review of real numbers - pearson chapter 1 review of real numbers • study skills builder. this...

The apparent magnitude of a star is the measure of its brightness as seen by some-one on Earth. The smaller the apparent magnitude, the brighter the star. Below,the apparent magnitudes of some stars are listed.Around 150 B.C., a Greek astronomer, Hipparchus, devised a system of classifying

the brightness of stars. Hipparchus’s system is the basis of the apparent magnitudescale used by modern astronomers. In Exercises 77 through 82, Section 1.2, we shall seehow this scale is used to describe the brightness of objects such as the sun, the moon,and some planets.

1

1C H A P T E R

Review of Real Numbers

1.1 Tips for Success inMathematics

1.2 Symbols and Sets ofNumbers

1.3 Fractions

1.4 Introduction to VariableExpressions and Equations

1.5 Adding Real Numbers

1.6 Subtracting Real Numbers

Integrated Review—Operations on RealNumbers

1.7 Multiplying and DividingReal Numbers

1.8 Properties of Real Numbers

The power of mathematics is itsflexibility. We apply numbers toalmost every aspect of our lives,from an ordinary trip to the gro-cery store to a rocket launchedinto space. The power of algebrais its generality. Using letters torepresentnumbers,wetie togetherthe trip to the grocery store andthe launched rocket.

In this chapter we reviewthe basic symbols and words—the language—of arithmetic andintroduce using variables in placeof numbers. This is our startingplace in the study of algebra.

Apparent Apparent Star Magnitude Star Magnitude

Arcturus -0.04 Spica 0.98

Sirius -1.46 Rigel 0.12

Vega 0.03 Regulus 1.35

Antares 0.96 Canopus -0.72

Sun -26.7 Hadar 0.61

(Source: Norton’s 2000.0: Star Atlas and Reference Handbook, 18th ed.,Longman Group, UK, 1989)

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2 CHAPTER 1 Review of Real Numbers

Before reading this section, remember that your instructor is your best source for in-formation. Please see your instructor for any additional help or information.

OBJECTIVE 1 � Getting ready for this course. Now that you have decided to take thiscourse, remember that a positive attitude will make all the difference in the world.Yourbelief that you can succeed is just as important as your commitment to this course.Make sure you are ready for this course by having the time and positive attitude that ittakes to succeed.

Next, make sure you have scheduled your math course at a time that will give youthe best chance for success. For example, if you are also working, you may want tocheck with your employer to make sure that your work hours will not conflict withyour course schedule. Also, schedule your class during a time of day when you aremore attentive and do your best work.

On the day of your first class period, double-check your schedule and allow your-self extra time to arrive in case of traffic problems or difficulty locating your classroom.Make sure that you bring at least your textbook, paper, and a writing instrument. Areyou required to have a lab manual, graph paper, calculator, or some other supply be-sides this text? If so, also bring this material with you.

OBJECTIVE 2 � General tips for success. Below are some general tips that will increaseyour chance for success in a mathematics class. Many of these tips will also help you inother courses you may be taking.

Exchange names and phone numbers with at least one other person in class. Thiscontact person can be a great help if you miss an assignment or want to discuss mathconcepts or exercises that you find difficult.

Choose to attend all class periods and be on time. If possible, sit near the front ofthe classroom. This way, you will see and hear the presentation better. It may also beeasier for you to participate in classroom activities.

Do your homework. You’ve probably heard the phrase “practice makes perfect”in relation to music and sports. It also applies to mathematics.You will find that the moretime you spend solving mathematics problems, the easier the process becomes. Be sureto schedule enough time to complete your assignments before the next class period.

Check your work. Review the steps you made while working a problem. Learn tocheck your answers in the original problems. You may also compare your answers withthe answers to selected exercises section in the back of the book. If you have made amistake, try to figure out what went wrong. Then correct your mistake. If you can’t findwhat went wrong, don’t erase your work or throw it away. Bring your work to your in-structor, a tutor in a math lab, or a classmate. It is easier for someone to find where youhad trouble if they look at your original work.

Learn from your mistakes and be patient with yourself. Everyone, even your in-structor, makes mistakes. (That definitely includes me—Elayn Martin-Gay.) Use yourerrors to learn and to become a better math student. The key is finding and under-standing your errors.

Was your mistake a careless one, or did you make it because you can’t read yourown math writing? If so, try to work more slowly or write more neatly and make a con-scious effort to carefully check your work.

Did you make a mistake because you don’t understand a concept? Take the timeto review the concept or ask questions to better understand it.

Did you skip too many steps? Skipping steps or trying to do too many steps men-tally may lead to preventable mistakes.

Know how to get help if you need it. It’s all right to ask for help. In fact, it’s agood idea to ask for help whenever there is something that you don’t understand.Make sure you know when your instructor has office hours and how to find his or heroffice. Find out whether math tutoring services are available on your campus. Check

OBJECTIVES

1 Get ready for this course.

2 Understand some general tipsfor success.

3 Understand how to use this text.

4 Get help as soon as you need it.

5 Learn how to prepare for andtake an exam.

6 Develop good timemanagement.

1.1 TIPS FOR SUCCESS IN MATHEMATICS

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Section 1.1 Tips for Success in Mathematics 3

out the hours, location, and requirements of the tutoring service. Videotapes and soft-ware are available with this text. Learn how to access these resources.

Organize your class materials, including homework assignments, gradedquizzes and tests, and notes from your class or lab. All of these items will be valuablereferences throughout your course especially when studying for upcoming testsand the final exam. Make sure that you can locate these materials when you needthem.

Read your textbook before class. Reading a mathematics textbook is unlikeleisure reading such as reading a novel or newspaper.Your pace will be much slower. Itis helpful to have paper and a pencil with you when you read.Try to work out exampleson your own as you encounter them in your text.You should also write down any ques-tions that you want to ask in class.When you read a mathematics textbook, some of theinformation in a section may be unclear. But when you hear a lecture or watch a video-tape on that section, you will understand it much more easily than if you had not readyour text beforehand.

Don’t be afraid to ask questions. Instructors are not mind readers. Many timeswe do not know a concept is unclear until a student asks a question. You are not theonly person in class with questions. Other students are normally grateful that someonehas spoken up.

Hand in assignments on time. This way you can be sure that you will not losepoints for being late. Show every step of a problem and be neat and organized. Also besure that you understand which problems are assigned for homework. You can alwaysdouble-check this assignment with another student in your class.

OBJECTIVE 3 � Using this text. There are many helpful resources that are available toyou in this text. It is important that you become familiar with and use these resources.They should increase your chances for success in this course.

• The main section of exercises in each exercise set is referenced by an example(s).Use this referencing if you have trouble completing an assignment from the exer-cise set.

• If you need extra help in a particular section, look at the beginning of the sectionto see what videotapes and software are available.

• Make sure that you understand the meaning of the icons that are beside manyexercises. The video icon tells you that the corresponding exercise may beviewed on the videotape that corresponds to that section. The pencil icon tellsyou that this exercise is a writing exercise in which you should answer in completesentences. The icon tells you that the exercise involves geometry.

• Practice exercises are located immediately after a worked-out Example in thetext. This exercise is similar to the previous Example and is a great way for you toimmediately reinforce a learned concept. All answers to the practice exercises arefound in the back of the text.

• Integrated Reviews in each chapter offer you a chance to practice—in oneplace—the many concepts that you have learned separately over several sections.

• There are many opportunities at the end of each chapter to help you understandthe concepts of the chapter.

Chapter Highlights contain chapter summaries and examples.

Chapter Reviews contain review problems organized by section.

Chapter Tests are sample tests to help you prepare for an exam.

Cumulative Reviews are reviews consisting of material from the beginningof the book to the end of that particular chapter.

• The Bigger Picture. This feature contains the directions for building an outlineto be used throughout the course. The purpose of this outline is to help you makethe transition from thinking “section by section” to thinking about how the math-ematics in this course is part of a bigger picture.

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• Study Skills Builder. This feature is found at the end of many exercise sets. Inorder to increase your chance of success in this course, please read and answerthe questions in the Study Skills Builder. For your convenience, the table belowcontains selected Study Skills Builder titles and their location.

Study Skills Builder Title Page of First Occurrence

Learning New Terms Page 57

What to Do the Day of an Exam Page 141

Have You Decided to Complete This Course Successfully? Page 81

Organizing a Notebook Page 122

How Are Your Homework Assignments Going? Page 92

Are You Familiar with Your Textbook Supplements? Page 198

How Well Do You Know Your Textbook? Page 311

Are You Organized? Page 329

AreYou Satisfied withYour Performance on a Particular Quiz or Exam? Page 390

Tips for Studying for an Exam Page 525

Are You Getting All the Mathematics Help That You Need? 02

How Are You Doing? Page 457

Are You Preparing for Your Final Exam? Page 790

See the Preface at the beginning of this text for a more thorough explanation of thefeatures of this text.

OBJECTIVE 4 � Getting help. If you have trouble completing assignments or under-standing the mathematics, get help as soon as you need it! This tip is presented as anobjective on its own because it is so important. In mathematics, usually the materialpresented in one section builds on your understanding of the previous section. Thismeans that if you don’t understand the concepts covered during a class period, there isa good chance that you will not understand the concepts covered during the next classperiod. If this happens to you, get help as soon as you can.

Where can you get help? Many suggestions have been made in the section onwhere to get help, and now it is up to you to do it.Try your instructor, a tutoring center,or a math lab, or you may want to form a study group with fellow classmates. If you dodecide to see your instructor or go to a tutoring center, make sure that you have a neatnotebook and are ready with your questions.

OBJECTIVE 5 � Preparing for and taking an exam. Make sure that you allow yourselfplenty of time to prepare for a test. If you think that you are a little “math anxious,” itmay be that you are not preparing for a test in a way that will ensure success. The waythat you prepare for a test in mathematics is important. To prepare for a test,

1. Review your previous homework assignments. You may also want to rework someof them.

2. Review any notes from class and section-level quizzes you have taken. (If this is afinal exam, also review chapter tests you have taken.)

3. Review concepts and definitions by reading the Highlights at the end of eachchapter.

4. Practice working out exercises by completing the Chapter Review found at the endof each chapter. (If this is a final exam, go through a Cumulative Review. There is

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Section 1.1 Tips for Success in Mathematics 5

one found at the end of each chapter except Chapter 1. Choose the review foundat the end of the latest chapter that you have covered in your course.) Don’t stophere!

5. It is important that you place yourself in conditions similar to test conditions tofind out how you will perform. In other words, as soon as you feel that you knowthe material, get a few blank sheets of paper and take a sample test. There is aChapter Test available at the end of each chapter. During this sample test, do notuse your notes or your textbook. Once you complete the Chapter Test, check youranswers in the back of the book. If any answer is incorrect, there is a CD availablewith each exercise of each chapter test worked. Use this CD or your instructor tocorrect your sample test. Your instructor may also provide you with a review sheet.If you are not satisfied with the results, study the areas that you are weak in andtry again.

6. Get a good night’s sleep before the exam.

7. On the day of the actual test, allow yourself plenty of time to arrive at where youwill be taking your exam.

When taking your test,

1. Read the directions on the test carefully.

2. Read each problem carefully as you take the test. Make sure that you answer thequestion asked.

3. Pace yourself by first completing the problems you are most confident with. Thenwork toward the problems you are least confident with. Watch your time so you donot spend too much time on one particular problem.

4. If you have time, check your work and answers.

5. Do not turn your test in early. If you have extra time, spend it double-checkingyour work.

OBJECTIVE 6 � Managing your time. As a college student, you know the demands thatclasses, homework, work, and family place on your time. Some days you probablywonder how you’ll ever get everything done. One key to managing your time isdeveloping a schedule. Here are some hints for making a schedule:

1. Make a list of all of your weekly commitments for the term. Include classes, work,regular meetings, extracurricular activities, etc. You may also find it helpful to listsuch things as laundry, regular workouts, grocery shopping, etc.

2. Next, estimate the time needed for each item on the list. Also make a note of howoften you will need to do each item. Don’t forget to include time estimates forreading, studying, and homework you do outside of your classes. You may want toask your instructor for help estimating the time needed.

3. In the following exercise set, you are asked to block out a typical week on theschedule grid given. Start with items with fixed time slots like classes and work.

4. Next, include the items on your list with flexible time slots. Think carefully abouthow best to schedule some items such as study time.

5. Don’t fill up every time slot on the schedule. Remember that you need to allowtime for eating, sleeping, and relaxing! You should also allow a little extra time incase some items take longer than planned.

6. If you find that your weekly schedule is too full for you to handle, you may need tomake some changes in your workload, classload, or in other areas of your life. Youmay want to talk to your advisor, manager or supervisor at work, or someone inyour college’s academic counseling center for help with such decisions.

Note: Don’t forget in this chapter we begin a feature called Study Skills Builder. Thepurpose of this feature is to remind you of some of the information given in this sectionand to further expand on some topics in this section.

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Monday Tuesday Wednesday Thursday Friday Saturday Sunday

4:00 A.M.

5:00 A.M.

6:00 A.M.

7:00 A.M.

8:00 A.M.

9:00 A.M.

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11:00 A.M.

12:00 P.M.

1:00 P.M.

2:00 P.M.

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Midnight

1:00 A.M.

2:00 A.M.

3:00 A.M.

1. What is your instructor’s name?

2. What are your instructor’s office location and office hours?

3. What is the best way to contact your instructor?

4. What does the icon mean?



7. Where are answers located in this text?

8. What Exercise Set answers are available to you in the answers section?

9. What Chapter Review, Chapter Test, and Cumulative Reviewanswers are available to you in the answer section?

10. Search after the solution of any worked example in this text.What are Practice exercises?

11. When might be the best time to work a Practice exercise?

12. Where are the answers to Practice exercises?

13. List any similarities between a Practice exercise and theworked example before it.

14. Go to the Highlights section at the end of this chapter. De-scribe how this section may be helpful to you when preparingfor a test.

15. Do you have the name and contact information of at leastone other student in class?

16. Will your instructor allow you to use a calculator in this class?

17. Are videotapes, CDs, and/or tutorial software available toyou? If so, where?

18. Is there a tutoring service available? If so, what are its hours?

19. Have you attempted this course before? If so, write downways that you might improve your chances of success duringthis second attempt.

20. List some steps that you can take if you begin having troubleunderstanding the material or completing an assignment.

21. Read or reread objective 6 and fill out the schedule grid below.

22. Study your filled-out grid from Exercise 21. Decide whetheryou have the time necessary to successfully complete thiscourse and any other courses you may be registered for.

1.1 EXERCISE SET

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Section 1.2 Symbols and Sets of Numbers 7

OBJECTIVES

1 Use a number line to ordernumbers.

2 Translate sentences intomathematical statements.

3 Identify natural numbers, wholenumbers, integers, rationalnumbers, irrational numbers,and real numbers.

4 Find the absolute value of a realnumber.

1.2 SYMBOLS AND SETS OF NUMBERS

OBJECTIVE 1 � Using a number line to order numbers. We begin with a review of theset of natural numbers and the set of whole numbers and how we use symbols tocompare these numbers. A set is a collection of objects, each of which is called amember or element of the set.A pair of brace symbols encloses the list of elementsand is translated as “the set of” or “the set containing.”

5 6

0 1 2 3 4 5

The three dots (an ellipsis) at the end of the list of elements of a set means that the listcontinues in the same manner indefinitely.

These numbers can be pictured on a number line. We will use number lines oftento help us visualize distance and relationships between numbers. Visualizing mathe-matical concepts is an important skill and tool, and later we will develop and exploreother visualizing tools.

To draw a number line, first draw a line. Choose a point on the line and label it 0.To the right of 0, label any other point 1. Being careful to use the same distance as from0 to 1, mark off equally spaced distances. Label these points 2, 3, 4, 5, and so on. Sincethe whole numbers continue indefinitely, it is not possible to show every whole numberon this number line. The arrow at the right end of the line indicates that the patterncontinues indefinitely.

Picturing whole numbers on a number line helps us to see the order of the num-bers. Symbols can be used to describe concisely in writing the order that we see.

These symbols may be used to form a mathematical statement. The statement might betrue or it might be false. The two statements below are both true.

If two numbers are not equal, then one number is larger than the other. Themeans “is greater than.” The “is less than.” For example,

On a number line, we see that a number to the right of another number is larger.Similarly, a number to the left of another number is smaller. For example, 3 is to the leftof 5 on a number line, which means that 3 is less than 5, or Similarly, 2 is to theright of 0 on a number line, which means 2 is greater than 0, or Since 0 is to theleft of 2, we can also say that 0 is less than 2, or

The symbols and are called inequality symbols.7Z , 6 ,0 6 2.

2 7 0.3 6 5.

3 6 5 states that “three is less than five”

2 7 0 states that “two is greater than zero”

symbol 6 meanssymbol 7

2 Z 6 states that “two is not equal to six”

2 = 2 states that “two is equal to two”

The symbol Z means “is not equal to.”

The equal symbol = means “is equal to.”

3 � 5

0 1 2 3 4 5

0

2 � 0 or 0 � 2

1 2 3 4 5

◗ Helpful HintNotice that has exactly the same meaning as Switching the order of the numbersand reversing the “direction of the inequality symbol” does not change the meaning of thestatement.

Also notice that, when the statement is true, the inequality arrow points to the smaller number.

5 7 3 has the same meaning as 3 6 5.

0 6 2.2 7 0

Natural NumbersThe set of natural numbers is {1, 2, 3, 4, 5, 6, Á }.

Whole NumbersThe set of whole numbers is {0, 1, 2, 3, 4, Á }.

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Two other symbols are used to compare numbers. The symbol means “is lessthan or equal to.” The symbol means “is greater than or equal to.” For example,

This statement is true since is true. If either or is true, thenis true.

This statement is true since is true. If either or is true, then istrue.

The statement is false since neither nor is true. The sym-bols and are also called inequality symbols.Ú…

6 = 106 7 106 Ú 10

3 Ú 33 = 33 7 33 = 3

3 Ú 3 states that “three is greater than or equal to three”

7 … 107 = 107 6 107 6 10

7 … 10 states that “seven is less than or equal to ten”

Ú

…

EXAMPLE 1 Insert the space between each pair of numbers tomake each statement true.

a. 2 3 b. 7 4 c. 72 27

Solution

a. since 2 is to the left of 3 on the number line.

b. since 7 is to the right of 4 on the number line.

c. since 72 is to the right of 27 on the number line.

Insert the space between each pair of numbers to make eachstatement true.

a. 5 8 b. 6 4 c. 16 82

6 , 7 , or = in1PRACTICE

72 7 27

7 7 4

2 6 3

6 , 7 , or = in

EXAMPLE 2 Tell whether each statement is true or false.

a. b. c. d.

Solution

a. True, since is true.

b. True, since is true.

c. False, since neither nor is true.

d. True, since is true.

Tell whether each statement is true or false.

a. b. c. d. 4 … 1425 … 253 Ú 89 Ú 3

2PRACTICE

23 7 0

23 = 023 6 0

8 = 8

8 = 8

23 Ú 023 … 08 … 88 Ú 8

OBJECTIVE 2 � Translating sentences. Now, let’s use the symbols discussed above totranslate sentences into mathematical statements.

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Translate each sentence into a mathematical statement.

a. Three is less than eight.

b. Fifteen is greater than or equal to nine.

c. Six is not equal to seven.

3PRACTICE

4Z3TTT

178TTT

11…9TTT


Solution

a. elevenis less thanor equal to

nine

0 4�4 5�5 2�2 3�3 1�1

0 4�4 5�5 2�2 3�3 1�1

negative integers positive integers

EXAMPLE 3 Translate each sentence into a mathematical statement.

a. Nine is less than or equal to eleven.

b. Eight is greater than one.

c. Three is not equal to four.

b. oneis greater

thaneight

c. fouris not

equal tothree

OBJECTIVE 3 � Identifying common sets of numbers. Whole numbers are notsufficient to describe many situations in the real world. For example, quantities smallerthan zero must sometimes be represented, such as temperatures less than 0 degrees.

We can picture numbers less than zero on a number line as follows:

Numbers less than 0 are to the left of 0 and are labeled and so on.A sign, such as the one in tells us that the number is to the left of 0 on a num-ber line. In words, is read “negative one.” A sign or no sign tells us that a num-ber lies to the right of 0 on the number line. For example, 3 and both meanpositive three.

The numbers we have pictured are called the set of integers. Integers to the leftof 0 are called negative integers; integers to the right of 0 are called positive integers.The integer 0 is neither positive nor negative.

+3+-1

-1,-

-1, -2, -3,

Notice the ellipses (three dots) to the left and to the right of the list for the integers.This indicates that the positive integers and the negative integers continue indefinitely.

IntegersThe set of integers is 5Á , -3, -2, -1, 0, 1, 2, 3, Á 6.

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1 unit

�2 units

irrationalnumber

0�2�3 1 2 3 4 5�1

�h �q q # t


Use an integer to express the number in the following: Fred overdrew his check-ing account and now owes his bank 52 dollars.

4PRACTICE

EXAMPLE 4 Use an integer to express the number in the following. “Pole ofInaccessibility, Antarctica, is the coldest location in the world, with an average annualtemperature of 72 degrees below zero.” (Source: The Guinness Book of Records)

Solution The integer represents 72 degrees below zero.-72

A problem with integers in real-life settings arises when quantities are smallerthan some integer but greater than the next smallest integer. On a number line, thesequantities may be visualized by points between integers. Some of these quantities be-tween integers can be represented as a quotient of integers. For example,

The point on a number line halfway between 0 and 1 can be represented by aquotient of integers.

The point on a number line halfway between 0 and can be represented by Other quotients of integers and their graphs are shown.

These numbers, each of which can be represented as a quotient of integers, are exam-ples of rational numbers. It’s not possible to list the set of rational numbers using thenotation that we have been using. For this reason, we will use a different notation.

-

12

.-1

12

,

We read this set as “the set of all numbers such that a and b are integers and b is not

equal to 0.” Notice that every integer is also a rational number since each integer can beexpressed as a quotient of integers. For example, the integer 5 is also a rational number

since

The number line also contains points that cannot be expressed as quotients ofintegers. These numbers are called irrational numbers because they cannot be repre-sented by rational numbers. For example, and are irrational numbers.p12

5 =

51

.

a

b

Both rational numbers and irrational numbers can be written as decimal numbers. Thedecimal equivalent of a rational number will either terminate or repeat in a pattern.For example, upon dividing we find that

23

= 0.66666Á 1decimal number repeats in a pattern2 34

= 0.75 1decimal number terminates or ends2 and

Rational Numbers

e a

b` a and b are integers and b Z 0 f

Irrational NumbersThe set of irrational numbers is

{Nonrational numbers that correspond to points on the number line}.

That is, an irrational number is a number that cannot be expressed as a quotient ofintegers.

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◗ Helpful HintFrom our previous definitions, we have that

Every real number is either

a rational number or an irrational number

�2 2 3 4 50�4�5 1�1�3

Negative numbersZero

Positive numbers

Natural Numbers orPositive Integers

Zero

0

Whole Numbers

0, 2, 56, 198

Negative Integers

�20, �13, �1

Noninteger RationalNumbers

Real Numbers

�18, �q, 0, �2, p, 4710

Rational Numbers

�35, �√, 0, 5,

Irrational Numbers

p, �7

1, 16, 170

Integers

�10, 0, 8

2711

3013�}, Ï,

On the following number line, we see that real numbers can be positive, negative,or 0. Numbers to the left of 0 are called negative numbers; numbers to the right of 0 arecalled positive numbers. Positive and negative numbers are also called signed numbers.

Several different sets of numbers have been discussed in this section. The follow-ing diagram shows the relationships among these sets of real numbers.

Common Sets of Numbers

The decimal representation of an irrational number will neither terminate nor repeat.For example, the decimal representations of irrational numbers and are

(For further review of decimals, see the Appendix.)Combining the natural numbers with the irrational numbers gives the set of real

numbers. One and only one point on a number line corresponds to each real number.

1decimal number does not terminate or repeat in a pattern2 p = 3.141592653 Á

1decimal number does not terminate or repeat in a pattern2 22 = 1.414213562 Á

p12

Real NumbersThe set of real numbers is

{All numbers that correspond to points on the number line}

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0 1 2�1�2

�1 � 0


We can now extend the meaning and use of inequality symbols such as and to apply to all real numbers.

76

Order Property for Real NumbersGiven any two real numbers a and b, if a is to the left of b on a number line.Similarly, if a is to the right of b on a number line.a 7 b

a 6 b

EXAMPLE 5 Given the set list the numbersin this set that belong to the set of:

a. Natural numbers b. Whole numbers

c. Integers d. Rational numbers

e. Irrational numbers f. Real numbers

Solution

a. The natural numbers are 11 and 112.

b. The whole numbers are 0, 11, and 112.

c. The integers are 0, 11, and 112.

d. Recall that integers are rational numbers also. The rational numbers are

11, and 112.

e. The irrational number is

f. The real numbers are all numbers in the given set.

Given the set , list the numbers in this

set that belong to the set of:




e25, 73

, -15, -34

, 25, -3.7, 8.8, -99 f5PRACTICE

12.

-1.5, 0, 14

,

-3, -2,

-3, -2,

e-2, 0, 14

, -1.5, 112, -3, 11, 22f ,

a b

a � b

b a

a � b

EXAMPLE 6 Insert the appropriate space to make eachstatement true.

a. b. c.

Solution

a. since is to the left of 0 on a number line.-1-1 6 0

-5 -67 142

-1 0

6 , 7 , or = in

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0 1 2 3�1�2�3

3 units 3 units

�5 � �6

�7 �6 �5 �4 �3


b. since simplifies to 7.

c. since is to the right of on the number line.-6-5-5 7 -6

142

7 =

142

Insert the appropriate space to make each statement true.

a. 0 3 b. 15 c. 3124

-5


For example, and since both 3 and are a distance of 3 units from 0on a number line.

-3ƒ -3 ƒ = 3ƒ 3 ƒ = 3

OBJECTIVE 4 � Finding the absolute value of a real number. A number line not onlygives us a picture of the real numbers, it also helps us visualize the distance betweennumbers.The distance between a real number a and 0 is given a special name called theabsolute value of a. “The absolute value of a” is written in symbols as ƒ a ƒ .

◗ Helpful HintSince is a distance, is always either positive or 0, never negative. That is, for any realnumber a, ƒ a ƒ Ú 0.

ƒ a ƒƒ a ƒ

EXAMPLE 7 Find the absolute value of each number.

a. b. c. d. e.

Solution

a. since 4 is 4 units from 0 on a number line.

b. since is 5 units from 0 on a number line.

c. since 0 is 0 units from 0 on a number line.

d. since is unit from 0 on a number line.

e. since 5.6 is 5.6 units from 0 on a number line.

Find the absolute value of each number.

a. b. c. d. e. ƒ 23 ƒ` 511`|-2.5||9||-8|

7PRACTICE

ƒ 5.6 ƒ = 5.6

12

-

12

` - 12` =

12

ƒ 0 ƒ = 0

-5ƒ -5 ƒ = 5

ƒ 4 ƒ = 4

ƒ 5.6 ƒ` - 12`ƒ 0 ƒƒ -5 ƒƒ 4 ƒ

Absolute ValueThe absolute value of a real number a, denoted by is the distance between a and0 on a number line.

ƒ a ƒ ,

M01_MART7317_04_SE_C01.QXD 12/3/07 3:44 PM


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212� F

100� C0� C

32� F


Insert the appropriate space to make the statementtrue. See Example 1.

1. 7 3 2. 9 15

3. 6.26 6.26 4. 2.13 1.13

5. 0 7 6. 20 0

7. 2 8.

9. The freezing point of water is 32° Fahrenheit. The boilingpoint of water is 212° Fahrenheit. Write an inequality state-ment using comparing the numbers 32 and 212.6 or 7

-6-4-2

6 , 7 , or = in 10. The freezing point of water is 0° Celsius. The boiling point ofwater is 100° Celsius. Write an inequality statement using

comparing the numbers 0 and 100.

11. The spring 2007 tuition and fees for a Texas resident under-graduate student at University of Texas at El Paso were ap-proximately $2631 for a 15-credit load. At the same time thetuition and fees for a Florida resident attending University ofFlorida were approximately $2456. Write an inequality state-ment using comparing the numbers 2631 and 2456.(Source: UTEP and UF)

12. The average salary in the San Jose, California, area for a chem-ical engineer is $67,841. The average salary for a database ad-ministrator in the same area is $75,657. Write an inequalitystatement using or comparing the numbers 67,841 and75,657. (Source: The Wall Street Journal)

Are the following statements true or false? See Example 2.

13. 14. 15.

16. 17. 18.

19. 20. 4 6 77 7 0

8 # 8 … 8 # 73 + 8 Ú 318217 7 16

10 7 114 Ú 711 … 11

76

6 or 7

6 or 7

EXAMPLE 8 Insert the appropriate space to make eachstatement true.

a. b. c. d. e.

Solution

a. since and b. since

c. since d. since

e. since

Insert the appropriate space to make each statement true.

a. b. 0 c. d. e. ƒ -4 ƒƒ 0 ƒƒ 2 ƒƒ 3 ƒƒ -11 ƒƒ -7 ƒƒ -3 ƒƒ -8 ƒƒ 8 ƒ


7 7 6.ƒ -7 ƒ 7 ƒ 6 ƒ

5 6 6.ƒ 5 ƒ 6 ƒ 6 ƒ3 7 2.ƒ -3 ƒ 7 ƒ -2 ƒ

5 = 5.ƒ -5 ƒ = 50 6 2.ƒ 0 ƒ = 0ƒ 0 ƒ 6 2

ƒ -7 ƒ ƒ 6 ƒƒ 5 ƒ ƒ 6 ƒƒ -3 ƒ ƒ -2 ƒƒ -5 ƒ 5ƒ 0 ƒ 2

6 , 7 , or = in

VOCABULARY & READINESS CHECK

Use the choices below to fill in each blank.

real natural whole irrational

inequality integers rational

1. The numbers are .


3. The symbols , , and are called symbols.

4. The are .

5. The numbers are {all numbers that correspond to points on the number line}.


7. The numbers are {nonrational numbers that correspond to points on the number line}.

8. The distance between a number and 0 on a number line is . b

e a

b` a and b are integers, b Z 0 f

{ Á , -3, -2, -1, 0, 1, 2, 3, Á }

7…Z

{1, 2, 3, 4, 5, Á } {0, 1, 2, 3, 4, Á }

ƒ b ƒ

1.2 EXERCISE SET

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21. An angle measuring 30° is shown and an angle measuring 45°is shown. Use the inequality symbol to write a state-ment comparing the numbers 30 and 45.

… or Ú

35. In which year(s) was the number of visitors the greatest?

36. What was the greatest number of visitors shown?

37. In what year(s) was the number of visitors greater than 280million?

38. In what year(s) was the number of visitors less than 270million?

39. Write an inequality statement comparing the number ofannual visitors in 2001 and 2006.

40. Do you notice any trends shown by this bar graph?

Tell which set or sets each number belongs to: natural numbers,whole numbers, integers, rational numbers, irrational numbers, andreal numbers. See Example 5.

41. 0 42.

43. 44.

45. 6 46. 5

47. 48.

49. 50.

Tell whether each statement is true or false.

51. Every rational number is also an integer.

52. Every negative number is also a rational number.

53. Every natural number is positive.

54. Every rational number is also a real number.

55. 0 is a real number.

56. Every real number is also a rational number.

57. Every whole number is an integer.

58. is an integer.

59. A number can be both rational and irrational.

60. Every whole number is positive.

Insert the appropriate space to make a true state-ment. See Examples 6 through 8.

61. 62.

63. 32 5.2 64. 7.1

65. 66.

67. 68.

69. 70. 0

71. 72.

73. 74.

75. 76.242

ƒ -12 ƒƒ -8 ƒƒ 0 ƒ

ƒ -50 ƒ-500ƒ -3 ƒƒ -2 ƒ

` - 25`` 2

5`ƒ 1 ƒƒ -1 ƒ

ƒ 0 ƒ-4ƒ -5 ƒ

-200ƒ -20 ƒ-50-51

123

82

243

183

-7

-20-200-100-10

6 , 7 , or = in

12

-1 59

- 25

2323

-

12

-2

14

Sum is 180�

Sumis 360�

45�30�

22. The sum of the measures of the angles of a triangle is 180°.The sum of the measures of the angles of a parallelogram is360°. Use the inequality symbol to write a statementcomparing the numbers 360 and 180.

… or Ú

Write each sentence as a mathematical statement. See Example 3.

23. Eight is less than twelve.

24. Fifteen is greater than five.

25. Five is greater than or equal to four.

26. Negative ten is less than or equal to thirty-seven.

27. Fifteen is not equal to negative two.

28. Negative seven is not equal to seven.

Use integers to represent the values in each statement.See Example 4.

29. Driskill Mountain, in Louisiana, has an altitude of 535 feet.New Orleans, Louisiana, lies 8 feet below sea level. (Source:U.S. Geological Survey)

30. During a Green Bay Packers football game, the team gained23 yards and then lost 12 yards on consecutive plays.

31. From 2005 to 2010, the population of Washington, D.C., is ex-pected to decrease by approximately 21,350. (Source: U.S.Census Bureau)

32. From 2005 to 2010, the population of Alaska is expectedto grow by about 33,000 people. (Source: U.S. Census Bureau)

33. Aaron Miller deposited $350 in his savings account. He laterwithdrew $126.

34. Aris Peña was deep-sea diving. During her dive, she ascen-ded 30 feet and later descended 50 feet.

The graph below is called a bar graph.This particular graph showsthe annual numbers of recreational visitors to U.S. National Parks.Each bar represents a different year, and the height of the bar rep-resents the number of visitors (in millions) in that year.

Total Annual Recreational Visitors toU.S. National Park System

1997 1998 1999 2000 2001 2002 2003 2004 2005 2006Year

250

260

270

280

290

300

Num

ber

of V

isit

ors

(in

mill

ions

)

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CONCEPT EXTENSIONS

The apparent magnitude of a star is the measure of its brightness asseen by someone on Earth. The smaller the apparent magnitude,the brighter the star. Use the apparent magnitudes in the table toanswer Exercises 77 through 82.

77. The apparent magnitude of the sun is The appar-ent magnitude of the star Arcturus is Write aninequality statement comparing the numbers and

78. The apparent magnitude of Antares is 0.96. The apparentmagnitude of Spica is 0.98. Write an inequality statementcomparing the numbers 0.96 and 0.98.

79. Which is brighter, the sun or Arcturus?

80. Which is dimmer, Antares or Spica?

81. Which star listed is the brightest?

82. Which star listed is the dimmest?

Rewrite the following inequalities so that the inequality symbolpoints in the opposite direction and the resulting statement has thesame meaning as the given one.

83. 84.

85. 86.

87. 88.

89. In your own words, explain how to find the absolute value ofa number.

90. Give an example of a real-life situation that can be describedwith integers but not with whole numbers.

-4 6 -2-10 7 -12

5 7 30 6 6

-13 … 1325 Ú 20

-26.7.-0.04

-0.04.-26.7.

Apparent ApparentStar Magnitude Star Magnitude

Arcturus Spica 0.98Sirius Rigel 0.12Vega 0.03 Regulus 1.35Antares 0.96 CanopusSun Hadar 0.61

(Source: Norton’s 2000: Star Atlas and Reference Handbook,18th ed., Longman Group, UK, 1989)

-26.7-0.72

-1.46-0.04

OBJECTIVES

1 Write fractions in simplest form.

2 Multiply and divide fractions.

3 Add and subtract fractions.

1.3 FRACTIONS

OBJECTIVE 1 � Writing fractions in simplest form. A quotient of two numbers such as

is called a fraction. In the fraction the top number, 2, is called the numerator and

the bottom number, 9, is called the denominator.A fraction may be used to refer to part of a whole. For example, of the circle to

the left is shaded. The denominator 9 tells us how many equal parts the whole circle isdivided into and the numerator 2 tells us how many equal parts are shaded.

To simplify fractions, we can factor the numerator and the denominator. In thestatement 3 and 5 are called factors and 15 is the product. (The raised dotsymbol indicates multiplication.)

To factor 15 means to write it as a product.The number 15 can be factored as or as

A fraction is said to be simplified or in lowest terms when the numerator and the

denominator have no factors in common other than 1. For example, the fraction is in lowest terms since 5 and 11 have no common factors other than 1.

To help us simplify fractions, we write the numerator and the denominator as aproduct of prime numbers.

511

1 # 15.3 # 5

3 # 5 = 15c c c

factor factor product

3 # 5 = 15,

29

29

,29

fl of the circleis shaded.

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Section 1.3 Fractions 17

Prime NumberA prime number is a natural number, other than 1, whose only factors are 1 and itself.The first few prime numbers are

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, and so on.

A natural number, other than 1, that is not a prime number is called a compositenumber. Every composite number can be written as a product of prime numbers. Wecall this product of prime numbers the prime factorization of the composite number.

EXAMPLE 1 Write each of the following numbers as a product of primes.

a. 40 b. 63

Solution

a. First, write 40 as the product of any two whole numbers, other than 1.

Next, factor each of these numbers. Continue this process until all of the factors areprime numbers.

40 = 4 # 10

63=

= 3 3 7

9 7#

# #

40=

= 2 2

4 #

# 2 5

10

##

All the factors are now prime numbers. Then 40 written as a product of primes is

b.

40 = 2 # 2 # 2 # 5

Write each of the following numbers as a product of primes.

a. 36 b. 75

1PRACTICE

To use prime factors to write a fraction in lowest terms, apply the fundamentalprinciple of fractions.

Fundamental Principle of Fractions

If is a fraction and c is a nonzero real number, thena # c

b # c=

a

b

a

b

To understand why this is true, we use the fact that since c is not zero, then

We will call this process dividing out the common factor of c.

a # c

b # c=

a

b#cc

=

a

b# 1 =

a

b

cc

= 1.

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OBJECTIVE 2 � Multiplying and dividing fractions. To multiply two fractions, multiplynumerator times numerator to obtain the numerator of the product; multiply denomi-nator times denominator to obtain the denominator of the product.

Answers to Concept Check:

answers may vary

Concept CheckExplain the error in the following steps.

a. b.67

=

5 + 15 + 2

=

12

1555

=

1 55 5

=

15

Multiplying Fractions

a

b#

c

d=

a # c

b # d, if b Z 0 and d Z 0

EXAMPLE 2 Write each fraction in lowest terms.

a. b. c.

Solution

a. Write the numerator and the denominator as products of primes; then apply thefundamental principle to the common factor 7.

b.

There are no common factors other than 1, so is already in lowest terms.

c.

Write each fraction in lowest terms.

a. b. c.725

6412

6372

2PRACTICE

8820

=

2 # 2 # 2 # 112 # 2 # 5

=

225

1127

1127

=

113 # 3 # 3

4249

=

2 # 3 # 77 # 7

=

2 # 37

=

67

8820

1127

4249

EXAMPLE 3 Multiply and Write the product in lowest terms.

Solution Multiply numerators.

Multiply denominators.

Next, simplify the product by dividing the numerator and the denominator by anycommon factors.

Multiply and . Write the product in lowest terms.79

38

3PRACTICE

=

239

=

2 # 53 # 5 # 13

215

#513

=

2 # 515 # 13

513

.215

Before dividing fractions, we first define reciprocals. Two fractions are reciprocals

of each other if their product is 1. For example and are reciprocals since

Also, the reciprocal of 5 is since 5 #15

=

51

#15

= 1.15

23

#32

= 1.32

23

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Dividing Fractions

a

b,

c

d=

a

b#dc

, if b Z 0, d Z 0, and c Z 0

To divide fractions, multiply the first fraction by the reciprocal of the secondfraction.

EXAMPLE 4 Divide. Write all quotients in lowest terms.

a. b. c.

Solution

a.

b.

c.

Divide. Write all quotients in lowest terms.

a. b. c.76

,

715

512

, 1534

,

49

4PRACTICE

38

,

310

=

38

#103

=

3 # 2 # 52 # 2 # 2 # 3

=

54

710

, 14 =

710

,

141

=

710

#114

=

7 # 12 # 5 # 2 # 7

=

120

.

45

,

516

=

45

#165

=

4 # 165 # 5

=

6425

38

,

310

710

, 1445

,

516

OBJECTIVE 3 � Adding and subtracting fractions. To add or subtract fractions with thesame denominator, combine numerators and place the sum or difference over thecommon denominator.

Adding and Subtracting Fractions with the Same Denominator

a

b-

c

b=

a - c

b, if b Z 0

a

b+

c

b=

a + c

b, if b Z 0

EXAMPLE 5 Add or subtract as indicated. Write each result in lowest terms.

a. b. c. d.

Solution

a. b.

c. d.53

-

13

=

5 - 13

=

43

97

-

27

=

9 - 27

=

77

= 1

310

+

210

=

3 + 210

=

510

=

52 # 5

=

12

27

+

47

=

2 + 47

=

67

53

-

13

97

-

27

310

+

210

27

+

47

Add or subtract as indicated. Write each result in lowest terms.

a. b. c. d.512

+

112

35

+

15

85

-

25

85

-

35

5PRACTICE

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Multiply by

2082

5=

2#

45 4

44

=

2 # 45 # 4

=

or 1.


Whole

�34

1216

To add or subtract fractions without the same denominator, first write the frac-tions as equivalent fractions with a common denominator. Equivalent fractions are

fractions that represent the same quantity. For example, and are equivalent

fractions since they represent the same portion of a whole, as the diagram shows.

Count the larger squares and the shaded portion is Count the smaller squares and

the shaded portion is Thus,

We can write equivalent fractions by multiplying a given fraction by 1, as shownin the next example. Multiplying a fraction by 1 does not change the value of thefraction.

34

=

1216

.1216

.

34

.

1216

34

EXAMPLE 6 Write as an equivalent fraction with a denominator of 20.

Solution Since multiply the fraction by Multiplying by does notchange the value of the fraction.

44

= 144

.5 # 4 = 20,

25

Write as an equivalent fraction with a denominator of 21.23

6PRACTICE

EXAMPLE 7 Add or subtract as indicated. Write each answer in lowest terms.

a. b. c.

Solution

a. Fractions must have a common denominator before they can be added or subtracted.Since 20 is the smallest number that both 5 and 4 divide into evenly, 20 is the leastcommon denominator. Write both fractions as equivalent fractions with denomina-tors of 20. Since

then

b. The least common denominator for denominators 2, 22, and 11 is 22. First, writeeach fraction as an equivalent fraction with a denominator of 22. Then add or sub-tract from left to right.

Then

12

+

1722

-

211

=

1122

+

1722

-

422

=

2422

=

1211

12

=

12

#1111

=

1122

, 1722

=

1722

, and 211

=

211

#22

=

422

25

+

14

=

820

+

520

=

1320

25

#44

=

2 # 45 # 4

=

820 and 1

4#55

=

1 # 54 # 5

=

520

3 16

- 1 1112

12

+

1722

-

211

25

+

14

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21412

11112

1312

or 114

3212

-1 -11112

Need toborrow

=

=

316

-11112

=

=

2+1 212


c. To find let’s use a vertical format.3 16

- 1 1112

,

Add or subtract as indicated. Write answers in lowest terms.

a. b. c.13

+

2930

-

45

9 113

- 5 12

511

+

17

7PRACTICE


Use the choices below to fill in each blank. Some choices may be used more than once.

simplified reciprocals equivalent denominator

product factors fraction numerator

1. A quotient of two numbers, such as , is called a .

2. In the fraction , the number 3 is called the and the number 11 is called the .

3. To factor a number means to write it as a .

4. A fraction is said to be when the numerator and the denominator have no common factors other than 1.

5. In , the numbers 7 and 3 are called and the number 21 is called the .

6. The fractions and are called .

7. Fractions that represent the same quantity are called fractions.

Represent the shaded part of each geometric figure by a fraction

92

29

7 # 3 = 21

311

58

8. 9. 10. 11.

Write each number as a product of primes. See Example 1.

1. 33 2. 60 3. 98

4. 27 5. 20 6. 56

7. 75 8. 32 9. 45

10. 24

Write the fraction in lowest terms. See Example 2.

11. 12. 13.

14. 15. 16.

17. 18.4245

1830

59

37

1520

1015

36

24

1.3 EXERCISE SET

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? E

Å

?fl

Z

q

?Z

a512

?~

≈

?

211

311

?

510

310

q meter

1~ meters1112 mile

E mile


Multiply or divide as indicated. Write the answer in lowest terms.See Examples 3 and 4.

19. 20. 21.

22. 23. 24.

25. 26. 27.

28. 29. 30.

The area of a plane figure is a measure of the amount of surface of

the figure. Find the area of each figure below. (The area of a rectan-

gle is the product of its length and width.The area of a triangle is

the product of its base and height.)

31. 32.

12

14

# 5 56

2 79

#13

335

#1063

710

#521

35

,

910

34

,

120

712

,

12

12

,

712

78

#321

23

#34

18

#35

12

#34

Each circle below represents a whole, or 1. Use subtraction to de-termine the unknown part of the circle.

55. 56.

Add or subtract as indicated. Write the answer in lowest terms. SeeExample 5.

33. 34.

35. 36.

37. 38.

39. 40.

Write each fraction as an equivalent fraction with the given de-nominator. See Example 6.

41. with a denominator of 30






Add or subtract as indicated. Write the answer in lowest terms. SeeExample 7.

47. 48.

49. 50.

51. 52.

53. 54. 2 -

38

125

- 1

710

-

815

522

-

533

5 29

- 3 16

2 1315

- 1 15

34

+

16

23

+

37

45

45

87

29

23

710

13132

+

35132

23105

+

4105

1835

-

1135

1721

-

1021

67

+

17

45

+

15

67

-

17

45

-

15

57. 58.

59. 60.

MIXED PRACTICE

Perform the following operations. Write answers in lowest terms.

61. 62. 63.

64. 65. 66.

67. 68. 69.

70. 71. 72.

73. 74. 75.

76. 77. 78.

79. 80.

81. 82.

The perimeter of a plane figure is the total distance around the fig-ure. Find the perimeter of each figure in Exercises 83 and 84.

83.

811

-

14

+

12

23

-

59

+

56

2 35

+ 4 710

1 12

+ 3 23

57132

-

13132

23105

-

2105

4 37

,

317

718

,

1436

3 ,

34

78

, 3 14

7 +

110

5 +

23

27

+

47

512

+

412

34

#712

34

,

712

23

,

35

23

#35

117

-

335

103

-

521

1135

+

335

1021

+

521

4Ωfeet

10q feet

4Ωfeet

15!feet

15!feet

5 feet

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84. 92. Engineering doctorates make up what fraction of all scienceand engineering doctorates awarded in the United States?

85. Yelena Isinbaeva currently holds the women’s pole vault

world record at meters.The men’s pole vault world record

is currently held by Sergei Bubka, at meters higher than

the women’s record. What is the current men’s pole vaultrecord? (Source: International Association of Athletics

Federations)

86. The Preakness, one of the horse races in the Triple Crown,

is a -mile race. The Belmont, another of the three races, is

of a mile longer than the Preakness. How long is the

Belmont? (Source: Sports Illustrated)

87. In your own words, explain how to add two fractions withdifferent denominators.

88. In your own words, explain how to multiply two fractions.

The following trail chart is given to visitors at the Lakeview ForestPreserve.

516

1316

1325

5150

89. How much longer is Red Falls Trail than Green Way Trail?

90. Find the total distance traveled by someone who hiked along all four trails.

93. What fraction of all science and engineering doctorates areawarded in the biological and agricultural sciences?

94. Social sciences and psychology doctorates together makeup what fraction of all science and engineering doctoratesawarded in the United States?

In 2006, Gap Inc. operated a total of 3054 stores worldwide. Thefollowing chart shows the store breakdown by brand. (Source:Gap Inc.)

95. What fraction of Gap-brand stores were Old Navy stores?Simplify this fraction.

96. What fraction of Gap-brand stores were either domestic orinternational Gap stores or Forth & Towne stores? Simplifythis fraction.

The area of a plane figure is a measure of the amount of surface of

the figure. Find the area of each figure. (The area of a triangle is

the product of its base and height. The area of a rectangle is theproduct of its length and width. Recall that area is measured insquare units.)

97.

12

12≈ feet

2≈ feet

9~feet

16qfeet

Trail Name Distance (miles)

Robin Path

Red Falls

Green Way

Autumn Walk 1 34

2 18

5 12

3 12

CONCEPT EXTENSIONS

The breakdown of science and engineering doctorate degreesawarded in the United States is summarized in the graph on thenext column, called a circle graph or a pie chart. Use the graph toanswer the questions. (Source: National Science Foundation)

91. What fraction of science and engineering doctorates areawarded in the physical sciences? W meter

meter311

√ foot

‡ foot

Science and Engineering Doctorates Awarded,by Field of Study

Biological/agriculturalsciences

Earth,atmospheric,

ocean sciences,3/100

Engineering,21/100

Physical sciences,7/50

Psychology,7/50

Social sciences,4/25

Mathematical/computer sciences,

7/100

Brand Number of Stores

Gap (Domestic) 1335

Gap (International) 256

Banana Republic 498

Old Navy 960

Forth & Towne 5

Total 3054

98.

M01_MART7317_04_SE_C01.QXD 12/17/07 11:35 AM


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Volume is (2 � 2 � 2)cubic centimeters.

2 cmexponent

base 23= 2 # 2 # 2=8

2 is a factor 3 times


OBJECTIVE 1 � Using exponents and the order of operations. Frequently in algebra,products occur that contain repeated multiplication of the same factor. For example,the volume of a cube whose sides each measure 2 centimeters is cubiccentimeters. We may use exponential notation to write such products in a morecompact form. For example,

The 2 in is called the base; it is the repeated factor. The 3 in is called theexponent and is the number of times the base is used as a factor. The expression iscalled an exponential expression.

232323

2 # 2 # 2 may be written as 23.

12 # 2 # 22OBJECTIVES

1 Define and use exponents andthe order of operations.

2 Evaluate algebraic expressions,given replacement values forvariables.

3 Determine whether a number isa solution of a given equation.

4 Translate phrases intoexpressions and sentences intoequations.

1.4 INTRODUCTION TO VARIABLE EXPRESSIONS AND EQUATIONS

EXAMPLE 1 Evaluate the following:

a. [read as “3 squared” or as “3 to the second power”]

b. [read as “5 cubed” or as “5 to the third power”]

c. [read as “2 to the fourth power”]

d. e.

Solution

a. b.

c. d.

e.

Evaluate:

a. b. c. d. e. a25b3

91a 110b2

5213

1PRACTICE

a37b2

= a37b a3

7b =

949

71= 724

= 2 # 2 # 2 # 2 = 16

53= 5 # 5 # 5 = 12532

= 3 # 3 = 9

a37b2

71

24

53

32

◗ Helpful Hintsince indicates repeated multiplication of the same factor.

23= 2 # 2 # 2 = 8, whereas 2 # 3 = 6.

2323Z 2 # 3

Using symbols for mathematical operations is a great convenience. However, the moreoperation symbols presented in an expression, the more careful we must be when per-forming the indicated operation. For example, in the expression do we addfirst or multiply first? To eliminate confusion, grouping symbols are used. Examples ofgrouping symbols are parentheses ( ), brackets [ ], braces and the fraction bar. Ifwe wish to be simplified by adding first, we enclose in parentheses.

If we wish to multiply first, may be enclosed in parentheses.

2 + 13 # 72 = 2 + 21 = 23

3 # 7

12 + 32 # 7 = 5 # 7 = 35

2 + 32 + 3 # 75 6,

2 + 3 # 7,

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Section 1.4 Introduction to Variable Expressions and Equations 25

Order of OperationsSimplify expressions using the order below. If grouping symbols such as parenthe-ses are present, simplify expressions within those first, starting with the innermostset. If fraction bars are present, simplify the numerator and the denominatorseparately.

1. Evaluate exponential expressions.

2. Perform multiplications or divisions in order from left to right.

3. Perform additions or subtractions in order from left to right.

To eliminate confusion when no grouping symbols are present, use the followingagreed upon order of operations.

Now simplify There are no grouping symbols and no exponents, so we multi-ply and then add.

Multiply.

Add. = 23

2 + 3 # 7 = 2 + 21

2 + 3 # 7.

EXAMPLE 2 Simplify each expression.

a. b. c. d. e.

Solution

a. Evaluate first.

Next divide, then add.

Divide.

Add.

b. First, simplify the numerator and the denominator separately.

Simplify numerator anddenominator separately.

Simplify.

c. Multiply and divide from left to right. Then subtract.

Subtract.

d. In this example, only the 4 is squared. The factor of 3 is not part of the base becauseno grouping symbol includes it as part of the base.

Evaluate the exponential expression.

Multiply. = 48

3 # 42= 3 # 16

= 29

3 # 10 - 7 , 7 = 30 - 1

= 2

=

3015

2112 + 32

ƒ -15 ƒ

=

2115215

= 27

= 2 + 25

6 , 3 + 52= 6 , 3 + 25

52

32

#12

-

12

3 # 423 # 10 - 7 , 72112 + 32

ƒ -15 ƒ

6 , 3 + 52

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e. The order of operations applies to operations with fractions in exactly the same wayas it applies to operations with whole numbers.

Multiply.

The least common denominator is 4.

Subtract.

Simplify each expression

a. b. c.

d. e.74

#14

-

14

9(14 - 6)

ƒ -2 ƒ

a23b2

# |-8|43, 8 + 36 + 3 # 9

2PRACTICE

=

14

=

34

-

24

32

#12

-

12

=

34

-

12

◗ Helpful HintBe careful when evaluating an exponential expression. In the exponent 2 applies onlyto the base 4. In we multiply first because of parentheses, so the exponent 2 applies tothe product

13 # 422 = 11222 = 1443 # 42= 3 # 16 = 48

3 # 4.13 # 422,

3 # 42,

Expressions that include many grouping symbols can be confusing. When simpli-fying these expressions, keep in mind that grouping symbols separate the expressioninto distinct parts. Each is then simplified separately.

EXAMPLE 3 Simplify

Solution The fraction bar serves as a grouping symbol and separates the numeratorand denominator. Simplify each separately. Also, the absolute value bars here serveas a grouping symbol. We begin in the numerator by simplifying within the absolutevalue bars.

Simplify the expression inside the absolute value bars.

Find the absolute value andsimplify the denominator.

Evaluate the exponential expression.

Simplify the numerator.

Simplify62

- 53 + | 6 - 5| # 8

3PRACTICE

=

83

=

3 + 1 + 43

=

3 + 1 + 22

3

3 + ƒ 4 - 3 ƒ + 22

6 - 3=

3 + ƒ 1 ƒ + 22

6 - 3

3 + ƒ 4 - 3 ƒ + 22

6 - 3.

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OBJECTIVE 2 � Evaluating algebraic expressions. In algebra, we use symbols, usuallyletters such as x, y, or z, to represent unknown numbers. A symbol that is used torepresent a number is called a variable. An algebraic expression is a collection ofnumbers, variables, operation symbols, and grouping symbols. For example,

are algebraic expressions. The expression 2x means Also, meansand means If we give a specific value to a variable, we can

evaluate an algebraic expression. To evaluate an algebraic expression means to find itsnumerical value once we know the values of the variables.

Algebraic expressions often occur during problem solving. For example, theexpression

gives the distance in feet (neglecting air resistance) that an object will fall in t seconds.(See Exercise 63 in this section.)

16t2

3 # y2.3y25 # 1p2+ 12 51p2

+ 122 # x.

2x, -3, 2x + 10, 51p2+ 12, and 3y2

- 6y + 15

EXAMPLE 4 Simplify

Solution Notice that both parentheses and brackets are used as grouping symbols.Start with the innermost set of grouping symbols.

Simplify the expression in parentheses.

Multiply.

Add.

Multiply.

Simplify 4325 - 3(5 + 3)4.4PRACTICE

= 66

= 3[22]

= 3[4 + 18]

3[4 + 2110 - 12] = 3[4 + 2192]

3[4 + 2110 - 12].

◗ Helpful HintBe sure to follow order ofoperations and resist the temp-tation to incorrectly add 4 and2 first.

EXAMPLE 5 Simplify

Solution

Simplify36 , 9 + 5

52- 3

.5PRACTICE

8 + 2 # 3

22- 1

=

8 + 64 - 1

=

143

8 + 2 # 3

22- 1

.

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Evaluate each expression if and .

a. b. c. d. x3+ y23

x+

xy

4x

3y2x + y

y = 5x = 26PRACTICE

EXAMPLE 6 Evaluate each expression if and

a. b. c. d.

Solution

a. Replace x with 3 and y with 2.

Let and

Multiply.

Subtract.

b. Let and

c. Replace x with 3 and y with 2. Then simplify.

d. Replace x with 3 and y with 2.

x2- y2

= 32- 22

= 9 - 4 = 5

xy

+

y

2=

32

+

22

=

52

y = 2.x = 33x

2y=

3 # 32 # 2

=

94

= 4

= 6 - 2

y = 2.x = 3 2x - y = 2132 - 2

x2- y2x

y+

y

23x

2y2x - y

y = 2.x = 3

OBJECTIVE 3 � Determining whether a number is a solution of an equation. Manytimes a problem-solving situation is modeled by an equation. An equation is amathematical statement that two expressions have equal value. The equal symbol

is used to equate the two expressions. For example,and are all equations.I = PRT

21x - 123

= 0,3 + 2 = 5, 7x = 35,

“=”

◗ Helpful HintAn equation contains the equal symbol An algebraic expression does not.“=”.

Concept CheckWhich of the following are equations? Which are expressions?

a. b. c. d. 12y = 3x12y + 3x5x - 85x = 8

When an equation contains a variable, deciding which values of the variablemake an equation a true statement is called solving an equation for the variable. Asolution of an equation is a value for the variable that makes the equation true. For ex-ample, 3 is a solution of the equation because if x is replaced with 3 thestatement is true.

Replace x with 3.

True

Similarly, 1 is not a solution of the equation because is not a truestatement.

1 + 4 = 7x + 4 = 7,

7 = 7

3 + 4 = 7T

x + 4 = 7

x + 4 = 7,


equations: a, d; expressions: b, c

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OBJECTIVE 4 � Translating phrases to expressions and sentences to equations. Nowthat we know how to represent an unknown number by a variable, let’s practicetranslating phrases into algebraic expressions and sentences into equations.Oftentimes solving problems requires the ability to translate word phrases andsentences into symbols. Below is a list of some key words and phrases to help ustranslate.

EXAMPLE 7 Decide whether 2 is a solution of

Solution Replace x with 2 and see if a true statement results.

Original equation

Replace x with 2.

Simplify each side.

True

Since we arrived at a true statement after replacing x with 2 and simplifying both sidesof the equation, 2 is a solution of the equation.

Decide whether 4 is a solution of .9x - 6 = 7x7PRACTICE

16 = 16

6 + 10 � 16

3122 + 10 � 8122 3x + 10 = 8x

3x + 10 = 8x.

◗ Helpful HintOrder matters when subtracting and also dividing, so be especially careful with thesetranslations.

Addition Subtraction Multiplication Division ( ) Equality (=)

Sum Difference of Product Quotient Equals

Plus Minus Times Divide Gives

Added to Subtracted from Multiply Into Is/was/should be

More than Less than Twice Ratio Yields

Increased by Decreased by Of Divided by Amounts to

Total Less Represents/ Is the same as

�( # )(�)(�)

EXAMPLE 8 Write an algebraic expression that represents each phrase. Letthe variable x represent the unknown number.

a. The sum of a number and 3 b. The product of 3 and a number

c. Twice a number d. 10 decreased by a number

e. 5 times a number, increased by 7

Solution

a. since “sum” means to add

b. and 3x are both ways to denote the product of 3 and x

c. or 2x

d. because “decreased by” means to subtract

e.5 times a number()*5x + 7

10 - x

2 # x

3 # x

x + 3

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Translate: 21

d. In words: 48is lessthan

triple anumber

Z17+4xTTTTT

Translate:

Care must be taken when the operation is subtraction.The expression wouldbe incorrect. Notice that

c. In words: 21is

not equal to17added to

four times a number

3 - 12 Z 12 - 3.3 - 12

x=12 - 3TTT

Translate:

b. In words: a numberisthree subtracted from 12

4=

15x

TTT


Write an algebraic expression that represents each phase. Let the variable xrepresent the unknown number.

a. Six times a number b. A number decreased by 8

c. The product of a number and 9 d. Two times a number, plus 3

e. The sum of 7 and a number

8PRACTICE

◗ Helpful HintMake sure you understand the difference when translating phrases containing “decreasedby,” “subtracted from,” and “less than.”

Phrase TranslationA number decreased by 10A number subtracted from 10

Notice the order.10 less than a numberA number less 10 x - 10

x - 1010 - xx - 10

MNow let’s practice translating sentences into equations.

EXAMPLE 9 Write each sentence as an equation or inequality. Let xrepresent the unknown number.

a. The quotient of 15 and a number is 4.

b. Three subtracted from 12 is a number.

c. Four times a number, added to 17, is not equal to 21.

d. Triple a number is less than 48.

Solution

a. In words: 4isthe quotient of 15

and a number

Translate: 3x 486

TTT

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Write each sentence as an equation or inequality. Let x represent theunknown number.

a. A number increased by 7 is equal to 13.

b. Two less than a number is 11.

c. Double a number, added to 9, is not equal to 25.

d. Five times 11 is greater than or equal to an unknown number.

9PRACTICE

Calculator Explorations

Exponents

To evaluate exponential expressions on a scientific calculator, find the key marked

or To evaluate, for example, press the following keys:

or

The display should read or

Order of Operations

Some calculators follow the order of operations, and others do not.To see whetheror not your calculator has the order of operations built in, use your calculator tofind To do this, press the following sequence of keys:

The correct answer is 14 because the order of operations is to multiply beforewe add. If the calculator displays then it has the order of operationsbuilt in.

Even if the order of operations is built in, parentheses must sometimes be

inserted. For example, to simplify press the keys

The display should read or

Use a calculator to evaluate each expression.

1. 2.

3. 4.

5. 6.

7. 8.

9. 10.956 - 45289 - 86

4623 + 12936 - 34

99 + 1401 + 9622241862 - 4552 + 89

3114 - 72 + 212120 - 528695

7454

� 5/112 - 721 �� 1 �

� 5 �� , �� 1 �� 1 �� 2 �� - �� 7 �� 2 �� = �. D or� ENTER �

512 - 7

,

� 14 �,

� 2 �� + �� 3 �� * �� 4 �� = �. D or� ENTER �

2 + 3 # 4.

� 3 ¿

5243

�� 243 �

� 3 �� ¿ �� 5 �� = �. D or� ENTER �

� 3 �� yx �� 5 �� = �35,� ¿ �.� yx �

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Evaluate. See Example 1.

1. 2.

3. 4.

5. 6.

7. 8.

9. 10.

11. 12.

13. 14.

15. 16.

17. 18.

MIXED PRACTICE

Simplify each expression. See Examples 2 through 5.

19. 20.

21. 22.

23. 24.

25. 26.

27. 28.

29. 30.

31. 32.

33. 34.

35. 36.4 # 3 + 24 + 3 # 2

19 - 3 # 56 - 4

3[4 + 316 - 42]2[5 + 218 - 32]8 - 5

24 - 206 - 49 - 2

34

#12

+

23

14

#23

-

16

2 # 525 # 32

6 - 2 # 2 + 252 + 15 - 22 + 42

516 - 22218 - 3212 # 5 - 3 # 64 # 8 - 6 # 2

8 + 5 # 35 + 6 # 2

10.072211.2224342

9272

a12b5a2

3b4

a 611b2a1

5b3

8151

1815

4433

2535

VOCABULARY & READINESS CHECKUse the choices below to fill in each blank.

add multiply equation variable base grouping

subtract divide expression solution solving exponent

1. In the expression , the 5 is called the and the 2 is called the .

2. The symbols ( ), [ ], and { } are examples of symbols.

3. A symbol that is used to represent a number is called a(n) .

4. A collection of numbers, variables, operation symbols, and grouping symbols is called a(n) .

5. A mathematical statement that two expressions are equal is called a(n) .

6. A value for the variable that makes an equation a true statement is called a(n) .

7. Deciding what values of a variable make an equation a true statement is called the equation.

8. To simplify the expression first .



11. To simplify the expression first . 20 - 4 , 2,

120 - 42 # 2,

11 + 32 # 6,

1 + 3 # 6,

52

37. 38.

39. 40.

41. 42.

43. Are parentheses necessary in the expression Explain your answer.

44. Are parentheses necessary in the expression Explain your answer.

For Exercises 45 and 46, match each expression in the first columnwith its value in the second column.

45. a. 19

b. 22

c. 64

d. 43

46. a. 15

b. 13

c. 27

d. 22

Evaluate each expression when and SeeExample 6.

47. 3y 48. 4x

49. 50.y

2z

z

5x

z = 5.x = 1, y = 3,

11 + 42 # 16 - 321 + 4 # 6 - 3

1 + 4 # 16 - 3211 + 42 # 6 - 3

6 + 2 # 15 + 326 + 2 # 5 + 3

16 + 22 # 5 + 3

16 + 22 # 15 + 32

12 + 32 # 5?

2 + 13 # 52?16 + ƒ 13 - 5 ƒ + 42

17 - 5

6 + ƒ 8 - 2 ƒ + 32

18 - 3

3 + 618 - 5242

+ 2

3 + 315 + 3232

+ 1

15 - ƒ 3 - 1 ƒ

12 - 3 # 2

ƒ 6 - 2 ƒ + 3

8 + 2 # 5

1.4 EXERCISE SET

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64. Does an object fall the same distance during each second?Why or why not? (See Exercise 63.)

Decide whether the given number is a solution of the given equa-tion. See Example 7.

65. Is 5 a solution of










Write each phrase as an algebraic expression. Let x represent theunknown number. See Example 8.

75. Fifteen more than a number

76. One-half times a number

77. Five subtracted from a number

78. The quotient of a number and 9

4 = 1 - x?

x = 5x + 15?

x + 6 = x + 6?

x + 6 = x + 6?

3x - 10 = 8?

2x - 5 = 5?

4x + 2 = x + 8?

2x + 6 = 5x - 1?

2x + 7 = 3x?

3x + 30 = 9x?

51. 52.

53. 54.

55. 56.

Evaluate each expression if and SeeExample 6.

57. 58.

59. 60.

61. 62.

Neglecting air resistance, the expression gives the distance infeet an object will fall in t seconds.

63. Complete the chart below. To evaluate remember tofirst find then multiply by 16.t2,

16t2,

16t2

y2+ x

x2+ 3y

x2+ z

y2+ 2z

y2- 3x + yx2

- 3y + x

y

z+ 8x

xz

+ 3y

z = 4.x = 12, y = 8,

2z25y2

ƒ 5z - 2y ƒƒ 2x + 3y ƒ

6y - 83x - 2 79. Three times a number, increased by 22

80. The product of 8 and a number, decreased by 10

Write each sentence as an equation or inequality. Use x to representany unknown number. See Example 9.

81. One increased by two equals the quotient of nine andthree.

82. Four subtracted from eight is equal to two squared.

83. Three is not equal to four divided by two.

84. The difference of sixteen and four is greater thanten.

85. The sum of 5 and a number is 20.

86. Twice a number is 17.

87. Thirteen minus three times a number is 13.

88. Seven subtracted from a number is 0.

89. The quotient of 12 and a number is

90. The sum of 8 and twice a number is 42.

91. In your own words, explain the difference between an ex-pression and an equation.

92. Determine whether each is an expression or an equation.

a. b.

c. d.

CONCEPT EXTENSIONS

93. Insert parentheses so that the following expression simplifiesto 32.

94. Insert parentheses so that the following expression simplifiesto 28.

Solve the following.

95. The perimeter of a figure is the distance around the figure.The expression represents the perimeter of a rec-tangle when l is its length and is its width. Find theperimeter of the following rectangle by substituting 8 for land 6 for w.

w2l + 2w

2 # 5 + 32

20 - 4 # 4 , 2

9y + x - 82x - 5 = 7x - 5

3x2- 26 = 13x2

- 26

12

.

yard¡ yard

™ yard

514

8 meters

6 meters

96. The expression represents the perimeter of a tri-angle when a, b, and c are the lengths of its sides. Find theperimeter of the following triangle.

a + b + c

Time t Distance 16t2

(in seconds) (in feet)

1

2

3

4

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�1 0 1 2 3 4 5�5 �4 �3 �2

Start End3 2

3 � 2 � 5

�1 0 1 2 3 4 5�5 �4 �3 �2

StartEnd�2 �1

�1 � (�2) � �3

OBJECTIVE 1 � Adding real numbers with the same sign. Real numbers can be added,subtracted, multiplied, divided, and raised to powers, just as whole numbers can. Weuse a number line to help picture the addition of real numbers.


99. The expression represents the rate of interest being

charged if a loan of P dollars for T years required I dollarsin interest to be paid. Find the interest rate if a $650 loan for3 years to buy a used IBM personal computer requires$126.75 in interest to be paid.

100. The expression represents the average speed r in miles per

hour if a distance of d miles is traveled in t hours. Find therate to the nearest whole number if the distance betweenDallas, Texas, and Kaw City, Oklahoma, is 432 miles, and ittakes Peter Callac 8.5 hours to drive the distance.

101. Verizon long-distance service offers a “Talk to the WorldPlan,” which offers lower rates on international phone callsto Verizon subscribers. This plan charges $3.00 per month,plus $0.12 per minute for calls to Japan.The expression 3.00 �0.12m represents the long-distance charge for a call to Japanlast month by a Verizon customer enrolled in this plan. Findthe monthly bill for the customer whose only call to Japanlasted 84 minutes.

102. In forensics, the density of a substance is used to help identify

it.The expression represents the density of an object with

a mass of M grams and a volume of V milliliters. Find thedensity of an object having a mass of 29.76 grams and a vol-ume of 12 milliliters.

M

V

d

t

I

PT97. The area of a figure is the total enclosed surface of the figure.

Area is measured in square units. The expression represents the area of a rectangle when l is its length andis its width. Find the area of the following rectangular-shaped lot.

wlw

7 inches

15 inches

5 inches

120 feet

100 feet

98. A trapezoid is a four-sided figure with exactly one pair of

parallel sides. The expression represents its area,

when B and b are the lengths of the two parallel sides and h is the height between these sides. Find the area if

and h = 5 inches.b = 7 inches,B = 15 inches,

12

h1B + b2

OBJECTIVES

1 Add real numbers with the samesign.

2 Add real numbers with unlikesigns.

3 Solve problems that involveaddition of real numbers.

4 Find the opposite of a number.

1.5 ADDING REAL NUMBERS

EXAMPLE 1 Add:

Solution Recall that 3 and 2 are called addends. We start at 0 on a number line, anddraw an arrow representing the addend 3. This arrow is three units long and points tothe right since 3 is positive. From the tip of this arrow, we draw another arrow repre-senting the addend 2. The number below the tip of this arrow is the sum, 5.

3 + 2

Add using a number line: .2 + 41PRACTICE

EXAMPLE 2 Add:

Solution Here, and are addends. We start at 0 on a number line, and draw anarrow representing This arrow is one unit long and points to the left since isnegative. From the tip of this arrow, we draw another arrow representing The num-ber below the tip of this arrow is the sum, -3.

-2.-1-1.

-2-1

-1 + 1-22

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0�

2�

�4�

Rises 6Degrees

Start

End

Section 1.5 Adding Real Numbers 35

Add using a number line: .-2 + (-3)2PRACTICE

Thinking of signed numbers as money earned or lost might help make additionmore meaningful. Earnings can be thought of as positive numbers. If $1 is earned andlater another $3 is earned, the total amount earned is $4. In other words,

On the other hand, losses can be thought of as negative numbers. If $1 is lost andlater another $3 is lost, a total of $4 is lost. In other words,

Using a number line each time we add two numbers can be time consuming. In-stead, we can notice patterns in the previous examples and write rules for addingsigned numbers. When adding two numbers with the same sign, notice that the sign ofthe sum is the same as the sign of the addends.

1-12 + 1-32 = -4.

1 + 3 = 4.

Adding Two Numbers with the Same SignAdd their absolute values. Use their common sign as the sign of the sum.

EXAMPLE 3 Add.

a. b. c.

Solution Notice that each time, we are adding numbers with the same sign.

-2 + 1-102-1 + 1-202-3 + 1-72

a. Add their absolute values: 3 � 7 � 10.

Use their common sign.

b. Add their absolute values: 1 � 20 � 21.

Common sign.

c. Add their absolute values.

Common sign.

-2 + 1-102 = -12

-1 + 1-202 = -21

-3 + 1-72 = -10

Add. a. b. -31 + (-1)-5 + (-8)3PRACTICE

OBJECTIVE 2 � Adding real numbers with unlike signs. Adding numbers whose signsare not the same can also be pictured on a number line.

EXAMPLE 4 Add:

Solution

-4 + 6

�1 0 1 2 3 4�5 �4 �3 �2

StartEnd�4

6

�4 � 6 � 2

Add using a number line: .-3 + 84PRACTICE

Using temperature as an example, if the thermometer registers 4 degrees below0 degrees and then rises 6 degrees, the new temperature is 2 degrees above 0 degrees.Thus, it is reasonable that

Once again, we can observe a pattern: When adding two numbers with differentsigns, the sign of the sum is the same as the sign of the addend whose absolute value islarger.

-4 + 6 = 2.

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a. Subtract their absolute values: 7 � 3 � 4.

The negative number, �7, has the larger absolute value so the sumis negative.

b.

c. 0.2 + 1-0.52 = -0.3

-2 + 10 = 8

3 + 1-72 = -4

Subtract their absolute values: 10 � 2 � 8.

The positive number, 10, has the larger absolute value so the sumis positive.

Subtract their absolute values: 0.5 � 0.2 � 0.3.

The negative number, �0.5, has the larger absolute valueso the sum is negative.


Add.

a. b. c. -0.6 + 0.4-19 + 2015 + 1-1825

PRACTICE

Adding Two Numbers with Different SignsSubtract the smaller absolute value from the larger absolute value. Use the sign ofthe number whose absolute value is larger as the sign of the sum.

EXAMPLE 5 Add.

a. b. c.

Solution Notice that each time, we are adding numbers with different signs.

0.2 + 1-0.52-2 + 103 + 1-72

EXAMPLE 6 Add.

a. b. c.

d. e. f.

Solution

a. Same sign.Add absolute values and use the common sign.

b. Different signs. Subtract absolute values and use the signof the number with the larger absolute value.

c. Different signs.

d. Same sign.

e.

f. -

38

+

25

= -

1540

+

1640

=

140

11.4 + 1-4.72 = 6.7

-

710

+ a-

110b = -

810

= -

45

0.6 + 1-1.12 = -0.5

-5 + 35 = 30

-8 + 1-112 = -19

-

38

+

25

11.4 + 1-4.72-

710

+ a-

110b

0.6 + 1-1.12-5 + 35-8 + 1-112

Add.

a. b.

c. d. -

27

+

310

2.2 + 1-1.72

3 + 1-92-

35

+ a-

25b

6PRACTICE

◗ Helpful HintDon’t forget that a commondenominator is needed whenadding or subtracting fractions.The common denominator hereis 40.

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Translate:

The overall loss is $2.

During a three-day period, a share of McDonald’s stock recorded the followinggain and losses:

8PRACTICE

1-32 = -2+1-12+2TTTTT


Add.

a. b. 3-8 + 54 + 3-5 + |-2|48 + 1-52 + 1-927

PRACTICE

EXAMPLE 7 Add.

a. b.

Solution

a. Perform the additions from left to right.

Adding numbers with different signs.

Adding numbers with like signs.

b. Simplify inside brackets first.

Add. = -1

= [-3] + [2]

[7 + 1-102] + [-2 + ƒ -4 ƒ] = [-3] + [-2 + 4]

= -12

3 + 1-72 + 1-82 = -4 + 1-82

[7 + 1-102] + [-2 + ƒ -4 ƒ ]3 + 1-72 + 1-82

◗ Helpful HintDon’t forget that brackets aregrouping symbols. We simplifywithin them first.

OBJECTIVE 3 � Solving problems by adding real numbers. Positive and negativenumbers are often used in everyday life. Stock market returns show gains and losses aspositive and negative numbers. Temperatures in cold climates often dip into thenegative range, commonly referred to as “below zero” temperatures. Bank statementsreport deposits and withdrawals as positive and negative numbers.

EXAMPLE 8 Finding the Gain or Loss of a Stock

During a three-day period, a share of Fremont General Corporation stock recordedthe following gains and losses:

Monday Tuesday Wednesday

a gain of $2 a loss of $1 a loss of $3

Find the overall gain or loss for the stock for the three days.

Solution Gains can be represented by positive numbers. Losses can be represented bynegative numbers. The overall gain or loss is the sum of the gains and losses.

In words: losspluslossplusgain

Monday Tuesday Wednesday

a loss of $5 a gain of $8 a loss of $2

Find the overall gain or loss for the stock for the three days.

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OBJECTIVE 4 � Finding the opposite of a number. To help us subtract real numbers inthe next section, we first review the concept of opposites. The graphs of 4 and areshown on a number line below.

-4

4 units 4 units

�1 0 1 2 3 4 5�5 �4 �3 �2

Notice that 4 and lie on opposite sides of 0, and each is 4 units away from 0.This relationship between and is an important one. Such numbers are

known as opposites or additive inverses of each other.+4-4

-4

Opposites or Additive InversesTwo numbers that are the same distance from 0 but lie on opposite sides of 0 arecalled opposites or additive inverses of each other.

Let’s discover another characteristic about opposites. Notice that the sum of anumber and its opposite is 0.

In general, we can write the following:

12

+ a-

12b = 0

-3 + 3 = 0

10 + 1-102 = 0

The sum of a number a and its opposite is 0.

a + 1-a2 = 0

-a

This is why opposites are also called additive inverses. Notice that this also means thatthe opposite of 0 is then 0 since 0 + 0 = 0.

EXAMPLE 9 Find the opposite or additive inverse of each number.

a. 5 b. c. d.

Solution

a. The opposite of 5 is Notice that 5 and are on opposite sides of 0 when plottedon a number line and are equal distances away.

b. The opposite of is 6.

c. The opposite of is

d. The opposite of is 4.5.

Find the opposite or additive inverse of each number.

a. b. 8 c. 6.2 d. -3-

59

9PRACTICE

-4.5

-

12

.12

-6

-5-5.

-4.512

-6

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This is true in general.

3=1-32-

TTTT


We use the symbol to represent the phrase “the opposite of” or “the additiveinverse of.” In general, if a is a number, we write the opposite or additive inverse of a as

We know that the opposite of is 3. Notice that this translates as

3is�3the opposite of

-3-a.

“-”

If a is a number, then -1-a2 = a.


a. b. c. d.

Solution

a. b. c.

d.

Simplify each expression.

a. b.

c. d. -1-82-1-5y2- a-

35b- ƒ -15 ƒ

10PRACTICE

-1-2x2 = 2x- a-

12b =

12

-1-102 = 10

- ƒ -6 ƒ-1-2x2- a-

12b-1-102

Since ƒ - 6 ƒ = 6, then - ƒ - 6 ƒ = - 6.

VOCABULARY & READINESS CHECKUse the choices below to fill in each blank.

positive number n opposites

negative number 0

1. Two numbers that are the same distance from 0 but lie on opposite sides of 0 are called .

2. The sum of a number and its opposite is always .

3. If n is a number, then .

Tell whether the sum is a positive number, a negative number, or 0. Do not actually find the sum.

4. .

5. .

6. .

7. .

8. .

9. . -

23

+

23

=

-3.68 + 0.27 =

-1.26 + 1-8.32 =

-162 + 162 =

-162 + 164 =

-80 + (-127) =

-(-n) =

-n

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MIXED PRACTICE

Add. See Examples 1 through 7.

1. 2.

3. 4.

5. 6.

7. 8.

9. 10.

11. 12.

13. 14.

15. 16.

17. 18.

19. 20.

21. 22.

23. 24.

25. 26.

27. 28.

29. 30.

31. 32.

33. 34.

35. 36.

37. 38.

39. 40.

41. 42.

43. 44.

45. 46.

47. 48.

49. 50.

51.

52.

53.

54.

55.

56.

Solve. See Example 8.

57. The low temperature in Anoka, Minnesota, was last night.During the day it rose only 9°. Find the high temperature for theday.

58. On January 2, 1943, the temperature was at 7:30 a.m. inSpearfish, South Dakota. Incredibly, it got 49° warmer in thenext 2 minutes. To what temperature did it rise by 7:32?

59. The deepest canyon in the world is the Great Canyon of theYarlungTsangpo inTibet.The bottom of the canyon is 17,657 feetbelow the surrounding terrain, called the rim. If you are standing

-4°

-15°

-3.7 + [0.1 + 1-0.62 + 8.1]

-1.3 + [0.5 + 1-0.32 + 0.4]

ƒ 43 + 1-732 ƒ + ƒ -20 ƒ

ƒ 9 + 1-122 ƒ + ƒ -16 ƒ

[-2 + 1-72] + [-11 + 22]

[-17 + 1-42] + [-12 + 15]

8 + 1-22 + 76 + 1-42 + 9

ƒ 7 + 1-172 ƒƒ 5 + 1-102 ƒ

-14 + 1-32 + 11-23 + 16 + 1-22-18 + 1-62 + 1-402-21 + 1-162 + 1-222-9 + 15 + 1-52-15 + 9 + 1-22-

56

+ a-

23b-

710

+ a-

35b

-

59

+

13

-

716

+

14

-

512

+

712

-

38

+

58

-6.7 + 1-7.62-9.6 + 1-3.52144 + 1-882117 + 1-792ƒ -6 ƒ + 1-612ƒ -8 ƒ + 1-1629.2 + 1-11.426.3 + 1-8.42-18 + 1-262-33 + 1-142-26 + 14-18 + 49

53 + 1-37227 + 1-46223 + 1-232-16 + 16

3 + 1-625 + 1-728 + 1-6210 + 1-32-5 + 9-7 + 3

7 + 1-52-9 + 1-32-7 + 1-42-2 + 1-32-10 + 5-14 + 2

6 + 1-428 + 1-72-6 + 1-142-6 + 1-829 + 1-1226 + 3

1230 feet above the bottom of the canyon, how far from therim are you?

1.5 EXERCISE SET

60. The lowest point in Africa is feet at Lake Assal inDjibouti. If you are standing at a point 658 feet above LakeAssal, what is your elevation? (Source: Microsoft Encarta)

-512

A negative net income results when a company’s expenses aremore than the money brought in.

61. The table below shows net incomes for Ford Motor Compa-ny’s Automotive sector for the years 2004, 2005, and 2006.Find the total net income for three years.

62. The table below shows net incomes for Continental Airlinesfor the years 2004, 2005, and 2006. Find the total net incomefor these years.

Year Net Income (in millions)

200420052006

(Source: Ford Motor Company)

-$5200- $3895- $155

Year Net Income (in millions)

200420052006

(Source: Continental Airlines)

$343- $68

- $409

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Section 1.6 Subtracting Real Numbers 41

In golf, scores that are under par for the entire round are shown asnegative scores; positive scores are shown for scores that are overpar, and 0 is par.

63. Paula Creamer was the winner of the 2007 LPGA SBS Openat Turtle Bay. Her scores were , and . What was heroverall score? (Source: Ladies Professional Golf Association)

64. During the 2007 PGA Buick Invitational Golf Tournament,Tiger Woods won with scores of , and . What washis overall score? (Source: Professional Golf Association)

Find each additive inverse or opposite. See Example 9.

65. 6 66. 4

67. 68.

69. 0 70.

71. 72.

73. In your own words, explain how to find the opposite of anumber.

74. In your own words, explain why 0 is the only number that isits own opposite.

Simplify each of the following. See Example 10.

75. 76.

77. 78.

79. 80.

81. Explain why adding a negative number to another negativenumber always gives a negative sum.

82. When a positive and a negative number are added, some-times the sum is positive, sometimes it is zero, and sometimesit is negative. Explain why and when this happens.

Decide whether the given number is a solution of the given equation.

83. Is a solution of



86. Is a solution of 1 = y + 7?-6

y + 1-32 = -7?-1

7 = -x + 3?

x + 9 = 5?-4

-1-72- ` - 23`

` - 23`- ƒ 0 ƒ

-1-32- ƒ -2 ƒ

ƒ -11 ƒƒ -6 ƒ

-

14

-8-2

-6-6, 0, -3

-2-5, -2

CONCEPT EXTENSIONS

The following bar graph shows each month’s average daily lowtemperature in degrees Fahrenheit for Barrow, Alaska. Use thisgraph to answer Exercises 87 through 92.

Ave

rage

Dai

ly L

ow T

empe

ratu

re(i

n de

gree

s Fa

hren

heit

)

Month

40

20

0

30

10

�10

�20

�30

Barrow, Alaska

Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

Source: National Climatic Data Center

�19.3�23.7

�21.1

�9.1

14.4

29.733.6 33.3

27.0

8.8

�6.9

�17.2

87. For what month is the graphed temperature the highest?

88. For what month is the graphed temperature the lowest?

89. For what month is the graphed temperature positive andclosest to 0°?

90. For what month is the graphed temperature negative andclosest to 0°?

91. Find the average of the temperatures shown for the monthsof April, May, and October. (To find the average of threetemperatures, find their sum and divide by 3.)

92. Find the average of the temperatures shown for the monthsof January, September, and October.

If a is a positive number and b is a negative number, fill in theblanks with the words positive or negative.

93. is . 94. is .

95. is . 96. is . b + b a + a

-b -a

OBJECTIVES

1 Subtract real numbers.

2 Add and subtract real numbers.

3 Evaluate algebraic expressionsusing real numbers.

4 Solve problems that involvesubtraction of real numbers.

1.6 SUBTRACTING REAL NUMBERS

OBJECTIVE 1 � Subtracting real numbers. Now that addition of signed numbers hasbeen discussed, we can explore subtraction. We know that Notice that

also. This means that

Notice that the difference of 9 and 7 is the same as the sum of 9 and the opposite of 7.In general, we have the following.

9 - 7 = 9 + 1-729 + 1-72 = 2,

9 - 7 = 2.

Subtracting Two Real NumbersIf a and b are real numbers, then a - b = a + 1-b2.

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In other words, to find the difference of two numbers, add the first number to the oppo-site of the second number.

EXAMPLE 1 Subtract.

a. b. c. d.

Solution

a. Add to the opposite of which is

b. Add 5 to the opposite of which is 6.

c. Add 3 to the opposite of 6, which is

d.

Subtract.

a. b. c. d. 5 - 79 - (-3)-8 - (-1)-7 - 6

1PRACTICE

-1 - 1-72 = -1 + 172 = 6

-6.

= -3

3 - 6 = 3 + 1-62 = 11

-6,

= -17

-4.+4,-13

-1 - 1-723 - 65 - 1-62-13 - 4

◗ Helpful HintStudy the patterns indicated.

7 - 1-12 = 7 + 112 = 8

-3 - 4 = -3 + 1-42 = -7

Change to addition.Change to opposite.

5 - 11 = 5 + 1-112 = -6

No change

EXAMPLE 2 Subtract.

a. b. c.

Solution

a.

b.

c. The common denominator is 15.

Subtract.

a. b. c. -

34

-

15

-

58

- a -

18b8.4 - (-2.5)

2PRACTICE

-

23

- a-

45b = -

23

+ a45b = -

1015

+

1215

=

215

-

310

-

510

= -

310

+ a-

510b = -

810

= -

45

5.3 - 1-4.62 = 5.3 + 14.62 = 9.9

-

23

- a-

45b-

310

-

510

5.3 - 1-4.62

add

opposite

-13 - 4 = -13 + 1-42

- 1-62 = 5 + 162add

5

opposite

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OBJECTIVE 2 � Adding and subtracting real numbers. If an expression containsadditions and subtractions, just write the subtractions as equivalent additions. Thensimplify from left to right.

EXAMPLE 3 Subtract 8 from

Solution Be careful when interpreting this: The order of numbers in subtraction isimportant. 8 is to be subtracted from

Subtract 5 from .-23PRACTICE

-4 - 8 = -4 + 1-82 = -12

-4.

-4.


a. b.

Solution

a.

b.


a. b. 3.5 + (-4.1) - (-6.7)-15 - 2 - (-4) + 7

4PRACTICE

= 6.3

1.6 - 1-10.32 + 1-5.62 = 1.6 + 10.3 + 1-5.62 = -6

-14 - 8 + 10 - 1-62 = -14 + 1-82 + 10 + 6

1.6 - 1-10.32 + 1-5.62-14 - 8 + 10 - 1-62

When an expression contains parentheses and brackets, remember the order ofoperations. Start with the innermost set of parentheses or brackets and work your wayoutward.


a. b.

Solution

a. Start with the innermost sets of parentheses. Rewrite as a sum.

Add:

Write as a sum.

Add.

Add.

b. Start simplifying the expression inside the brackets by writing as a sum.

= -3 = -2 + 1-12 = 8 + 1-102 + 1-12 = 8 - 10 + 1-12 = 23

- ƒ 10 ƒ + [-1]

23- ƒ 10 ƒ + [-6 - 1-52] = 23

- ƒ 10 ƒ + [-6 + 5]

-6 - 1-52 = -12

= -3 + [-9]

-7 - 2 = -3 + [-7 + 1-22]-2 + 1-52. = -3 + [1-72 - 2]

-3 + [1-2 - 52 - 2] = -3 + [1-2 + 1-522 - 2]

-2 - 5

23- ƒ 10 ƒ + [-6 - 1-52]-3 + [1-2 - 52 - 2]

Add.

Evaluate and

Write as a sum.

Add.

Add.

8 - 10

ƒ 10 ƒ .23


a. b. |-13| - 32+ 32 - (-7)4-4 + 3(-8 - 3) - 54

5PRACTICE

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OBJECTIVE 3 � Evaluating algebraic expressions. Knowing how to evaluate expres-sions for given replacement values is helpful when checking solutions of equations andwhen solving problems whose unknowns satisfy given expressions. The next exampleillustrates this.

EXAMPLE 6 Find the value of each expression when and

a. b.

Solution

a. Replace x with 2 and y with Be sure to put parentheses around to separatesigns. Then simplify the resulting expression.

b. Replace the x with 2 and y with and simplify.

Find the value of each expression when and .

a. b. y2+ x

7 - x

2y + x

y = 4x = -36PRACTICE

= 19

= 4 + 15

= 4 - 1-152 = 4 - 31-52

x2- 3y = 22

- 31-52-5

=

714

=

12

=

2 + 514

x - y

12 + x=

2 - 1-5212 + 2

-5-5.

x2- 3y

x - y

12 + x

y = -5.x = 2

OBJECTIVE 4 � Solving problems by subtracting real numbers. One use of positive andnegative numbers is in recording altitudes above and below sea level, as shown in thenext example.

EXAMPLE 7 Finding the Difference in Elevations

The lowest point on the surface of the Earth is the Dead Sea, at an elevation of 1349 feetbelow sea level. The highest point is Mt. Everest, at an elevation of 29,035 feet. Howmuch of a variation in elevation is there between these two world extremes? (Source:National Geographic Society)

Mt. Everest

Dead Sea

29,035 feetabove sea level

(�29,035)

1349 feetbelow sea level

(�1349)

Sea level (0)

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43�y


Solution To find the variation in elevation between the two heights, find the differenceof the high point and the low point.

In words:

Translate: 29,035

Thus, the variation in elevation is 30,384 feet.

On Tuesday morning, a bank account balance was $282. On Thursday the account balance had dropped to . Find the overall change in this accountbalance.

- $757

PRACTICE

= 30,384 feet

= 29,035 + 1349(-1349)-

TTT

low pointminushigh point

A knowledge of geometric concepts is needed by many professionals, such as doctors,carpenters, electronic technicians, gardeners, machinists, and pilots, just to name a few.With this in mind, we review the geometric concepts of complementary and supple-mentary angles.

EXAMPLE 8 Find each unknown complementary or supplementary angle.

a. b.

Solution

a. These angles are complementary, so their sum is 90°. This means that x is

b. These angles are supplementary, so their sum is 180°. This means that y is

Find each unknown complementary or supplementary angle.

a. b.

8PRACTICE

y = 180° - 62° = 118°

180° - 62°.

x = 90° - 38° = 52°

90° - 38°.

Complementary and Supplementary AnglesTwo angles are complementary if their sum is 90°.

Two angles are supplementary if their sum is 180°.

x

x � y � 180�

y

x � y � 90�

xy

38�

x62�

y

62�

x

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Translate each phrase. Let x represent “a number.” Use the choices below to fill in each blank.

1. 7 minus a number 4. 7 less a number 2. 7 subtracted from a number 5. A number less than 7 3. A number decreased by 7 6. A number subtracted from 7

x - 77 - x

MIXED PRACTICE

Subtract. See Examples 1 through 5.

1. 2.

3. 4.

5. 6.

7. 8.

9. 10.

11. 12.

13. 14.

15. 16.

17. 18.

19. 20.

21. 22.

23. 24.

25. 26.

27. 28.

29. 30.

31. 32.

Perform the operation. See Example 3.

33. Subtract from 8. 34. Subtract 3 from

35. Subtract from 36. Subtract 17 from 1.

37. Subtract 8 from 7. 38. Subtract 9 from

39. Decrease by 15. 40. Decrease 11 by

41. In your own words, explain why simplifies to a negativenumber.

42. Explain why is the same as

Simplify each expression. (Remember the order of operations.)See Examples 4 and 5.

43.

44.

45. 5 - 9 + 1-42 - 8 - 8

-16 - 1-32 + 1-112 - 14

-10 - 1-82 + 1-42 - 20

6 + 1-112.6 - 11

5 - 8

-14.-8

-4.

-6.-1

-2.-5

4.3 - 1-0.8728.3 - 1-0.622-

110

-

78

-

16

-

34

-

47

- a-

17b-

311

- a-

511b

-6.1 - 1-5.32-2.6 - 1-6.72-17 - 1-172-21 - 1-212-36 - 51-44 - 27

8.3 - 11.29.7 - 16.1

15 - 1-33216 - 1-212-4 - 1-162-6 - 1-1123 - 1-627 - 1-42-8 - 4-6 - 5

-20 - 1-482-16 - 1-18234

-

78

12

-

13

12 - 1-5216 - 1-328 - 114 - 9

-12 - 8-6 - 4

46.

47. 48.

49. 50.

51. 52.

53. 54.

55.

56.

57.

58.

Evaluate each expression when and SeeExample 6.

59. 60.

61. 62.

63. 64.

65. 66.

67. 68.

Solve. See Example 7.

69. Within 24 hours in 1916, the temperature in Browning,Montana, fell from 44 degrees to degrees. How large adrop in temperature was this?

70. Much of New Orleans is below sea level. If George descends12 feet from an elevation of 5 feet above sea level, what is hisnew elevation?

71. In a series of plays, the San Francisco 49ers gain 2 yards, lose5 yards, and then lose another 20 yards. What is their totalgain or loss of yardage?

72. In some card games, it is possible to have a negative score.Lavonne Schultz currently has a score of 15 points. She thenloses 24 points. What is her new score?

-56

ƒ 5y - x ƒ

6t

ƒ x - 1-102 ƒ

2t

t2- xy2

- x

15 - x

y + 29 - x

y + 6

ƒ x + t - 7y ƒƒ x ƒ + 2t - 8y

y - xx - y

t = 10.x = -5, y = 4,

ƒ -2 ƒ + 62+ 1-3 - 82

ƒ -3 ƒ + 22+ [-4 - 1-62]

-5 + [14 - 152 - 1-62 - 8]

-2 + [18 - 112 - 1-2 - 92]12 - 32 + 5213 - 62 + 42

4 - 617 - 322 - 318 - 6223

- 6 # 333- 8 # 9

-9 - 13 - 82-6 - 12 - 1127 - 12 + 1-52 - 2 + 1-22


1.6 EXERCISE SET

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Find each unknown complementary or supplementary angle. SeeExample 8.

79. 80.

73. Pythagoras died in the year (or 475 B.C.). When was heborn, if he was 94 years old when he died?

74. The Greek astronomer and mathematician Geminus died in60 A.D. at the age of 70. When was he born?

75. A commercial jet liner hits an air pocket and drops 250 feet.After climbing 120 feet, it drops another 178 feet. What is itsoverall vertical change?

76. Tyson Industries stock posted a loss of 1.625 points yesterday.If it drops another 0.75 point today, find its overall change forthe two days.

77. The highest point in Africa is Mt. Kilimanjaro,Tanzania, at anelevation of 19,340 feet. The lowest point is Lake Assal,Djibouti, at 512 feet below sea level. How much higher isMt. Kilimanjaro than Lake Assal? (Source: National Geo-graphic Society)

-475

78. The airport in Bishop, California, is at an elevation of 4101feet above sea level. The nearby Furnace Creek Airport inDeath Valley, California, is at an elevation of 226 feet belowsea level. How much higher in elevation is the Bishop Air-port than the Furnace Creek Airport? (Source: National Cli-matic Data Center)

105�y60�

x

50�

x50�y

Ave

rage

Dai

ly L

ow T

empe

ratu

re(i

n de

gree

s Fa

hren

heit

)

Month

40

20

0

30

10

�10

�20

�30

Barrow, Alaska

Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

Source: National Climatic Data Center

�19.3�23.7

�21.1

�9.1

14.4

29.733.6 33.3

27.0

8.8

�6.9

�17.2

89. Record the monthly increases and decreases in the low tem-perature from the previous month.

Month Monthly Increase or Decrease

FebruaryMarchAprilMayJuneJulyAugustSeptemberOctoberNovemberDecember

81. 82.








CONCEPT EXTENSIONS

Recall from the last section the bar graph below that showseach month’s average daily low temperature in degrees Fahren-heit for Barrow, Alaska. Use this graph to answer Exercises 89through 91.

4 = 1 - x?

-x - 13 = -15?

-x - 6 = -x - 1?-10

-x + 6 = -x - 1?-2

x - 10 = -7?

x - 9 = 5?-4

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90. Which month had the greatest increase in temperature?

91. Which month had the greatest decrease in temperature?

If a is a positive number and b is a negative number, determinewhether each statement is true or false.

92. is always a positive number.

93. is always a negative number.b - a

a - b



Without calculating, determine whether each answer is positive ornegative. Then use a calculator to find the exact difference.

96.

97. 4.362 - 7.0086

56,875 - 87,262

ƒ b - a ƒ

ƒ b ƒ - ƒ a ƒ

INTEGRATED REVIEW OPERATIONS ON REAL NUMBERSSections 1.1–1.6

Answer the following with positive, negative, or 0.

1. The opposite of a positive number is a number.

2. The sum of two negative numbers is a number.

3. The absolute value of a negative number is a number.

4. The absolute value of zero is .

5. The reciprocal of a positive number is a number.

6. The sum of a number and its opposite is .

7. The absolute value of a positive number is a number.

8. The opposite of a negative number is anumber.

Fill in the chart:

Number Opposite Absolute Value

9.

10.

11.

12.

Perform each indicated operation and simplify.

13. 14.

15. 16.

17. 18.

19. 20.

21. 22. -8.6 - 1.24.5 - 7.9

5 - 14-14 - (-12)

-2 + (-37)18 + (-25)

-8 - 10-15 + 17

7 - (-3)-19 + (-23)

911

-3

-

125

17

23.

24.

25.

26.

27.

28.

29.

30.

31.

32.

33.

34.

35. Subtract 5 from 1.

36. Subtract from .

37. Subtract from .

38. Subtract from

39.

40.

Evaluate each expression when and.

41. 42.

43. 44.

45. 46.ƒ -x - y + z ƒ

2z

ƒ5z - x ƒ

y - x

z - yy + z

x + yx - y

z = 9x = -2, y = -1,

3(10 - 9)2+ 6(20 - 19)3

2(19 - 17)3- 3(-7 + 9)2

-

58

.110

14

-

25

-3-2

-6 + [(-3 + 7) + (4 - 15)]

-7 + [(1 - 2) + (-2 - 9)]

ƒ -4 - 5 ƒ + 52+ (-50)

ƒ -9 ƒ + 32+ (-4 - 20)

92+ (10 - 30)

(7 - 17) + 42

30 - 5(10 - 8)

24 - 6(14 - 11)

11 - 20 + (-3) - 12

-9 - (-7) + 4 - 6

23

-

78

-

34

-

17

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Section 1.7 Multiplying and Dividing Real Numbers 49

OBJECTIVE 1 � Multiplying and dividing real numbers. In this section, we discoverpatterns for multiplying and dividing real numbers. To discover sign rules for multipli-cation, recall that multiplication is repeated addition. Thus means that 2 is anaddend 3 times. That is,

which equals 6. Similarly, means is an addend 3 times. That is,

Since then This suggests that the productof a positive number and a negative number is a negative number.

What about the product of two negative numbers? To find out, consider the fol-lowing pattern.

3 # 1-22 = -6.1-22 + 1-22 + 1-22 = -6,

1-22 + 1-22 + 1-22 = 3 # 1-22-23 # 1-22

2 + 2 + 2 = 3 # 2

3 # 2

Multiplying Real Numbers1. The product of two numbers with the same sign is a positive number.

2. The product of two numbers with different signs is a negative number.

OBJECTIVES

1 Multiply and divide real numbers.

2 Evaluate algebraic expressionsusing real numbers.

1.7 MULTIPLYING AND DIVIDING REAL NUMBERS

-3 #-1 = 3

-3 #-2 = 6

Product increases by 3 each time.

Factor decreases by 1 each time

Factor decreases by 1 each time

-3 # 2 = -6-3 # 1 = -3-3 # 0 = 0

Product increases by 3 each time.

This pattern continues as

This suggests that the product of two negative numbers is a positive number.

EXAMPLE 1 Multiply.

a. b. c.

Solution

a. b. c.

Multiply.

a. b. c. (-6)(9)(-3)(-4)8(-5)

1PRACTICE

-91-102 = 90141-12 = -14-8142 = -32

-91-102141-121-82142

We know that every whole number multiplied by zero equals zero. This remainstrue for real numbers.

Zero as a FactorIf b is a real number, then Also, 0 # b = 0.b # 0 = 0.

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Multiplying signed decimals or fractions is carried out exactly the same way asmultiplying by integers.

EXAMPLE 2 Perform the indicated operations.

a. b. c. d.

Solution

a. By the order of operations, we multiply from left to right. Notice that, because oneof the factors is 0, the product is 0.

b. Multiply two factors at a time, from left to right.

Multiply

c. Multiply from left to right.

Multiply

d. Follow the rules for order of operation.

Find each product.

Add 44 to the opposite of

Add.

Perform the indicated operations.

a. b.

c. d. (-2)(-3) - (-4)(5)(-4)(0)(5)

(-3)(-2)(4)(-1)(-5)(-6)

2PRACTICE

= 54

-10.= 44 + 10

1-421-112 - 1521-22 = 44 - 1-102

= 45

(-1)(5). 1-121521-92 = 1-521-92

= -24

(-2)(-3). 1-221-321-42 = 1621-42

1721021-62 = 01-62 = 0

1-421-112 - 1521-221-121521-921-221-321-421721021-62

◗ Helpful HintYou may have noticed from the example that if we multiply:

• an even number of negative numbers, the product is positive.

• an odd number of negative numbers, the product is negative.

EXAMPLE 3 Multiply.

a. b. c.

Solutiona. The product of two numbers with different signs is negative.

b.

c. a -

45b1-202 =

4 # 205 # 1

=

4 # 4 # 55 # 1

=

161

or 16

23

# a -

710b = -

2 # 73 # 10

= -

2 # 73 # 2 # 5

= -

715

= -0.06

1-1.2210.052 = -[11.2210.052]

a -

45b1-2022

3# a -

710b1-1.2210.052

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Multiply.

a. b. c. a -

712b(-24)a -

35b # a4

9b(0.23)(-0.2)

3PRACTICE

Now that we know how to multiply positive and negative numbers, let’s see howwe find the values of and for example.Although these two expressions looksimilar, the difference between the two is the parentheses. In the parenthesestell us that the base, or repeated factor, is In only 4 is the base. Thus,

The base is

The base is 4. -42= -14 # 42 = -16

-4. 1-422 = 1-421-42 = 16

-42,-4.1-422,-42,1-422

EXAMPLE 4 Evaluate.

a. b. c. d.

Solution

a. The base is

b. The base is 2.

c. The base is

d. The base is 3.

Evaluate.

a. b. c. d. -43(-4)3-62(-6)2

4PRACTICE

-32= -13 # 32 = -9

-3.1-322 = 1-321-32 = 9

-23= -12 # 2 # 22 = -8

-2.1-223 = 1-221-221-22 = -8

-321-322-231-223

Just as every difference of two numbers can be written as the sumso too every quotient of two numbers can be written as a product. For example, the

quotient can be written as Recall that the pair of numbers 3 and has a

special relationship. Their product is 1 and they are called reciprocals or multiplicativeinverses of each other.

13

6 #13

.6 , 3

a + 1-b2,a - b

◗ Helpful HintBe careful when identifying the base of an exponential expression.

-32

Base is 3-32

= -13 # 32 = -9

1-322Base is -3

1-322 = 1-321-32 = 9

Notice that 0 has no multiplicative inverse since 0 multiplied by any number is never 1but always 0.

Reciprocals or Multiplicative InversesTwo numbers whose product is 1 are called reciprocals or multiplicative inverses ofeach other.

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We may now write a quotient as an equivalent product.

EXAMPLE 5 Find the reciprocal of each number.

a. 22 b. c. d.

Solutiona. The reciprocal of 22 is since

b. The reciprocal of is since

c. The reciprocal of is

d. The reciprocal of is

Find the reciprocal of each number.

a. b. 15 c. d. -5-

27

83

5PRACTICE

-

139

.-

913

-

110

.-10

316

#163

= 1.163

316

22 #122

= 1.122

-

913

-10316

In other words, the quotient of two real numbers is the product of the first number andthe multiplicative inverse or reciprocal of the second number.

Since the quotient can be written as the product it follows that sign

patterns for dividing two real numbers are the same as sign patterns for multiplyingtwo real numbers.

a #1b

,a , b

EXAMPLE 6 Use the definition of the quotient of two numbers to divide.

a. b. c.

Solution

a. b.

c.

Use the definition of the quotient of two numbers to divide.

a. b. c.-35-7

24 , (-6)16-2

6PRACTICE

20-4

= 20 #-

14

= -5

-14-2

= -14 #-

12

= 7-18 , 3 = -18 #13

= -6

20-4

-14-2

-18 , 3

Multiplying and Dividing Real Numbers1. The product or quotient of two numbers with the same sign is a positive number.

2. The product or quotient of two numbers with different signs is a negativenumber.

Quotient of Two Real NumbersIf a and b are real numbers and b is not 0, then

a , b =

a

b= a #

1b

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The definition of the quotient of two real numbers does not allow for division by0 because 0 does not have a multiplicative inverse.There is no number we can multiply

0 by to get 1. How then do we interpret We say that division by 0 is not allowed or

not defined and that does not represent a real number.The denominator of a fraction can never be 0.

Can the numerator of a fraction be 0? Can we divide 0 by a number? Yes. Forexample,

In general, the quotient of 0 and any nonzero number is 0.

03

= 0 #13

= 0

30

30

?

EXAMPLE 7 Divide.

a. b. c. d.

Solution

a. b. c.

d.

Divide.

a. b. c. d. -

49

, 835

, a -

12b-48

3-18-6

7PRACTICE

-

32

, 9 = -

32

#19

= -

3 # 12 # 9

= -

3 # 12 # 3 # 3

= -

16

23

, a -

54b =

23

# a -

45b = -

815

-363

= -12-24-4

= 6

-

32

, 923

, a -

54b-36

3-24-4

Zero as a Divisor or Dividend1. The quotient of any nonzero real number and 0 is undefined. In symbols, if

is undefined.

2. The quotient of 0 and any real number except 0 is 0. In symbols, if a Z 0, 0a

= 0.

a Z 0, a

0

EXAMPLE 8 Perform the indicated operations.

a. b. c.

Solution

a. is undefined b. c.


a. b. c.-5

6(0)-40

0-2

8PRACTICE

01-822

=

02

= 00

-3= 0

10

01-822

0-3

10

Notice that and This means that

12-2

= -

122

=

-122

-122

= -6.12-2

= -6, -122

= -6,

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In words, a single negative sign in a fraction can be written in the denominator, inthe numerator, or in front of the fraction without changing the value of the fraction.Thus,

1-7

=

-17

= -

17

In general, if a and b are real numbers, b Z 0, a

-b=

-a

b= -

a

b.

Examples combining basic arithmetic operations along with the principles oforder of operations help us to review these concepts.


a. b.

Solutiona. First, simplify the numerator and denominator separately, then divide.

b. Simplify the numerator and denominator separately, then divide.


a. b.3(-2)3

- 9-6 + 3

(-8)(-11) - 4-9 - (-4)

9PRACTICE

21-322 - 20

-5 + 4=

2 # 9 - 20-5 + 4

=

18 - 20-5 + 4

=

-2-1

= 2

=

39-5

or -395

1-1221-32 + 3

-7 - 1-22 =

36 + 3-7 + 2

21-322 - 20

-5 + 4

1-1221-32 + 3

-7 - 1-22

OBJECTIVE 2 � Evaluating algebraic expressions using real numbers. Using what wehave learned about multiplying and dividing real numbers, we continue to practiceevaluating algebraic expressions.

EXAMPLE 10 If and evaluate each expression.

a. b. c.

Solution

a. Replace x with and y with and simplify.

b. Replace x with and y with

Substitute the given values for the variables.

Evaluate exponential expressions.

Subtract.

c. Replace x with and y with and simplify.

3x

2y=

31-2221-42 =

-6-8

=

34

-4-2

= 0

= 16 - 1162 x4

- y2= 1-224 - 1-422

-4.-2

5x - y = 51-22 - 1-42 = -10 - 1-42 = -10 + 4 = -6

-4-2

3x

2yx4

- y25x - y

y = -4,x = -2

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Calculator Explorations

Entering Negative Numbers on a Scientific Calculator

To enter a negative number on a scientific calculator, find a key marked(On some calculators, this key is marked for “change sign.”) To enter

for example, press the keys The display will read

Entering Negative Numbers on a Graphing Calculator

To enter a negative number on a graphing calculator, find a key marked

Do not confuse this key with the key which is used for subtraction. To enter

for example, press the keys The display will read

Operations with Real Numbers

To evaluate on a calculator, press the keys

or

The display will read or .

Use a calculator to simplify each expression.

1. 2.

3. 4.

5. 6.

7. 8.9. (Be careful.) 10. (Be careful.)-12521-12522

58- 625995

- 4550

-444 - 444.8-181 - 324

-5012942175 - 265

451322 - 812182134 + 25168 - 912-591-82 + 1726-38126 - 272

� -217 - 92 - 20-16

�� -16 �

� 1-2 �� 2 �� 1 �� 7 �� - �� 9 �� 2 �� - �� 2 �� 0 �� ENTER �.� 2 �� + /- �� * �� 1 �� 7 �� - �� 9 �� 2 �� - �� 2 �� 0 �� = �,

-217 - 92 - 20

� -8 �.� 1-2 � � 8 �.-8,

� - �,� 1-2 �.

� -8 �.� 8 � � + /- �.-8,� CHS �

� + /- �.


positive 0 negative undefined

1. If n is a real number, then and .

2. If n is a real number, but not 0, then and we say is .n

0

0n

=

0 # n = n # 0 =

If and , evaluate each expression.

a. b. c.2x

3yx2

- y37y - x

y = -2x = -510PRACTICE


3. The product of two negative numbers is a number.

4. The quotient of two negative numbers is a number.

5. The quotient of a positive number and a negative number is a number.

6. The product of a positive number and a negative number is a number.

7. The reciprocal of a positive number is a number.

8. The opposite of a positive number is a number.

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1.7 EXERCISE SET

Multiply. See Examples 1 through 3.

1. 2.

3. 4.

5. 6.

7. 8.

9. 10.

11. 12.

13. 14.

15. 16.

17. 18.

19. 20.

21. 22.

23. 24.

25. 26.

27. 28.

29. 30.

31. 32.

33. 34.

Perform the indicated operations. See Example 2.

35. 36.

37. 38.

Decide whether each statement is true or false.

39. The product of three negative integers is negative.

40. The product of three positive integers is positive.

41. The product of four negative integers is negative.

42. The product of four positive integers is positive.

Evaluate. See Example 4.

43. 44.

45. 46.

47. 48.

49. 50.

Find each reciprocal or multiplicative inverse. See Example 5.

51. 9 52. 100 53.

54. 55. 56.

57. 58. 59. 0.2

60. 1.5 61. 62.1

-8.91

-6.3

-

613

-

311

-8-1417

23

1-722-72

-521-5221-125-15

-241-224

20 - 1-421321-221-621-121-22 - 1-5281-32 - 41-521-22152 - 1-112132

1-221-321-421-221-121221-321-52-

2536

a 615b-

2025

a 516b

-121122-11111227

a-

211b2

3 a-

49b

1-721-721-521-5251-9241-72-201602-101802-0.51-0.32-0.21-0.7261-2.5251-1.42-

56a-

310b-

34a-

89b

-

18a-

13b-

12a-

35b

31-5221-92-6 # 0-7 # 0

-2 # 8-3 # 4

-61-112-51-10271-4221-12-8152-6142

Divide. See Examples 6 through 8.

63. 64. 65.

66. 67. 68.

69. 70. 71.

72. 73. 74.

75. 76. 77.

78. 79. 80.

81. 82.

83. 84.

MIXED PRACTICE

Simplify. See Example 9.

85. 86.

87. 88.

89. 90.

91. 92.

93. 94.

95. 96.

97. 98.

99. 100.

101. 102.

If and evaluate each expression. See Example 10.

103. 104.

105. 106.

107. 108.

109. 110.

111. 112.

113. At the end of 2006, Delta Airlines posted a net loss of$6203 million, which we will write as Ifthis continues, what will Delta’s income be after four years?(Source: Delta Airlines)

- $6203 million.

4 - 2x

y + 3

-3 - y

x - 4

2y - 12

x - 42x - 5y - 2

y3+ 3xx3

+ 3y

x2- 2y22x2

- y2

4x + 5y3x + 2y

y = -3,x = -5

ƒ -3 + 6 ƒ + ƒ -2 + 7 ƒ

ƒ -2 # 2 ƒ

ƒ 5 - 9 ƒ + ƒ 10 - 15 ƒ

ƒ 21-32 ƒ

-4 - 81-22-9 - 21-32

-3 - 21-92-15 - 31-42

8 - 31-222 - 51-42

6 - 21-324 - 31-22

-2 - 42

31-62-3 - 52

21-72

-20 + 1-421321 - 5

22 + 1321-22-5 - 2

6 + 1-2224 - 9

8 + 1-4224 - 12

32+ 45

-62+ 4

-2

-151 - 4

129 - 12

-61-32-4

-91-32-6

-

512

,

512

-

49

,

49

-

110

, a-

811b-

59

, a-

34b

45

, a-

12b6

7, a-

13b14

-2

30-2

-45-9

-12-4

30

50

-

248

-

153

0-9

0-4

-605

-4812

-18-6

-16-4

20-10

18-2


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THE BIGGER PICTURE SIMPLIFYING EXPRESSIONS


114. At the end of the third quarter of 2006, General Motorsreported a net loss of $115 million. If this continued, whatwould General Motor’s income be after four more quarters?(Source: General Motors)








CONCEPT EXTENSIONS

121. Explain why the product of an even number of negativenumbers is a positive number.

2x + 4 = x + 8?-4

-3x - 5 = -20?

45x

= -15?-3

x

10= 2?-20

2x = x - 1?-4

-5x = -35?

122. If a and b are any real numbers, is the statement always true? Why or why not?

123. Find any real numbers that are their own reciprocal.

124. Explain why 0 has no reciprocal.

If q is a negative number, r is a negative number, and t is a positivenumber, determine whether each expression simplifies to a positiveor negative number. If it is not possible to determine, state so.

125. 126.

127. 128.

129. 130.

Write each of the following as an expression and evaluate.

131. The sum of and the quotient of and 3

132. The sum of 1 and the product of and

133. Twice the sum of and

134. 7 subtracted from the quotient of 0 and 5

-3-5

-5-8

-15-2

r1q - t2t1q + r2t + rq + t

q2 # r # tq

r # t

a # b = b # a

This is a special feature that we introduce in this section.Among other concepts introduced later in this text, it isvery important for you to be able to simplify expressionsand solve equations and to know the difference betweenthe two. To help with this, we began an outline below andexpand this outline throughout the text. Although sugges-tions are given, this outline should be in your own words.Once you complete the new portion of your outline, try theexercises to the right. Remember: Study your outline oftenas you proceed through this text.

I. Simplifying Expressions

A. Real Numbers

1. Add:

-1.7 + (-0.21) = -1.91

3. Multiply or Divide: Multiply or divide as usual. Ifthe signs of the two numbers are the same, theanswer is positive. If the signs of the two numbersare different, the answer is negative.


1. 2.

3. 4.

5. 6.

7. 8.

9. 10.

11. 12.

13. 14.

15. 16. -23(1 - 5) - (7 - 17)420 , 2 # 5

-2 12

, a -3 14b2 + 3(8 - 11)3

-

89a -

316b-8.3 - 8.3

-25 - (-13)0

8.6

8.60

|-42|- |-2|

-72(-7)2

-40-5

-

17

+ a -

35b

86 - 100-0.2(25)

-

38

#711

= -

2188

, -42 , (-10) = 4.2

Adding like signs.Add absolute values.Attach the common sign.

Adding unlike signs.Subtract absolute values. Attach the sign of the number with the larger absolute value.

-7 + 3 = -4

2. Subtract: Add the first number to the opposite ofthe second number.17

-

13

=

321

+ a -

721b = -

421

STUDY SKILLS BUILDER

Answer the following.

1. Name one way you might place a word and its defini-tion on a card.

2. How do new terms stand out in this text so that theycan be found?

3 * 5

Learning New Terms

Many of the terms used in this text may be new to you. Itwill be helpful to make a list of new mathematical terms andsymbols as you encounter them and to review them fre-quently. Placing these new terms (including page refer-ences) on index cards might help you later whenyou’re preparing for a quiz.

3 * 5

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Concept CheckWhich of the following pairs of actions are commutative?

a. “raking the leaves” and “bagging the leaves”b. “putting on your left glove” and “putting on your right glove”c. “putting on your coat” and “putting on your shirt”d. “reading a novel” and “reading a newspaper”

Answers to Concept Check:b, d


OBJECTIVE 1 � Using the commutative and associative properties. In this sectionwe give names to properties of real numbers with which we are already familiar.Throughout this section, the variables a, b, and c represent real numbers.

We know that order does not matter when adding numbers. For example,we know that is the same as This property is given a special name—thecommutative property of addition. We also know that order does not matter whenmultiplying numbers. For example, we know that This propertymeans that multiplication is commutative also and is called the commutative propertyof multiplication.

-5162 = 61-52.5 + 7.7 + 5

OBJECTIVES

1 Use the commutative andassociative properties.

2 Use the distributive property.

3 Use the identity and inverseproperties.

1.8 PROPERTIES OF REAL NUMBERS

Commutative Properties

Addition:

Multiplication: a # b = b # a

a + b = b + a

These properties state that the order in which any two real numbers are addedor multiplied does not change their sum or product. For example, if we let and

then the commutative properties guarantee that

3 + 5 = 5 + 3 and 3 # 5 = 5 # 3

b = 5,a = 3

◗ Helpful HintIs subtraction also commutative? Try an example. Does No! The left side ofthis statement equals 1; the right side equals There is no commutative property of sub-traction. Similarly, there is no commutative property for division. For example, doesnot equal 2 , 10.

10 , 2-1.

3 - 2 = 2 - 3?

EXAMPLE 1 Use a commutative property to complete each statement.

a. b.

Solutiona. By the commutative property of addition

b. By the commutative property of multiplication

Use a commutative property to complete each statement.

a. b. x + 17 = x # 8 =

1PRACTICE

3 # x = x # 3

x + 5 = 5 + x

3 # x = x + 5 =

Let’s now discuss grouping numbers. We know that when we add three numbers,the way in which they are grouped or associated does not change their sum. For

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Section 1.8 Properties of Real Numbers 59

example, we know that This result is the same ifwe group the numbers differently. In other words, also.Thus, This property is called the associative property ofaddition.

We also know that changing the grouping of numbers when multiplying does notchange their product. For example, (check it). This is theassociative property of multiplication.

2 # 13 # 42 = 12 # 32 # 4

= 12 + 32 + 4.2 + 13 + 42 12 + 32 + 4 = 5 + 4 = 9,2 + 13 + 42 = 2 + 7 = 9.

These properties state that the way in which three numbers are grouped does notchange their sum or their product.

Associative Properties

Addition:

Multiplication: 1a # b2 # c = a # 1b # c2 1a + b2 + c = a + 1b + c2

◗ Helpful HintRemember the difference between the commutative properties and the associative proper-ties. The commutative properties have to do with the order of numbers, and the associativeproperties have to do with the grouping of numbers.

EXAMPLE 2 Use an associative property to complete each statement.

a. b.

Solutiona. By the associative property of addition

b. By the associative property of multiplication

Use an associative property to complete each statement.

a. b. -4 # (2 # 7) = 12 + 92 + 7 =

2PRACTICE

1-1 # 22 # 5 = -1 # 12 # 525 + 14 + 62 = 15 + 42 + 6

1-1 # 22 # 5 = 5 + 14 + 62 =

Let’s now illustrate how these properties can help us simplify expressions.


a. b.

Solution

a. By the commutative property of addition

By the associative property of addition

Add.

b. By the associative property of multiplication

Multiply.


a. b. 5(-6x)(5 + x) + 9

3PRACTICE

= -21x

-317x2 = 1-3 # 72x = 22 + x

= 110 + 122 + x

10 + 1x + 122 = 10 + 112 + x2

-317x210 + 1x + 122

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5(x+3y-z)=5(x)+5(3y)-5(z)

–5(–3+2z)=–5(–3)+(–5)(2z)

2(x+y)=2 x+2# # y

Solution

a.

b.

c.

= 5x + 15y - 5y

= 15 - 10z

= 2x + 2y


OBJECTIVE 2 � Using the distributive property. The distributive property of multi-plication over addition is used repeatedly throughout algebra. It is useful because itallows us to write a product as a sum or a sum as a product.

We know that Compare that with Since both original expressions equal 42, they must equal each other, or= 42.14 + 28

7122 + 7142 =712 + 42 = 7162 = 42.

7(2+4)=7(2)+7(4)

This is an example of the distributive property.The product on the left side of the equalsign is equal to the sum on the right side. We can think of the 7 as being distributed toeach number inside the parentheses.

Distributive Property of Multiplication Over Addition

(b+c)a=ba+ca

Since multiplication is commutative, this property can also be written as

The distributive property can also be extended to more than two numbers insidethe parentheses. For example,

2(x-y)=2(x)-2(y)

3(x+y+z)=3(x)+3(y)+3(z)

a(b+c)=ab+ac

Since we define subtraction in terms of addition, the distributive property is also truefor subtraction. For example

= 2x - 2y

= 3x + 3y + 3z

EXAMPLE 4 Use the distributive property to write each expression withoutparentheses. Then simplify if possible.

a. b. c.d. e. f. 413x + 72 + 10-13 + x - w2-112 - y2

51x + 3y - z2-51-3 + 2z221x + y2

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◗ Helpful HintNotice in part (e) that is first rewritten as -113 + x - w2.

-13 + x - w2

We can use the distributive property in reverse to write a sum as a product.

EXAMPLE 5 Use the distributive property to write each sum as a product.

a. b.

Solution

a. b.

Use the distributive property to write each sum as a product.

a. b. 9w + 9z5 # w + 5 # 3

5PRACTICE

7s + 7t = 71s + t28 # 2 + 8 # x = 812 + x2

7s + 7t8 # 2 + 8 # x

OBJECTIVE 3 � Using the identity and inverse properties. Next, we look at the identityproperties.

The number 0 is called the identity for addition because when 0 is added to anyreal number, the result is the same real number. In other words, the identity of the realnumber is not changed.

The number 1 is called the identity for multiplication because when a real num-ber is multiplied by 1, the result is the same real number. In other words, the identity ofthe real number is not changed.

d.

e.

f. Apply the distributive property.

Multiply.

Add.

Use the distributive property to write each expression without parentheses.Then simplify, if possible.

a. b.

c. d.

e. f. 2(7x + 4) + 6-(x - 7 + 2s)

(3 - y) # (-1)2(3x - 4y - z)

-6(4 + 2t)5(x - y)

4PRACTICE

= 12x + 38

= 12x + 28 + 10

413x + 72 + 10 = 413x2 + 4172 + 10

= -3 - x + w

= 1-12132 + 1-121x2 - 1-121w2 -13 + x - w2 = -113 + x - w2

= -2 + y

-112 - y2 = 1-12122 - 1-121y2

Identities for Addition and Multiplication0 is the identity element for addition.

1 is the identity element for multiplication.

a # 1 = a and 1 # a = a

a + 0 = a and 0 + a = a

Notice that 0 is the only number that can be added to any real number with theresult that the sum is the same real number.Also, 1 is the only number that can be mul-tiplied by any real number with the result that the product is the same real number.

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Additive inverses or opposites were introduced in Section 1.5. Two numbers arecalled additive inverses or opposites if their sum is 0. The additive inverse or oppositeof 6 is because The additive inverse or opposite of is 5 because

Reciprocals or multiplicative inverses were introduced in Section 1.3.Two nonzeronumbers are called reciprocals or multiplicative inverses if their product is 1.The reciprocal

or multiplicative inverse of is because Likewise, the reciprocal of is

because -5 A - 15B = 1.-

15

-523

#32

= 1.32

23

-5 + 5 = 0.-56 + 1-62 = 0.-6

Concept Check

Which of the following, is the

a. opposite of ? b. reciprocal of ? -

310

-

310

-

310

,103

,310

, 0,-

103

,1,

Additive or Multiplicative InversesThe numbers a and are additive inverses or opposites of each other becausetheir sum is 0; that is,

The numbers b and (for ) are reciprocals or multiplicative inverses of each

other because their product is 1; that is,

b #1b

= 1

b Z 01b

a + 1-a2 = 0

-a

EXAMPLE 6 Name the property or properties illustrated by each truestatement.

Solution

a. Commutative property of multiplication (order changed)

b. Associative property of addition (grouping changed)

c. Identity element for addition

d. Commutative property of multiplication (order changed)

e. Multiplicative inverse property

f. Additive inverse property

g. Commutative and associative properties of multiplication(order and grouping changed)

Name the property or properties illustrated by each true statement.

a.

b.

c.

d.

e.

f.

g. (-3b) # 7 = (-3 # 7) # b

3 + (-3) = 0

(2 + x) + 5 = 5 + (2 + x)

-

34

# a -

43b = 1

8 + (t + 0) = 8 + t

6 + (3 + y) = (6 + 3) + y

(7 # 3x) # 4 = (3x # 7) # 4

6PRACTICE

-6 # 1y # 22 = 1-6 # 22 # y

-2 + 2 = 0

-2 # a -

12b = 1

0.2 # 1z # 52 = 0.2 # 15 # z21b + 02 + 3 = b + 3

1x + 72 + 9 = x + 17 + 923 # y = y # 3


a. b. -

103

310

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distributive property associative property of multiplication commutative property of addition

opposites or additive inverses associative property of addition

reciprocals or multiplicative inverses commutative property of multiplication

1. is a true statement by the .


3. is true by the .



6. The numbers and are called .

7. The numbers and are called .23

-

23

-

32

-

23

x + (7 + y) = (x + 7) + y

2 # (x # y) = (2 # x) # y

3(y + 6) = 3 # y + 3 # 6

x # 5 = 5 # x

x + 5 = 5 + x


Use a commutative property to complete each statement. SeeExample 1.

1. 2.

3. 4.

5. 6.

7. 8.

Use an associative property to complete each statement. SeeExample 2.

9. 10.

11. 12.

13. 14.

15.

16.

Use the commutative and associative properties to simplify eachexpression. See Example 3.

17. 18.

19. 4(6y) 20. 2(42x)

21. 22.

23. 24.

25. 26.

27. 28.

29. Write an example that shows that division is not commu-tative.

30. Write an example that shows that subtraction is not commu-tative.

Use the distributive property to write each expression withoutparentheses. Then simplify the result. See Example 4.

31. 32. 71a + b241x + y2

27

a 72

rb34

a43

sb-3112y2-918x27 + 1x + 42113 + a2 + 13

18

18z215

15y2

1r + 32 + 118 + 19 + b2

6 + 1r + s2 =

1a + b2 + c =

1-3y2 # z =4 # 1ab2 =

1y + 42 + z = 2 + 1a + b2 =

3 # 1xy2 =1xy2 # z =

19 + 3y =2x + 13 =

ab =xy =

-2 # x =-4 # y =

4 + y =x + 16 =

33. 34.

35. 36.

37. 38.

39. 40.

41. 42.

43. 44.

45.

46.

47.

48.

49.

50.

51.

52.

53.

54.

55.

56.

57.

58.

59.

60.

61.

62. -612x + 12 - 1

-414x + 52 - 5

-1115x + 32 + 10

-914x + 82 + 2

1014s + 62 - 40

312r + 52 - 7

-

15

110a - 25b2-

13

13x - 9y2

14

14x - 22

12

16x + 82-1q - 2 + 6r2-1r - 3 - 7p2-19r + 52-15x + 22-414 + 2p + 52-411 - 2m + n2813y + z - 6251x + 4m + 22

-512r + 112-713y + 52-31z - y2-21y - z221x + 52316 + x2318x - 12714x - 32517 + 8y2213x + 52111y - 4291x - 62

1.8 EXERCISE SET

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Use the distributive property to write each sum as a product. SeeExample 5.

63. 64.

65. 66.

67. 68.

69. 70.

Name the properties illustrated by each true statement. SeeExample 6.

71.

72.

73.

74.

75.

76.

77.

78.

79.

80.

81.

82.

83.

84. r + 0 = r

-4 # 18 # 32 = 18 #-42 # 3

111 + r2 + 8 = 1r + 112 + 8

-41y + 72 = -4 # y + 1-42 # 7

1a + 92 + 6 = a + 19 + 620 + 6 = 6

6 #16

= 1

14 # y2 # 9 = 4 # 1y # 921 # 9 = 9

913 + 72 = 9 # 3 + 9 # 7

1x + 92 + 3 = 19 + x2 + 3

2 + 1x + 52 = 12 + x2 + 5

413 + 82 = 4 # 3 + 4 # 8

3 # 5 = 5 # 3

25x + 25y30a + 30b

1-32a + 1-32b1-12 # 5 + 1-12 # x

9a + 9b11x + 11y

14 # z + 14 # 54 # 1 + 4 # y

CONCEPT EXTENSIONS

Fill in the table with the opposite (additive inverse), and the reci-procal (multiplicative inverse). Assume that the value of eachexpression is not 0.

Expression Opposite Reciprocal

85. 8

86.

87. x

88. 4y

89.

90. 7x

Determine which pairs of actions are commutative.

91. “taking a test” and “studying for the test”

92. “putting on your shoes” and “putting on your socks”

93. “putting on your left shoe”and“putting on your right shoe”

94. “reading the sports section”and“reading the comics section”

95. Explain why 0 is called the identity element for addition.

96. Explain why 1 is called the identity element for multiplication.

12x

-

23

Sections 1.3, 1.4, 1.5Magic SquaresA magic square is a set of numbers arranged in a square table sothat the sum of the numbers in each column, row, and diagonal isthe same. For instance, in the magic square below, the sum of eachcolumn, row, and diagonal is 15. Notice that no number is usedmore than once in the magic square.

2 9 4

7 5 3

6 1 8

CHAPTER 1 GROUP ACTIVITYtitled Melencolia I, created by German artist Albrecht Dürer in1514, features the following four-by-four magic square on thebuilding behind the central figure.

16 3 2 13

5 10 11 8

9 6 7 12

4 15 14 1

The properties of magic squares have been known for avery long time and once were thought to be good luck charms.Theancient Egyptians and Greeks understood their patterns. A magicsquare even made it into a famous work of art. The engraving

Group Exercises1. Verify that what is shown in the Dürer engraving is, in fact, a

magic square. What is the common sum of the columns, rows,and diagonals?

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Chapter 1 Highlights 65

2. Negative numbers can also be used in magic squares. Completethe following magic square:

3. Use the numbers 0, 3, 6, 9, and 12 to form amagic square.

-12, -9, -6, -3,

CHAPTER 1 VOCABULARY CHECK

Fill in each blank with one of the words or phrases listed below.set inequality symbols opposites absolute value numeratordenominator grouping symbols exponent base reciprocalsvariable equation solution

1. The symbols and are called .

2. A mathematical statement that two expressions are equal is called an .

3. The of a number is the distance between that number and 0 on the number line.

4. A symbol used to represent a number is called a .

5. Two numbers that are the same distance from 0 but lie on opposite sides of 0 are called .

6. The number in a fraction above the fraction bar is called the .

7. A of an equation is a value for the variable that makes the equation a true statement.

8. Two numbers whose product is 1 are called .

9. In the 2 is called the and the 3 is called the .

10. The number in a fraction below the fraction bar is called the .

11. Parentheses and brackets are examples of .

12. A is a collection of objects.

23,

7Z , 6 ,

◗ Helpful HintAre you preparing for your test? Don’t forget to takethe Chapter 1 Test on page 72.Then check your answersat the back of the text and use the Chapter Test PrepVideo CD to see the fully worked-out solutions to any ofthe exercises you want to review.

CHAPTER 1 HIGHLIGHTS

Irrational Numbers: {real numbers that cannot beexpressed as a quotient of integers}

Rational Numbers: {real numbers that can be expressedas a quotient of integers}

Whole Numbers:

Integers: 5Á , -3, -2, -1, 0, 1, 2, 3, Á 650, 1, 2, 3, 4, Á 6

SECTION 1.2 SYMBOLS AND SETS OF NUMBERS

DEFINITIONS AND CONCEPTS EXAMPLES

5a, c, e6Natural Numbers: 51, 2, 3, 4, Á 6 Given the set list the numbers

that belong to the set of

e -3.4, 23, 0, 23

, 5, -4 f ,

Natural numbers: 5

Rational numbers:

Irrational Numbers: (continued)13

-4, -3.4, 0, 23

, 5

Whole numbers: 0, 5Integers: -4, 0, 5

A set is a collection of objects, called elements, enclosedin braces.

0 -4

-1

-2

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The absolute value of a real number a, denoted by isthe distance between a and 0 on the number line.

ƒ a ƒ ,

A quotient of two integers is called a fraction. Thenumerator of a fraction is the top number. Thedenominator of a fraction is the bottom number.

If then a and b are factors and c is the product.a # b = c,

A fraction is in lowest terms when the numerator and thedenominator have no factors in common other than 1.

To write a fraction in lowest terms, factor the numerator andthe denominator; then apply the fundamental principle.

To add fractions with the same denominator, add thenumerators and place the sum over the commondenominator.

Fractions that represent the same quantity are calledequivalent fractions.

To subtract fractions with the same denominator,subtract the numerators and place the difference overthe common denominator.

To divide fractions, multiply the first fraction by thereciprocal of the second fraction.

The reciprocal of is

To multiply fractions, numerator times numerator isthe numerator of the product and denominator timesdenominator is the denominator of the product.

ba

.a

b

Two fractions are reciprocals if their product is 1.

Order Property for Real Numbers

For any two real numbers a and b, a is less than b if a isto the left of b on a number line.

SECTION 1.2 SYMBOLS AND SETS OF NUMBERS (continued)

SECTION 1.3 FRACTIONS


ƒ 5 ƒ = 5 ƒ 0 ƒ = 0 ƒ -2 ƒ = 2

1317

; numerator; denominator

7 # 9 = 63T T T

factor factor product

is in lowest terms.1317

Write in lowest terms.

614

=

2 # 32 # 7

=

37


25

#37

=

635

1315

-

315

=

1015

=

23

and are equivalent fractions.420

15

15

=

1 # 45 # 4

=

420

511

+

311

=

811

59

,

27

=

59

#72

=

3518

The reciprocal of .625

is 256

�3 � 0 0 � �3 0 � 2.5 2.5 � 0

0 2�2 3�3 1�1

A line used to picture numbers is called a number line.

18 Ú -

13

Ú is greater than or equal to

6 … 6 … is less than or equal to

1 6 4 6 is less than

4 7 1 7 is greater than

3 Z -3 Z is not equal to

-7 = -7Symbols: = is equal to

Real Numbers: {all numbers that correspond to a pointon the number line}

0 2�2 3�3 1�1

Real numbers: -4, -3.4, 0, 23

, 13, 5

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Chapter 1 Highlights 67

The expression is an exponential expression. Thenumber a is called the base; it is the repeated factor.

an

A symbol used to represent a number is called a variable.

1. Simplify exponential expressions.

2. Multiply or divide in order from left to right.

3. Add or subtract in order from left to right.

The number n is called the exponent; it is the numberof times that the base is a factor.

Order of Operations

Simplify expressions in the following order. Ifgrouping symbols are present, simplify expressionswithin those first, starting with the innermost set.Also, simplify the numerator and the denominatorof a fraction separately.

To evaluate an algebraic expression containing avariable, substitute a given number for the variable andsimplify.

A mathematical statement that two expressions areequal is called an equation.

To Add Two Numbers with the Same Sign

To Add Two Numbers with Different Signs

1. Subtract their absolute values.

2. Use the sign of the number whose absolute valueis larger as the sign of the sum.

1. Add their absolute values.

2. Use their common sign as the sign of the sum.

A solution of an equation is a value for the variable thatmakes the equation a true statement.

An algebraic expression is a collection of numbers,variables, operation symbols, and grouping symbols.

SECTION 1.4 INTRODUCTION TO VARIABLE EXPRESSIONS AND EQUATIONS

SECTION 1.5 ADDING REAL NUMBERS


72= 7 # 7 = 49

43= 4 # 4 # 4 = 64

= 4

=

8421

=

64 + 2021

=

64 + 514221

82

+ 517 - 323 # 7

=

82+ 514221

Equations:

A = pr2

3x - 9 = 20

Add.

14 + 1-92 = 5

-25 + 5 = -20

-3 + 1-82 = -11

10 + 7 = 17

Determine whether 4 is a solution of

True

4 is a solution.

27 = 27

20 + 7 � 27

5142 + 7 � 27

5x + 7 = 27

5x + 7 = 27.

Examples of algebraic expressions are:

5x, 21y - 62, q2- 3q + 1

6

Examples of variables are:

q, x, z

Evaluate if and

= 16 = 25 - 9

x2- y2

= 1522 - 32

y = 3.x = 5x2- y2

(continued)

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Two numbers that are the same distance from 0 but lieon opposite sides of 0 are called opposites or additiveinverses. The opposite of a number a is denoted by -a.

Commutative Properties

Addition:

Multiplication:

Associative Properties

Addition:

Multiplication: 1-3 # 22 # 11 = -3 # 12 # 1121a # b2 # c = a # 1b # c2 15 + 102 + 20 = 5 + 110 + 2021a + b2 + c = a + 1b + c2

-8 # 5 = 5 # 1-82a # b = b # a

3 + 1-72 = -7 + 3a + b = b + a

The quotient of 0 and any nonzero number is 0.

0b

= 0

The quotient of a nonzero number and 0 is undefined.

b

0 is undefined.

Products and Quotients Involving Zero

The product of 0 and any number is 0.

b # 0 = 0 and 0 # b = 0

Multiplying and Dividing Real Numbers

The product or quotient of two numbers with the samesign is a positive number. The product or quotient oftwo numbers with different signs is a negative number.

The sum of a number a and its opposite, is 0.

a + 1-a2 = 0

-a,

If a is a number, then -1-a2 = a.

To subtract two numbers a and b, add the first number ato the opposite of the second number b.

a - b = a + 1-b2

Quotient of two real numbers

a

b= a #

1b

SECTION 1.5 ADDING REAL NUMBERS (continued)

SECTION 1.6 SUBTRACTING REAL NUMBERS

SECTION 1.7 MULTIPLYING AND DIVIDING REAL NUMBERS


The opposite of is 7.The opposite of 123 is -123.

-7

12 + 1-122 = 0

-4 + 4 = 0

Subtract.

-30 - 1-302 = -30 + 30 = 0

-5 - 22 = -5 + 1-222 = -27

3 - 1-442 = 3 + 44 = 47

-1-142 = 14

-1-82 = 8

-4 # 0 = 0 0 # a - 34b = 0

-850

is undefined.

018

= 0 0

-47= 0

42-6

= -7 -42

6= -7

9010

= 9 -90-10

= 9

-2 # 4 = -8 2 # 1-42 = -8

7 # 8 = 56 -7 # 1-82 = 56

Multiply or divide.

422

= 42 #12

= 21

SECTION 1.8 PROPERTIES OF REAL NUMBERS

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Chapter 1 Review 69

(1.2) Insert the appropriate space to make thefollowing statements true.

1. 8 10 2. 7 2

3. 4.

5. 6.

7. 8.

9. 1.2 1.02 10. -

34

-

32

-1-142ƒ -14 ƒ-1- ƒ -1 ƒ

-9ƒ -9 ƒƒ -8 ƒƒ -7 ƒ

-8122

-5-4

6 , 7 , or = in

CHAPTER 1 REVIEW

Are You Preparing for a Test on Chapter 1?

Below I have listed some common trouble areas for topicscovered in Chapter 1. After studying for your test—butbefore taking your test—read these.

• Do you know the difference between and

(Section 1.2)

• Evaluate if and

(Sections 1.4 and 1.6)

• Make sure you are familiar with order of operations.Sometimes the simplest-looking expressions can give youthe most trouble.

(Section 1.4)

• Do you know the difference between and

(Section 1.7)1-322 = 9 and -32= -9

-32?1-3221 + 213 + 62 = 1 + 2192 = 1 + 18 = 19

x - y = 7 - 1-32 = 10

y = -3.x = 7x - y

ƒ -3 ƒ = 3; - ƒ -3 ƒ = -3; and -1-32 = 3

-1-32?ƒ -3 ƒ , - ƒ -3 ƒ ,

• Do you know that these fractions are equivalent?

(Section 1.7)

• Do you know the difference between opposite and recip-rocal? If not, study the table below.

(Sections 1.5 and 1.7)

-

13

=

-13

=

1-3

Remember: This is simply a checklist of selected topicsgiven to check your understanding. For a review of Chapter 1in the text, see the material at the end of Chapter 1.

Number Opposite Reciprocal

5

-313

-

13

-

74

47

-

47

15

-5

Two numbers whose product is 1 are calledmultiplicative inverses or reciprocals. The reciprocal

of a nonzero number a is because a #1a

= 1.1a

Distributive Property a1b + c2 = a # b + a # c

Identities

a # 1 = a 1 # a = a

a + 0 = a 0 + a = a

InversesAddition or opposite: a + 1-a2 = 0

Multiplication or reciprocal: b #1b

= 1

SECTION 1.8 PROPERTIES OF REAL NUMBERS (continued)


The reciprocal of 3 is

The reciprocal of is -52

.-

25

13

.

-213 + x2 = -2 # 3 + 1-221x2 516 + 102 = 5 # 6 + 5 # 10

7 + 1-72 = 0

-14 # 1 = -14 1 # 27 = 27

5 + 0 = 5 0 + 1-22 = -2

3 #13

= 1

STUDY SKILLS BUILDER

Translate each statement into symbols.

11. Four is greater than or equal to negative three.

12. Six is not equal to five.

13. 0.03 is less than 0.3.

14. New York City has 155 museums and 400 art galleries. Writean inequality comparing the numbers 155 and 400. (Source:Absolute Trivia.com)

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Given the following sets of numbers, list the numbers in each setthat also belong to the set of:




15. e -6, 0, 1, 1 12

, 3, p, 9.62 f

Day Gain or Loss in Dollars

Monday

Tuesday

Wednesday

Thursday

Friday -4

+1

+5

-2

+1

16. e -3, -1.6, 2, 5, 112

, 15.1, 25, 2p f

32. A trim carpenter needs a piece of quarter round molding

feet long for a bathroom. She finds a piece feet long.

How long a piece does she need to cut from the molding in order to use it in the bathroom?

In December 1998, Nkem Chukwu gave birth to the world’s firstsurviving octuplets in Houston, Texas. The following chart givesthe octuplets’ birthweights. The babies are listed in order of birth.

7 12

-foot-long

7 12

6 18

17. Which day showed the greatest loss?

18. Which day showed the greatest gain?

(1.3) Write the number as a product of prime factors.

19. 36 20. 120

Perform the indicated operations. Write results in lowest terms.

21. 22.

23. 24.

25. 26.

27. 28.

29. Determine the unknown part of the given circle.

Find the area and the perimeter of each figure.

30. 31.

?

~Z

2 # 8 34

5 ,

13

7 16

- 2 23

2 34

+ 6 58

34

-

320

715

+

56

78

,

2132

815

#2730

1a meter

√ meterin.3

11

in.311

in.511

in.511

Baby’s Name Gender Birthweight (pounds)

Ebuka girl

Chidi girl

Echerem girl

Chima girl

Odera girl

Ikem boy

Jioke boy

Gorom girl

(Source: Texas Children’s Hospital, Houston, Texas)

1 18

1 1316

1 18

1116

1 58

1 34

1 1116

1 12

33. What was the total weight of the boy octuplets?

34. What was the total weight of the girl octuplets?

35. Find the combined weight of all eight octuplets.

36. Which baby weighed the most?

37. Which baby weighed the least?

38. How much more did the heaviest baby weigh than the lightestbaby?

39. By March 1999,Chima weighed pounds.How much weighthad she gained since birth?

40. By March 1999, Ikem weighed pounds. How muchweight had he gained since birth?

(1.4) Simplify each expression.

41. 42.

43. 44.

45. 46.47. 48.

49. 50.

Translate each word statement to symbols.

51. The difference of twenty and twelve is equal to the productof two and four.

52. The quotient of nine and two is greater than negative five.

5[312 + 52 - 5]4 + ƒ 6 - 2 ƒ + 82

4 + 6 # 4

8 + 312 # 6 - 12311 + 2 # 52 + 468 - 5 # 236 # 32

+ 2 # 8

a34b3a2

7b2

5224

4 532

5 12

The following chart shows the gains and losses in dollars of DensityOil and Gas stock for a particular week.

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Chapter 1 Review 71

Evaluate each expression if and

53. 54.

55. 56.

57. The expression represents the measure of theunknown angle of the given triangle. Replace a with 37 and bwith 80 to find the measure of the unknown angle.

Decide whether the given number is a solution to the given equation.



(1.5) Find the additive inverse or the opposite.

60. 61.

62. 63.

Find the following sums.

64. 65.

66. 67.

68. 69.

70. The lowest elevation in North America is feet at DeathValley in California. If you are standing at a point 728 feetabove Death Valley, what is your elevation? (Source: NationalGeographic Society)

(1.6) Perform the indicated operations.

71. 72.73. 74.

75. 76.

If and evaluate each expression.

77. 78.

(1.7) Find the multiplicative inverse or reciprocal.

79. 80.


81. 82.

83. 84.

85. 86.

87. 88.0

-2-60

31-222 - 5

-14

41-32 + 1-822 + 1-22

42-3

-18-6

1-221-14261-82

35

-6

y - x + 5x

2x2x2

- y + z

z = -9,x = 3, y = -6,

11 - 1-92 + 618 - 222 + 3 # 4

-21 - 16 + 318 - 224 - 15-6 - 1-112-3.1 - 8.46 - 20

-282

-2.8 + 6.7-4.6 + 1-9.32-8 + ƒ -3 ƒ

116

+ a-

14b

-6 + 1-112-15 + 4

- ƒ -7 ƒƒ -2 ƒ

23

-9

3x2+ 4 = x - 1?x = 1

7x - 3 = 18?x = 3

?

80� 37�

180 - a - b

x2- 3y2x

y+

z

2y

x1y + 2z22x + 3y

z = 8.x = 6, y = 2, 89.

90.


91. 92.

93. During the 1999 LPGA Sara Lee Classic, Michelle McGannhad scores of and in three rounds of golf. Findher average score per round. (Source: Ladies ProfessionalGolf Association)

94. During the 1999 PGA Masters Tournament, Bob Esteshad scores of and 0 in four rounds of golf. Findhis average score per round. (Source: Professional GolfAssociation)

(1.8) Name the property illustrated.

95.

96.

97.

98.

99.

100.

101.

102.

103.

104.

Use the distributive property to write each expression withoutparentheses.

105.

106.

107.

108.

109.

110.

MIXED REVIEW

Insert or in the space between each pair of numbers.

111. 112.


113. 114.

115. 116.

117. 118.

119. 120.

121. 122.

123. 124.-15 + (-4)2

+ ƒ -9 ƒ

10 - 2 # 5-

58

,

34

5 + 2[(7 - 5)2+ (1 - 3)]

-3 - 2(-9)

-15 - 3(-4)

(-2)48 , 2 # 4

-0.5(-0.3)-

45

a 516b

-204

4(-20)

14 - 20-7.2 + (-8.1)

-1 12

-2 12

- ƒ -11 ƒ ƒ11.4 ƒ

=6 , 7 ,

-8(2y + 9) - 1

-4(3x + 5) - 7

12

(6z - 10)

-(7 - x + 4z)

-3(z + y)

5(y - 2)

8 + 0 = 8

4 #14

= 1

13 # 82 # 4 = 3 # 18 # 42618 + 52 = 6 # 8 + 6 # 5

2 # 8 = 8 # 2

2 + 13 + 92 = 12 + 32 + 9

4 + 1-42 = 0

318 - 52 = 3 # 8 + 3 # 1-526 # 1 = 6

-6 + 5 = 5 + 1-62

-1, 0, -3,

+1-9, -7,

x2- y3x2

- y4

y = -2,x = -5

-52- 12 - 202 , 1-32 # 3

-42- 1-3 + 52 , 1-12 # 2

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34. During which down did the greatest loss of yardageoccur?

35. Was a touchdown scored?

36. The temperature at the Winter Olympics was a frigid14 degrees below zero in the morning, but by noon it hadrisen 31 degrees.What was the temperature at noon?

37. United HealthCare is a health insurance provider. In 3consecutive recent years, it had net incomes of $356 mil-lion, $460 million, and million. What was UnitedHealthCare’s total net income for these three years?(Source: United HealthCare Corp.)

38. Jean Avarez decided to sell 280 shares of stock, which de-creased in value by $1.50 per share yesterday. How muchmoney did she lose?

- $166

0� 31�

?�

�14�

Gains and Losses in Yards

First Down 5

Second Down

Third Down

Fourth Down 29

-2

-10

Remember to use the Chapter Test Prep Video CD to see the fully worked-outsolutions to any of the exercises you want to review.CHAPTER 1 TEST

Translate the statement into symbols.

1. The absolute value of negative seven is greater thanfive.

2. The sum of nine and five is greater than or equal tofour.

Simplify the expression.

3. 4.

5. 6.

7. 8.

9. 10.

11. 12.

13.

14. 15.

16. 17.

Insert the appropriate space to make each of thefollowing statements true.

18. 19.20. 21.

22. In the state of Massachusetts, there are 2221 licensedchild care centers and 10,993 licensed home-based childcare providers. Write an inequality statement comparingthe numbers 2221 and 10,993. (Source: Children’s Foun-dation)

23. Given list the num-

bers in this set that also belong to the set of:a. Natural numbersb. Whole numbersc. Integersd. Rational numberse. Irrational numbersf. Real numbers


24. 25.

26. 27.

Identify the property illustrated by each expression.

28.29. 6 # 8 = 8 # 6

8 + 19 + 32 = 18 + 92 + 3

y + z - 1

x2 + 3x - y

x + yzx2+ y2

z = -3,x = 6, y = -2,

e -5, -1, 0, 14

, 1, 7, 11.6, 17, 3p f ,

ƒ -2 ƒ -1 - 1-322 ƒ -3 ƒ

4 -8-3 -7

6 , 7 , or = in

1-221021-32-6

-12 + 3 # 84

6[5 + 213 - 82 - 3]31-422 - 80

12 - 62 ,

-2 - 6-3 - 1

-

12

-1 18

+ 5 34

12

-

56

ƒ -6 ƒ + 2

5 - 6-80

ƒ -16 ƒ

-81-621-22

11321-3212 , 4 # 3 - 6 # 2

-13 - 1-22-13 + 8

30.

31.

32. Find the opposite of

33. Find the reciprocal of

The New Orleans Saints were 22 yards from the goal when thefollowing series of gains and losses occurred.

-

13

.

-9.

16162 = 1

-612 + 42 = -6 # 2 + 1-62 # 4

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