nn2

� m � n

ExampleExample


See pages666–671.

12-512-5

See pages672–677.

12-612-6

Dividing PolynomialsConcept Summary

• To divide a polynomial by a monomial, divide each term of thepolynomial by the monomial.

• To divide a polynomial by a binomial, use long division.

Find (x3 � 2x2 � 22x � 21) � (x � 3).

x2 � x � 19

x � 3�x�3�� 2�x�2�� 2�2�x� �� 2�1�(�)x3 � 3x2 Multiply x2 and x � 3.

x2 � 22x Subtract.(�)x2 � 3x Multiply x and x � 3.

� 19x � 21 Subtract.(�)� 19x � 57 Multiply �19 and x � 3.

� 36 Subtract. The quotient is x3 � x � 19 � �x

3�6

3�.

Exercises Find each quotient. See Examples 1–5 on pages 666–668. 25–28. See margin.25. (4a2b2c2 � 8a3b2c � 6abc2) � 2ab2 26. (x3 � 7x2 � 10x � 6) � (x � 3)

27. �x3 �

x �7x

2� 6

� 28. (48b2 � 8b � 7) � (12b � 1)

Exercises Find each quotient. See Examples 1–4 on pages 660 and 661.

21. �2

pq

3

� � �4

pq

2

� 2p 22. �y

y�

2

4� � �

y2

3

�

y16

� �y(y

3� 4)�

23. �3yy

�

�

1

4

2� � (y2 � 6y � 8) 24. �2m2 �

m7�m

5� 15

� � �93mm

2

��

24

� �23mm �

�23

�

�(y � 4)

3(y � 2)�

1




Rational Expressions with Unlike Denominators Concept Summary

• Rewrite rational expressions with unlike denominators using the leastcommon denominator (LCD). Then add or subtract.

Find �x �

x3

� � �x �

52

� .

�x �

x3

� � �x �

52

� � �xx

��

22

� � �x �

x3

� � �xx

��

33

� � �x �

52

� The LCD is (x � 3)(x � 2).

� �(x �

x2

3�)(x

2x� 2)

� � �(x �

5x3�)(x

1�5

2)� Multiply.

� �(xx

2

��

33)x(x

��

125)

� Add.

Exercises Find each sum or difference. See Examples 3–5 on pages 679 and 680.

35. �32dc2� � �

23cd� �

4c6

2

c�d2

9c� 36. �

r2

r2��

291r

� � �r �

3r3

� �r �

4r3

� 37. �a

3�a

2� � �

a5�a

1� �

(a �8a2

2)�(a

7�a

1)�

38. �73n� � �

97n� �

2221n

� 39. �37a� � �

63a2� �

14a6a

�2

3� 40. �

2x2�x

8� � �

5x �4

20� �

55xx��

240

�


Mixed Expressions and Complex Fractions Concept Summary

• Write mixed expressions as rational expressions in the same way as mixednumbers are changed to improper fractions.

• Simplify complex fractions by writing them as division problems.

Simplify .

� The LCD in the numerator is y � 3.

� Add in the numerator.

� �y2 �

y3

�

y3

� 40� � (y � 5) Rewrite as a division sentence.

� �y2 �

y3

�

y3

� 40� � �

y �

1

5� Multiply by the reciprocal of y � 5.

� �(y �

y8

�

)(y3

� 5)�

1

� �y �

1

5�

1

or �yy

�

�

8

3� Factor.

Exercises Write each mixed expression as a rational expression.See Example 1 on page 684.

41. 4 � �x �

x2

� �5xx��

28

� 42. 2 � �xx2��

24

� �2xx��

25

� 43. 3 � �xx

2

2

�

�

yy

2

2� �4x

x

2

2�

�

2y2y2

�

�y2 �

y3

�

y3

� 40�

��y � 5

�y(

(

yy

�

�

3

3

)

)� � �

y4

�

0

3�

��y � 5

y � �y

4

�0

3�

��y � 5

y � �y

4�0

3�

��y � 5

See pages678–683.

12-712-7

ExampleExample

ExampleExample

See pages684–689.

12-812-8



Solving Rational EquationsConcept Summary

• Use cross products to solve rational equations with a single fraction oneach side of the equal sign.

• Multiply every term of a more complicated rational equation by the LCDto eliminate fractions.

Solve �56n� � �

n �1

2� � �

3(nn

��

12)

�.

�56n� � �

n �1

2� � �

3(nn

��

12)

� Original equation

6(n � 2)��56n� � �

n �1

2�� 6(n � 2)�

3(nn

��

12)

� The LCD is 6(n � 2)

�6(n �

62)(5n)� � �

6(nn

��

22)

� � �6(n

3�(n

2�)(n

2�)

1)� Distributive Property

(n � 2)(5n) � 6 � 2(n � 1) Simplify.

5n2 � 10n � 6 � 2n � 2 Multiply.

5n2 � 12n � 4 � 0 Subtract.

(5n � 2)(n � 2) � 0 Factor.

n � �25

� or n � 2

CHECK Let n � �25

�. Let n � 2.

� � �3(

22

��

12)

� � � �2 �

12

�

��274� � ��

274� � �

3(30)� � �

160� � �

10

�

Exercises Solve each equation. State any extraneous solutions.See Examples 6 and 7 on page 693.

47. �43x� � �

72

� � �71x2� � �

14

� �5 48. �21x1� � �

32x� � �

16

� 29

49. �32r� � �

r �3r

2� � �3 ��

14

� 50. �x �

x2

� � �xx

��

36

� � �1x

� 3

51. �x2 �

33x

� � �xx

��

23

� � �1x

� 52. �n �

14

� � �n �

11

� � �n2 � 3

2n � 4�

�1; extraneous 0 no solution

When you check the value 2, youget a zero in the denominator. So, 2 is an extraneous solution.

5(2)�

6

1��2

5� � 2

5��25

��

6

�2

5� � 1

�3��

2

5� � 2�


• Extra Practice, see pages 846–849.• Mixed Problem Solving, see page 864.

ExampleExample

See pages690–695.

12-912-9

Simplify each expression. See Examples 3 and 4 on pages 685 and 686.

44. �3yx� 45. �

220aa2 �

�31a6

� 46. �y2

y2�

�

116

yy

�

�

18

0�

y � 9 � �y �

6

4�

��y � 4 � �

y �2

1�

5 � �4

a�

��2a� � �3

4�

�xy

2

3�

��93yx2

�

1 1 12

1 1 1 1


Practice Test

Chapter 12 Practice Test 701

Skills and ApplicationsSkills and Applications

Write an inverse variation equation that relates x and y. Assume that y varies inversely as x. Then solve.

4. If y � 21 when x � 40, find y when x � 84. xy � 840; 105. If y � 22 when x � 4, find x when y � 16. xy � 88; 5.5


6. �65m

��

2m15

� ��13

�; �52

� 7. �2x2

3�

�5x

x� 3

� �2x

1� 1�; �3, �

12

� 8. �24cc2

2

��

1112cc��

291

� �2cc

��

73

�; ��32

�, 7

9. �t �

t9

�; �9, 0, 9 10. �162uu

��

51t8t

� ; �32

�t, 0 11. �xx

��

11

�; �3, �1, 2

Perform the indicated operations.

12. �x

2�x

7� � �

x1�4

7� 2 13. �

2nn

��

38

� � �62nn

��

214

� �32

nn

��

91

�

14. (10m2 � 9m � 36) � (2m � 3) 5m � 12 15. �x2 �

x4�x

5� 32� � �

x2 �x

7�x

3� 12� �

xx

��

85

�

16. �zz

2

2��

29zz

��

1250

� � (z � 3) �z �

14

� 17. �4xx

2

2��

1x1x

��6

6� � �

xx

2

2��

8xx

��

1126

� �4xx��

43

�

18. (10z4 � 5z3 � z2) � 5z3 2z � 1 � �51z� 19. �

7y �

y14

� � �6 �

63y

� �7y(

2

x�

�

126)y(x

�

�

228)

�

20. �xx

��

52

� � 6 �7xx

��

127

� 21. �xx

2

��

11

� � �xx

2

��

11

� ��x

2�x

1�


22. �n

2�n

4� � 2 � �

n �4

5� �14 23. �

x2 � 53x � 6� � �

x �7

3� � ��

xx

��

12

� 7; extraneous �2

24. FINANCE Barrington High School is raising money for Habitat for Humanity bydoing lawn work for friends and neighbors. Scott can rake a lawn and bag theleaves in 5 hours, while Kalyn can do it in 3 hours. If Scott and Kalyn worktogether, how long will it take them to rake a lawn and bag the leaves? 1�7

8� h

25. STANDARDIZED TEST PRACTICE Which expression can be usedto represent the area of the triangle? B

�12

�(x � y) �32

�(x � y)

�14

�(x � y) �x1�08

y�DC

BA x2 � y 2

12

36x � y

24x � y

48x � y

x � 4 � �x �

52

�

��x � 6 � �

x1�5

2�

�56

� � �ut�

��2tu� � 3

1 � �9t�

�1 � �8

t21�

www.algebra1.com/chapter_test

Vocabulary and ConceptsVocabulary and Concepts

Choose the letter that best matches each algebraic expression.

1. a 2. 3 � �aa

��

11

� c 3. �x2 � 2

2x � 4� b

�ba�

��xy

�

a. complex fraction

b. rational expression

c. mixed expression

Chapter 12 Practice Test 701

Introduction In this chapter, you learned the many different operations usedwith rational algebraic expressions. Think back to when you first learned aboutfractions. Fractions are also rational expressions.Ask Students Pick several of the concepts that you learned in this chapter.Demonstrate them with rational numbers and fractions. How are fractionssimilar to rational expressions?

Portfolio Suggestion

Assessment Options

Vocabulary Test A vocabularytest/review for Chapter 12 can befound on p. 772 of the Chapter 12Resource Masters.

Chapter Tests There are sixChapter 12 Tests and an Open-Ended Assessment task availablein the Chapter 12 Resource Masters.

Open-Ended AssessmentPerformance tasks for Chapter 12can be found on p. 771 of theChapter 12 Resource Masters. Asample scoring rubric for thesetasks appears on p. A34.

Unit 4 Test A unit test/reviewcan be found on pp. 779–780 ofthe Chapter 12 Resource Masters.

TestCheck andWorksheet Builder

This networkable software hasthree modules for assessment.

• Worksheet Builder to makeworksheets and tests.

• Student Module to take testson-screen.

• Management System to keepstudent records.

Chapter 12 TestsForm Type Level Pages

1 MC basic 759–760

2A MC average 761–762

2B MC average 763–764

2C FR average 765–766

2D FR average 767–768

3 FR advanced 769–770

MC = multiple-choice questionsFR = free-response questions



Standardized Test PracticeStudent Record Sheet (Use with pages 702–703 of the Student Edition.)

Select the best answer from the choices given and fill in the corresponding oval.

1 4 7

2 5 8

3 6

Solve the problem and write your answer in the blank.

For Question 9, also enter your answer by writing each number or symbol in abox. Then fill in the corresponding oval for that number or symbol.

9 (grid in) 9

10

11

12

13

Select the best answer from the choices given and fill in the corresponding oval.

14

15

16

17

Record your answers for Question 18 on the back of this paper.

DCBA

DCBA

DCBA

DCBA

0 0 0

.. ./ /

.

99 9 987654321

87654321

87654321

87654321

DCBADCBA

DCBADCBADCBA

DCBADCBADCBA

NAME DATE PERIOD

1212

An

swer

s

Part 2 Short Response/Grid InPart 2 Short Response/Grid In

Part 3 Quantitative ComparisonPart 3 Quantitative Comparison

Part 4 Open-EndedPart 4 Open-Ended

Part 1 Multiple ChoicePart 1 Multiple Choice

Standardized Test PracticeStudent Recording Sheet, p. A1

Teaching Tip In Question 6,remind students that an orderedpair must satisfy both equationsto be a solution.

Additional Practice

See pp. 777–778 in the Chapter 12Resource Masters for additionalstandardized test practice.

Record your answers on the answer sheetprovided by your teacher or on a sheet ofpaper.

1. A cylindrical container is 8 inches in heightand has a radius of 2.5 inches. What is thevolume of the container to the nearest cubicinch? (Hint: V � �r2h) (Lesson 3-8) D

63 126

150 157

2. Which function includes all of the orderedpairs in the table? (Lesson 4-8) B

y � �2x y � �3x � 1

y � 2x � 4 y � 3x � 1

3. Which equation describes the graph below?(Lesson 5-4) B

4x � 5y � 40

4x � 5y � �40

4x � 5y � �8

rx � 5y � 10

4. Which equation represents the line that passes

through (�12, 5) and has a slope of ��14

�?(Lesson 5-5) A

x � 4y � 8 �x � 4y � 20

�4x � y � 65 x � 4y � 5

5. Which inequality represents the shadedregion? (Lesson 6-6) D

y � ��12

�x � 2

y ��12

�x � 2

y � �2x � 2

y �2x � 2

6. Which ordered pair is the solution of thefollowing system of equations? (Lesson 7-4) B

3x � y � �2�2x � y � 8

(�6, 16) (�2, 4)

(�3, 2) (2, �8)

7. The length of a rectangular door is 2.5 times its width. If the area of the door is 9750 squareinches, which equation will determine thewidth w of the door? (Lesson 8-1) B

w2 � 2.5w � 9750

2.5w2 � 9750

2.5w2 � 9750 � 0

7w � 9750

8. A scientist monitored a 144-gram sample of a radioactive substance, which decays into anonradioactive substance. The table shows the amount, in grams, of the radioactive substance remaining at intervals of 20 hours. How many grams of the radioactive substanceare likely to remain after 100 hours?(Lessons 10-6 and 10-7) C

1 g 2.25 g

4.5 g 9 gDC

BA

D

C

B

A

DC

BA

D

C

B

A y

xO

DC

BA

D

C

B

y

xO 4�4�8 8

8

4

�4

�8

A

DC

BA

DC

BA

Part 1 Multiple Choice


Test-Taking TipQuestions 2, 4, 8 Sometimes sketching the graph of a function can help you to see the relationshipbetween x and y and answer the question.

�3 �1 1 3 5

10 4 �2 �8 �14

x

y

0 20 40 60 80 100

144 72 36

Time (h)

Mass (g)


These two pages contain practicequestions in the various formatsthat can be found on the mostfrequently given standardizedtests.

A practice answer sheet for thesetwo pages can be found on p. A1of the Chapter 12 Resource Masters.

Log On for Test Practice The Princeton Review offersadditional test-taking tips and

practice problems at their web site. Visitwww.princetonreview.com orwww.review.com

TestCheck andWorksheet Builder

Special banks of standardized testquestions similar to those on the SAT,ACT, TIMSS 8, NAEP 8, and Algebra 1End-of-Course tests can be found onthis CD-ROM.

http://www.princetonreview.com

http://www.review.com

Evaluating Open-EndedAssessment Questions

Open-Ended Assessmentquestions are graded by using amultilevel rubric that guides youin assessing a student’s knowl-edge of a particular concept.

Goal: Students examine arelationship to see if it is inverse.

Sample Scoring Rubric: Thefollowing rubric is a samplescoring device. You may wish toadd more detail to this sample tomeet your individual scoringneeds.

Answers

Part 2 Short Response/Grid In

Part 4 Open Ended

Part 3 Quantitative Comparison

www.algebra1.com/standardized_test Chapter 12 Standardized Test Practice 703

Aligned and verified by

Column A Column B

�x2 �

12x

� �y2 �

12y

�

5 � �x

3�x

1�

�24y

3� 15�

��

�6y

6� 6�

the excluded value the excluded value

of a in �16a32

�a

24� of b in �5bb

��

63�

�500� � �20� � �125� � �45��180� � �720�

Record your answers on the answer sheetprovided by your teacher or on a sheet ofpaper.

9. A family drove an average of 350 miles per day during three days of their trip. They drove 360 miles on the first day and270 miles on the second day. How manymiles did they drive on the third day?(Lesson 3-4) 420

10. The area of the rectangular playground atHillcrest School is 750 square meters. Thelength of the playground is 5 meters greaterthan its width. What are the length andwidth of the playground in meters?(Lesson 9-5) � � 30, w � 25

11. Use the Quadratic Formula or factoring to determine whether the graph of y � 16x2 � 24x � 9 intersects the x-axis in zero, one, or two points. (Lesson 10-4) one

12. Express �xx

2

3�� x

9� � �

x3�x

3� as a quotient of

two polynomials written in simplest form.(Lesson 11-3) �3

xx2 �

�19�

13. Express the following quotient in simplestform. (Lesson 11-4)

�x �

x4

� � �x2

4�x

16�

�x �

44

� or �14

�x � 1

Compare the quantity in Column A and the quantity in Column B. Then determinewhether:

the quantity in Column A is greater,

the quantity in Column B is greater,

the two quantities are equal, or

the relationship cannot be determinedfrom the information given.

14. x � �14

�, y � 4

B (Lesson 1-3)

15.

C (Lesson 11-2)

16.

A (Lesson 12-2)

17.

D (Lesson 12-8)

Record your answers on a sheet of paper.Show your work. 18a–c. See margin.

18. A 12-foot ladder is placed against the side of a building so that the bottom of the ladder is 6 feet from the base of thebuilding. (Lesson 12-1)

a. Suppose the bottom of the ladderis moved closer tothe base of thebuilding. Does theheight that the ladderreaches increase ordecrease?

b. What conclusion can you make about theheight the ladder reaches and thedistance between the bottom of theladder and the base of the building?

c. Does this relationship form an inverseproportion? Explain your reasoning.

12 ft

6 ft

D

C

B

A

Chapter 12 Standardized Test Practice 703

Score Criteria4 A correct solution that is supported

by well-developed, accurateexplanations

3 A generally correct solution, butmay contain minor flaws inreasoning or computation

2 A partially correct interpretationand/or solution to the problem

1 A correct solution with nosupporting evidence or explanation

0 An incorrect solution indicating nomathematical understanding of the concept or task, or no solution is given

18a. The height increases.

18b. Sample answer: As the distancebetween the bottom of the ladderand the base of the buildingdecreases, the height that theladder reaches increases.

18c. No; sample answer: When thebottom of the ladder is 6 feet fromthe base of the building, it reaches aheight of about 10.4 feet. When thebottom of the ladder is 4 feet fromthe base of the building, it reachesa height of about 11.3 feet. In order

to form an inverse proportion, 6 � 10.4 mustequal 4 � 11.3. However, 6 � 10.4 � 63.6 and4 � 11.3 � 45.2. Because the products arenot equal, this relationship does not form aninverse proportion.


· 640a chapter 12 rational expressions and equations pacing suggestions for the entire year can...

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