mil52817 ch07 493-570...the denominator is a perfect square trinomial. skill practice answers 7. 8....

493

In Chapter 7 we define a rational

expression as the ratio of two

polynomials. Then we will learn how

to add, subtract, multiply, and divide

rational expressions. Then we study

equations and applications involving

rational expressions.

Working with rational expressions

is similar to working with fractions so it

is worthwhile to review operations of

fractions first.

Fill in the appropriate operations

(�, �, �, �) into the equations to make

them true. Use each of the symbols

�, �, �, � in each column only once.

RationalExpressions

7.1 Introduction to Rational Expressions

7.2 Multiplication and Division of Rational Expressions

7.3 Least Common Denominator

7.4 Addition and Subtraction of Rational Expressions

Problem Recognition Exercises—Operations on Rational

Expressions

7.5 Complex Fractions

7.6 Rational Equations

Problem Recognition Exercises—Comparing Rational

Equations and Rational Expressions

7.7 Applications of Rational Equations and Proportions

77

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Column 1 Column 2

12

16

23

143

103

57

34

76

12

12

18

634

18

18

56

310

122

23

��

�

�

�

�

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494 Chapter 7 Rational Expressions

1. Definition of a Rational ExpressionIn Section 1.1, we defined a rational number as the ratio of two integers, where

Examples of rational numbers:

In a similar way, we define a rational expression as the ratio of two polynomials,where

Examples of rational expressions:

2. Evaluating Rational Expressions

Evaluating Rational Expressions

Evaluate the rational expression (if possible) for the given values of x:

a. b. c. d.

Solution:

Substitute the given value for the variable. Then use the order of operationsto simplify.

a. b.

Substitute Substitute

c. d.

Substitute Substitute

1. Evaluate the expression for the given values of x.

a. b. c. d. x � �5x � 3x � 0x � 2

x � 3x � 5

Skill Practice

� �2

�120

�12�6

x � 3.12

132 � 3x � �3.

121�32 � 3

12x � 3

12x � 3

� �6� �4

�12�2

�12�3

x � 1.12

112 � 3x � 0.

12102 � 3

12x � 3

12x � 3

x � 3x � �3x � 1x � 0

12x � 3

Example 1

3x � 6x2 � 4

, 34

, 6r 5 � 2r

7

q � 0.pq,

23

, �15

, 9

q � 0.

pq,

Undefined.Recall thatdivision by zerois undefined.

Section 7.1 Introduction to Rational Expressions

Concepts

1. Definition of a Rational

Expression

2. Evaluating Rational

Expressions

3. Domain of a Rational

Expression

4. Simplifying Rational

Expressions to Lowest Terms

5. Simplifying a Ratio of �1

Skill Practice Answers

1a. b.

c. 0 d. Undefined

�35

�17


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Section 7.1 Introduction to Rational Expressions 495

3. Domain of a Rational ExpressionIn Example 1(d), the expression 12�(x � 3) is undefined for x � 3. The fact thata rational expression may be defined for some values of the variable but not forothers leads us to an important concept called the domain of an expression.

Informal Definition of the Domain of an Algebraic Expression

Given an algebraic expression, the domain of the expression is the set of realnumbers that when substituted for the variable makes the expression resultin a real number.

For a rational expression, the domain will consist of the real numbers that whensubstituted into the expression does not make the denominator equal to zero.Therefore, in Example 1(d), because 12�(x � 3) is undefined for x � 3, the domainis all real numbers except 3. We write this in set-builder notation as

5x 0x is a real number and x � 36

Steps to Find the Domain of a Rational Expression

1. Set the denominator equal to zero and solve the resulting equation.2. The domain is the set of real numbers excluding the values found in

step 1.

Finding the Domain of Rational Expressions

Find the domain of the expressions.

a. b. c. d.

Solution:

a.

Set the denominator equal to zero.

Solve the equation.

The domain is the set of realnumbers except

b.

Set the denominator equal to zero.

Domain: The domain is the set of realnumbers except 0.

5x 0 x is a real number and x � 06

x � 0

�5x

Domain: 5y 0 y is a real number and y � �726

�72.

y � �72

2y

2�

�72

2y � �7

2y � 7 � 0

y � 3

2y � 7

2x3 � 5x2 � 9

a � 10a2 � 25

�5x

y � 3

2y � 7

Example 2


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c.

Set the denominator equal to zero.The equation is quadratic.

Factor the equation.

Set each factor equal to zero.

The domain is the set of realnumbers except 5 and

Domain:

d.

The quantity cannot be negative for any real number, x, so the denomina-tor cannot equal zero. Therefore, no numbers are excluded from the do-main. The domain is the set of all real numbers.

Find the domain.

2. 3. 4. 5.

4. Simplifying Rational Expressions to Lowest Terms

In many cases, it is advantageous to simplify or reduce a fraction to lowestterms. The same is true for rational expressions.

The method for simplifying rational expressions mirrors the process for sim-plifying fractions. In each case, factor the numerator and denominator. Commonfactors in the numerator and denominator form a ratio of 1 and can be reduced.

factor1

Simplifying a fraction:

factor1

Simplifying arational expression:

Informally, to simplify a rational expression to lowest terms, we reduce commonfactors whose ratio is 1. Formally, this is accomplished by applying the fundamentalprinciple of rational expressions.

2x � 6x2 � 9

21x � 32

1x � 32 1x � 32�

21x � 32

112 �2

x � 3

2135

3 � 75 � 7

�35

� 112 �35

8z4 � 1

w � 4w 2 � 9

tt 2 � 10t

a � 22a � 8

Skill Practice

x2 � 9x2

2x3 � 5x2 � 9

5a 0 a is a real number and a � 5, a � �56

�5. a � 5 or a � �5

a � 5 � 0 or a � 5 � 0

1a � 52 1a � 52 � 0

a2 � 25 � 0

a � 10a2 � 25


2.

3.

4.

5. The set of all real numbersw � 3, w � �365w 0w is a real number andt � 0, t � �1065t 0 t is a real number and 5a 0a is a real number and a � 46

Fundamental Principle of Rational Expressions

Let p, q, and r represent polynomials. Then

prqr

�pq

for q � 0 and r � 0

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Simplifying a Rational Expression to Lowest Terms

Given the expression:

a. Factor the numerator and denominator.

b. Determine the domain of the expression.

c. Simplify the expression to lowest terms.

Solution:

a.Factor out the GCF in the numerator.Factor the denominator as adifference of squares.

b. To find the domain restrictions, setthe denominator equal to zero. Theequation is quadratic.

Set each factor equal to 0.

The domain is all real numbersexcept and 7.

Domain:

1

c. Reduce common factors whose ratiois 1.

6. Given:

a. Factor the numerator and the denominator.b. Determine the domain of the expression.c. Simplify the rational expression to lowest terms.

In Example 3, it is important to note that the expressions

are equal for all values of p that make each expression a real number. Therefore,

for all values of p except and (At and the originalexpression is undefined.) This is why the domain of an expression is alwaysdetermined before the expression is simplified.

p � �7,p � 7p � �7.p � 7

2p � 14

p2 � 49�

2p � 7

2p � 14

p2 � 49 and

2p � 7

5z � 25z 2 � 3z � 10

Skill Practice

� 2

p � 7 1provided p � 7 and p � �72

21p � 72

1p � 72 1p � 72

5p 0 p is a real number and p � �7, p � 76

�7or p � 7p � �7

p � 7 � 0 or p � 7 � 0

1p � 72 1p � 72 � 0

� 21p � 72

1p � 72 1p � 72

2p � 14

p2 � 49

2p � 14

p2 � 49

Example 3


6a.

b.

c.5

z � 2

z � �5, z � 265z 0z is a real number and

51z � 52

1z � 52 1z � 22

Avoiding Mistakes:

The domain of a rational expres-sion is always determined beforesimplifying the expression tolowest terms.


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From this point forward, we will write statements of equality between tworational expressions with the assumption that they are equal for all values of thevariable for which each expression is defined.

Simplifying Rational Expressions to Lowest Terms

Simplify the rational expressions to lowest terms.

a. b.

Solution:

a.

Factor the numerator and denominator.

1

Reduce common factors to lowest terms.

b.

Factor out the GCF.

1 1

Reduce common factors whose ratio is 1.

Simplify to lowest terms.

7. 8.

The process to simplify a rational expression to lowest terms is based on theidentity property of multiplication. Therefore, this process applies only to fac-tors (remember that factors are multiplied). For example,

3x3y

�31

� x3 � y

� 1 �xy

�xy

x 2 � 12x 2 � x � 3

15q 3

9q 2

Skill Practice

� 1

51c � 42

� 21c � 42

2 � 51c � 42 1c � 42

� 21c � 42

101c � 422

� 21c � 42

101c2 � 8c � 162

2c � 810c2 � 80c � 160

� 2a

� 2 � 13 � 3 � a � a � a � a2

13 � 3 � a � a � a � a2 � a

� 2 � 3 � 3 � a � a � a � a3 � 3 � a � a � a � a � a

18a4

9a5

2c � 810c2 � 80c � 160

18a4

9a5

Example 4

The denominator is a perfect squaretrinomial.


7. 8.x � 12x � 3

5q

3

Simplify

Avoiding Mistakes:

Given the expression

do not be tempted to reducebefore factoring. The terms 2cand 10c2 cannot be “canceled”because they are terms, notfactors.

Similarly, the �8 in thenumerator cannot be “canceled”with the �80c or 160 in thedenominator because they areterms, not factors.

2c � 8

10c 2 � 80c � 160


Terms that are added or subtracted cannot be reduced to lowest terms. Forexample,

The objective of simplifying a rational expression to lowest terms is to createan equivalent expression that is simpler to use. Consider the rational expres-sion from Example 4(b) in its original form and in its simplified form. If wechoose an arbitrary value of c from the domain of the original expression andsubstitute that value into each expression, we see that the simplified form is easierto evaluate. For example, substitute

Original Expression Simplified Expression

Substitute .

� �210

or �15

� �15

� �2

90 � 240 � 160

� 1

51�12�

6 � 810192 � 240 � 160

� 1

513 � 42�

2132 � 8

101322 � 80132 � 160c � 3

151c � 42

2c � 810c2 � 80c � 160

c � 3:

x � 3y � 3

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5. Simplifying a Ratio ofWhen two factors are identical in the numerator and denominator, they form aratio of 1 and can be reduced. Sometimes we encounter two factors that are op-posites and form a ratio of For example,

Simplified Form Details/Notes

The ratio of a number and its opposite is

The ratio of a number and its opposite is

1

Factor out 1

Recognizing factors that are opposites is useful when simplifying rationalexpressions.

2 � xx � 2

��11�2 � x2

x � 2�

�11x � 22

x � 2�

�11

� �12 � x

x � 2� �1

�1

x � 7�x � 7

�x � 7

�11x � 72�

x � 7

�11x � 72�

1�1

� �1x � 7

�x � 7� �1

�1.100

�100� �1

�1.�55

� �1

�1.

�1

Cannot be simplified


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Simplifying Rational Expressions to Lowest Terms

Simplify the rational expressions to lowest terms.

a. b.

Solution:

a.


Notice that and are opposites andform a ratio of

Details:

b.


Notice that and are opposites andform a ratio of

Details:

� �1

y � 5 or

1�1y � 52

or �

1y � 5

� �11y � 52

1y � 52 1y � 52

5 � y

1y � 52 1y � 52�

�11�5 � y2

1y � 52 1y � 52�

5 � y

1y � 52 1y � 52

�1�1.

y � 55 � y

� 5 � y

1y � 52 1y � 52

5 � y

y2 � 25

� �3

� 31�12

31c � d2

d � c�

31c � d2

�11�d � c2�

31c � d2

�11c � d2�

31c � d2

d � c

�1

�1.1d � c21c � d2

� 31c � d2

d � c

3c � 3dd � c

5 � y

y2 � 253c � 3dd � c

Example 5

TIP: It is important to recognize that a rational expression may be written in

several equivalent forms, particularly when a negative factor is present. For

example, since two numbers with opposite signs form a negative quotient, the

number can be written as or as The � sign can be written in the

numerator, in the denominator, or out in front of the fraction.

For this reason, the expression in Example 5(b) can be written in a variety

of forms.

3�4.

�34�3

4


9. 10.20 � 5y

y 2 � 16b � a

a2 � b2

Skill PracticeSkill Practice Answers

9.

10.�5

y � 4 or

5� 1y � 42

or �5

y � 4

�1a � b

or 1

� 1a � b2 or �

1a � b


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Study Skills Exercises

1. Review Section R.2 in this text. Write an example of how to simplify (reduce) a fraction, multiply twofractions, divide two fractions, add two fractions, and subtract two fractions. Then as you learn aboutrational expressions, compare the operations on rational expressions with those on fractions. This is a greatplace to use 3 � 5 cards again. Write an example of an operation with fractions on one side and the sameoperation with rational expressions on the other side.

2. Define the key terms:

a. rational expression b. domain

Concept 1: Definition of a Rational Expression

3. a. What is a rational number?

b. What is a rational expression?

4. a. Write an example of a rational number. (Answers will vary.)

b. Write an example of a rational expression. (Answers will vary.)

Concept 2: Evaluating Rational Expressions

For Exercises 5–11, substitute the given number into the expression and simplify (if possible).

5. 6. 7.

8. 9. 10.

11.

12. A bicyclist rides 24 miles against a wind and returns 24 miles with the samewind. His average speed for the return trip traveling with the wind is 8 mphfaster than his speed going out against the wind. If x represents the bicyclist’sspeed going out against the wind, then the total time, t, required for the roundtrip is given by

where and t is measured in hours.

a. Find the time required for the round trip if the cyclist rides 12 mph against the wind.

b. Find the time required for the round trip if the cyclist rides 24 mph against the wind.

x 7 0t �24x

�24

x � 8

1a � 42 1a � 12

1a � 42 1a � 12 let a � 1

1a � 72 1a � 12

1a � 22 1a � 52 let a � 2

y � 3

3y2 � 25y � 18 let y � �3

y � 8

2y2 � y � 1 let y � 8

w � 42w � 8

let w � 0w � 10w � 6

let w � 01

x � 6 let x � �2

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Section 7.1 Practice Exercises


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13. The manufacturer of mountain bikes has a fixed cost of $56,000, plus a variable cost of $140 per bike. Theaverage cost per bike, y (in dollars), is given by the equation:

where x represents the number of bikes produced and

a. Find the average cost per bike if the manufacturer produces 1000 bikes.

b. Find the average cost per bike if the manufacturer produces 2000 bikes.

c. Find the average cost per bike if the manufacturer produces 10,000 bikes.

Concept 3: Domain of a Rational Expression

For Exercises 14–23, write the domain.

14. 15. 16.

17. 18. 19.

20. 21. 22.

23.

24. Construct a rational expression that is undefined for (Answers will vary.)

25. Construct a rational expression that is undefined for (Answers will vary.)

26. Construct a rational expression that is undefined for and (Answers will vary.)

27. Construct a rational expression that is undefined for and (Answers will vary.)

28. Evaluate the expressions for 29. Evaluate the expressions for

a. b. a. b.

30. Evaluate the expressions for 31. Evaluate the expressions for

a. b. a. b.

Concept 4: Simplifying Rational Expressions to Lowest Terms

For Exercises 32–41,

a. Write the domain in set-builder notation. b. Simplify the expression to lowest terms.

32. 33. 34. 35.r 2 � 4r � 2

t 2 � 1t � 1

8x � 84x � 4

3y � 6

6y � 12

x � 5x � 1

1x � 522

x2 � 6x � 5x � 12x � 3

3x2 � 2x � 16x2 � 7x � 3

x � 4.x � �1.

x � 1x � 3

2x2 � 4x � 62x2 � 18

5x � 1

5x � 5x2 � 1

x � 3.x � 4.

x � 4.x � �1

x � 7.x � �3

x � 5.

x � 2.

z2 � 10z � 99

y2 � y � 12

12x � 1x2 � 4

x � 4x2 � 9

c � 11c2 � 5c � 6

b � 12b2 � 5b � 6

4y � 1

13y � 72 1y � 32

x � 512x � 52 1x � 82

�3h � 4

5k � 2

x 7 0.y �56,000 � 140x

x


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36. 37. 38. 39.

40. 41.

For Exercises 42–83, simplify the expression to lowest terms.

42. 43. 44. 45.

46. 47. 48. 49.

50. 51. 52. 53.

54. 55. 56. 57.

58. 59. 60. 61.

62. 63. 64. 65.

66. 67. 68. 69.

70. 71. 72. 73.

74. 75. (Hint: Factor by grouping.)

76. 77.

78. 79.

80. 81.

82. 83.

Concept 5: Simplifying a Ratio of �1

84. What is the relationship between and

85. What is the relationship between and �w � p?w � p

2 � x?x � 2

2c2 � cd � d2

5c2 � 3cd � 2d2

2x2 � xy � 3y2

2x2 � 11xy � 12y2

x2 � 1ax3 � bx2 � ax � b

5x3 � 4x2 � 45x � 36x2 � 9

3x � 3y

2x2 � 4xy � 2y2

49p2 � 28pq � 4q2

14p � 4q

4m3 � 3m2

4m3 � 7m2 � 3m2t2 � 3t

2t4 � 13t3 � 15t2

3pr � ps � 3qr � qs

3pr � ps � 3qr � qsac � ad � 2bc � 2bd2ac � ad � 4bc � 2bd

4t2 � 16t4 � 16

5q2 � 5

q4 � 1x2 � 5x � 142x2 � x � 10

3x2 � 7x � 6x2 � 7x � 12

h2 � h � 6h2 � 2h � 8

y2 � 6y � 9

2y2 � y � 15r 2 � 36r � 6

q2 � 25q � 5

b2 � 64b � 8

a2 � 49a � 7

5z � 15z2 � 4z � 21

2x � 4x2 � 3x � 10

6p2 � 12p

2pq � 4p3x2 � 6x

9xy � 18x3x � 15x2 � 25

4w � 8w2 � 4

714c � 21

520a � 25

1p � 22 1p � 324

1p � 2221p � 322x12x � 122

4x312x � 12

1n � 72

91n � 22 1n � 72

1m � 112

41m � 112 1m � 112

1c � 42 1c � 12

1c � 42 1c � 22

1p � 32 1p � 52

1p � 52 1p � 42

81x � 12

41x � 12

31y � 22

61y � 2260rs4t2

�12r4s2t3

�24x2y5z

8xy4z3

20a4b2

25ab2

18st5

12st3

15c3

3c5

7b2

21b

t2 � 3t � 10t2 � t � 20

a2 � 3a � 10a2 � a � 6

8b � 204b2 � 25

9x2 � 46x � 4

12a2

24a2 � 18a7w

21w2 � 35w

(Hint: Factor bygrouping.)


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For Exercises 86–99, simplify to lowest terms.

86. 87. 88. 89.

90. 91. 92. 93.

94. 95. 96. 97.

98. 99.

Expanding Your Skills

For Exercises 100–103, factor and simplify to lowest terms.

100. 101. 102. 103.x2 � 25

x3 � 125z2 � 16z3 � 64

y3 � 27

y2 � 3y � 9w3 � 8

w2 � 2w � 4

49 � b2

b2 � 10b � 21x2 � x � 12

16 � x2

4t � 1616 � 4t

10x � 1210x � 12

2 � n2 � n

k � 55 � k

3a2 � 55 � 3a2

2m � 7n7n � 2m

4q � 412 � 12q

3y � 6

12 � 6y

z � 10�z � 10

�4 � y

4 � y

8 � p

p � 8x � 55 � x

1. Multiplication of Rational ExpressionsRecall from Section R.2 that to multiply fractions, we multiply the numerators andmultiply the denominators. The same is true for multiplying rational expressions.

Section 7.2 Multiplication and Division of Rational Expressions

Concepts

1. Multiplication of Rational

Expressions

2. Division of Rational

Expressions

Multiplication of Rational Expressions

Let p, q, r, and s represent polynomials, such that Then,

pq

�rs

�prqs

q � 0, s � 0.

For example:

Multiply the Fractions Multiply the Rational Expressions

Sometimes it is possible to simplify a ratio of common factors to 1 before mul-tiplying. To do so, we must first factor the numerators and denominators of eachfraction.

1 11514

�2110

�3 � 52 � 7

�3 � 72 � 5

�94

2x3y

�5z7

�10xz21y

23

�57

�1021


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Section 7.2 Multiplication and Division of Rational Expressions 505

Steps to Multiply Rational Expressions

1. Factor the numerators and denominators of all rational expressions.2. Simplify the ratios of common factors to 1 or �1.3. Multiply the remaining factors in the numerator, and multiply the

remaining factors in the denominator.

The same process is also used to multiply rational expressions.

Multiplying Rational Expressions

Multiply.

a. b. c.

Solution:

a.

Factor into prime factors.

1 1 1

Simplify.

Multiply remaining factors.

b.

Factor into prime factors.

1 1 1

Simplify.

c.

Factor the numerators anddenominators completely.

1 1

Simplify the ratios of common factorsto 1 or

or or �x � 4x � 7

x � 4�1x � 72

��1x � 42

x � 7

��11x � 42

x � 7

�1.�

517 � x2

51x � 12�1x � 42 1x � 12

1x � 72 1x � 72

�1

�517 � x2

51x � 12�1x � 42 1x � 12

1x � 72 1x � 72

35 � 5x5x � 5

�x2 � 5x � 4

x2 � 49

�1

c1c � d2

�31c � d2

2 � 3 � c�

21c � d2 1c � d2

�31c � d2

2 � 3 � c�

21c � d2 1c � d2

3c � 3d6c

�2

c2 � d2

�3a3

2

�5 � a � a � b

2�

2 � 3 � a2 � 5 � b

�5 � a � a � b

2�

2 � 3 � a2 � 5 � b

5a2b2

�6a

10b

35 � 5x5x � 5

�x2 � 5x � 4

x2 � 493c � 3d

6c�

2c2 � d2

5a2b2

�6a

10b

Example 1

Avoiding Mistakes:

If all the factors in the numeratorreduce to a ratio of 1, do not for-get to write the factor of 1 in thenumerator.

TIP: The ratio

because and

are opposites.x � 7

7 � x�1

7 � xx � 7 �


Multiply.

1. 2.

3.

2. Division of Rational ExpressionsRecall that to divide fractions, multiply the first fraction by the reciprocal ofthe second.

Multiply by the reciprocal Factor1 1

of the second fraction

The same process is used to divide rational expressions.

Dividing Rational Expressions

Divide.

a. b. c.

Solution:

a.

Multiply the first fraction by thereciprocal of the second.

Factor each polynomial.

1 1

Reduce common factors.

�25

t � 3

�51t � 32

2�

2 � 51t � 32 1t � 32

�51t � 32

2�

2 � 51t � 32 1t � 32

�5t � 15

2�

10t2 � 9

5t � 152

�t2 � 9

10

3x4y5x6y

p2 � 11p � 30

10p2 � 250�

30p � 5p2

2p � 45t � 15

2�

t2 � 910

Example 2

3 � 72 � 5

�3 � 57 � 7

�9

142110

�1549

2110

�4915

p2 � 4p � 35p � 10

�p

2 � p � 6

9 � p 2

4x � 8x � 6

�x

2 � 6x2x

7a3b

�15b14a

2

Skill Practice

Division of Rational Expressions

Let p, q, r, and s represent polynomials, such that Then,

pq

�rs

�pq

�sr

�psqr

q � 0, r � 0, s � 0.



1. 2.

3.� 1p � 12

5 or

p � 1

�5 or �

p � 1

5

21x � 225

2a

B&ImiL52817_ch07_493-570 11/8/06 11:37 PM Page 506CONFIRMING PAGES

b.

Multiply the first fraction by thereciprocal of the second.

Factor the trinomial.

Factor out the GCF.

Factor out the GCF. Then factor the difference of squares.

Factor out the GCF.

1 1


c. This fraction bar denotes division

Multiply by the reciprocal of the secondfraction.

1 1 1


Divide the rational expressions.

4. 5.

6.

4x 2 � 92x2 � x � 3

�20x � 30

x 2 � 7x � 6

7y � 14

y � 1�

y 2 � 2y � 8

2y � 2

Skill Practice

�910

�3 � x

2 � 2 � y�

2 � 3 � y

5 � x

�3x4y

�6y

5x

�3x4y

�5x6y

1�2.

3x4y5x6y

� �1p � 22

25p1p � 52

�1p � 52 1p � 62

2 � 51p � 52 1p � 52�

21p � 22

5p16 � p2

�1

� 2 � 51p � 52 1p � 52 10p2 � 250 � 101p2 � 252

2p � 4 � 21p � 22

p2 � 11p � 30 � 1p � 52 1p � 62

�p2 � 11p � 30

10p2 � 250�

2p � 4

30p � 5p2

p2 � 11p � 30

10p2 � 250�

30p � 5p2

2p � 4

TIP: are opposites and form a

ratio of

�p � 6

�11p � 62� �1

p � 6

6 � p�

p � 6

�11�6 � p2

�1.

1p� 62 and 16 � p2

TIP: A fraction with

one or more rational

expressions in its

numerator or denominator

is called a complex

fraction, for example,

30p � 5p2 � 5p16 � p2

�1p � 52 1p � 62

2 � 51p � 52 1p � 52�

21p � 22

5p16 � p2


4ab3c3

a3b9c


4. 5. 6.a2c 2

12x � 6

1014

y � 4

3x

4y

5x

6y

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎭


Review Exercises

1. Explain the difference between multiplying the fractions and dividing the fractions .

For Exercises 2–9, multiply or divide the fractions.

2. 3. 4. 5.

6. 7. 8. 9.

Concept 1: Multiplication of Rational ExpressionsFor Exercises 10–21, multiply.

10. 11. 12. 13.

14. 15. 16. 17.

18. 19. 20. 21.

Concept 2: Division of Rational ExpressionsFor Exercises 22–35, divide.

22. 23. 24. 25.

26. 27. 28. 29.

30. 31. 32.

33. 34. 35.4t2 � 1t2 � 5t

�2t2 � 5t � 2t2 � 3t � 10

p2 � 2p � 3

p2 � p � 6�

p2 � 1

p2 � 2p

x2 � 2xy � y2

2x � y�

x � y

5

3p � 4q

p2 � 4pq � 4q2 �4

p � 2q9 � b2

15b � 15�

b � 35b

m2 � n2

9�

3n � 3m27m

4a2 � 4a9a � 9

�5

12a3x � 21

6x2 � 42x�

712x

8b � 163b � 3

�5b � 102b � 2

4a � 126a � 18

�3a � 9

5a � 15

10a3b3a

�5b9ab

8m4n5

5n6 �24mn15m3

6cd5d2 �

8c3

10d4x7y

�2x2

21xy

1x � y22

x2 � xy�

xy � x

b2 � a2

a � b�

aa2 � ab

b � 36

�20

3 � b10

2 � a�

a � 216

2p � 46p

�4p2

8p � 16

3y � 18

y2 �4y

6y � 365a � 20

a�

3a10

4x � 2420x

�5x8

10a2b15b2 �

30b2a3

6x3

9x6y2 �18x4y7

4y7st

2 �t

2

14s2

2xy

5x2 �154y

9234

21475

725

� 56 �512

185

�25

34

�38

67

�5

1235

�12

23

�59

23

�59





• NetTutor




Mixed Exercises

For Exercises 36–61, multiply or divide as indicated.

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. 63.

64. 65.

66. 67.r3 � s3

r � s�

r2 � 2rs � s2

r2 � s2

p3 � q3

p � q�

p � q

2p2 � 2pq � 2q2

t2 � t � 2t2 � 5t � 6

�t � 1

t�

5t � 5t � 3

a2 � 5aa2 � 7a � 12

�a3 � 7a2 � 10aa2 � 9a � 18

�a � 6a � 4

x2 � 253x2 � 3xy

�x2 � 4x � xy � 4y

x2 � 9x � 20�

x � 5x

b3 � 3b2 � 4b � 12b4 � 16

�3b2 � 5b � 2

3b2 � 10b � 3�

36b � 12

4m2 � 4mn � 3n2

8m2 � 18n2 �3m � 3n

6m2 � 15mn � 9n2

x2 � xy � 2y2

x � 2y�

x2 � 4xy � 4y2

x2 � 4y2

x4 � 166x2 � 24

�x2 � 2x

3x

y4 � 1

2y2 � 3y � 1�

2y2 � 2

8y2 � 4y

3my � 9m � ny � 3n

9m2 � 6mn � n2 �30m � 10n5y2 � 15y

ax � a � bx � b2x2 � 4x � 2

�4x � 4a2 � ab

4h2 � 5h � 1h2 � h � 2

�6h2 � 7h � 22h2 � 3h � 2

k2 � 3k � 2k2 � 5k � 4

�k2 � 5k � 6

k2 � 10k � 24

p2 � 6p � 824

16 � p2

6p � 6

w2 � 6w � 98

9 � w2

4w � 12

y2 � y � 12

y2 � y � 20�

y2 � y � 30

y2 � 2y � 3

x2 � 2x � 3x2 � 3x � 2

�x2 � 2x � 8x2 � 4x � 3

12q � 32 �2q2 � 5q � 12

q � 715t � 12 �

5t2 � 9t � 23t � 8

p2 � 3p � 2

p2 � 4p � 3�

p � 1p � 2

t2 � 4t � 5t2 � 7t � 10

�t � 4t � 1

z2 � 11z � 28z � 1

�z � 1

z2 � 6z � 7

y2 � 5y � 36

y2 � 2y � 8�

y � 2

y � 6

3y2 � 20y � 7

5y � 35� 13y � 12

2a2 � 13a � 248a � 12

� 1a � 82

1r � 52 �4r

2r2 � 7r � 15

q � 1

5q2 � 28q � 12� 15q � 22

6m � 65

3m � 310

5t � 1012

4t � 88

5t � 15t2 � 31t � 6

� 1t � 621w � 32 �w

2w2 � 5w � 3



TIP: When multiplying both the numerator and denominator of the

fraction by yz, we are actually multiplying the fraction by 1. This is

because Hence,

5

x�

5

x� 1 �

5

x� a

yz

yzb �

5 � yz

x � yz�

5yz

xyz

yz

yz � 1.

5x

1. Writing Equivalent Rational ExpressionsIn Sections 7.1 and 7.2, we learned how to simplify, multiply, and divide rationalexpressions. Our next goal is to add and subtract rational expressions. As withfractions, rational expressions can be added or subtracted only if they have thesame denominator. Therefore, we must first learn how to identify a common de-nominator between two or more rational expressions. Then we must learn howto convert a rational expression into an equivalent rational expression with theindicated denominator.

Using the identity property of multiplication, we know that for

This principle is used to convert a rational expression into an equivalent ex-pression with a different denominator. For example, can be converted into anequivalent expression with a denominator of 12 as follows:

In this example, we multiplied by a convenient form of 1. The ratio waschosen so that the product produced a new denominator of 12. Notice thatmultiplying is equivalent to multiplying the numerator and denominatorof the original expression by 6. In general, if the numerator and denominatorof a rational expression are both multiplied by the same nonzero quantity, thevalue of the expression remains unchanged.

Creating Equivalent Fractions

Convert each expression into an equivalent expression with the indicateddenominator.

a. b.

c. d.

Solution:

a. To convert to an equivalent fraction with adenominator of xyz, multiply both numeratorand denominator by the missing factor of yz.

5 � yzx � yz

�5yzxyz

5x

5x

�xyz

15

�5x � 20

ww � 5

�1w � 52 1w � 22

75p2 �

20p6

5x

�xyz

Example 1

12 by 66

66

12

12

�12

�66

�1 � 62 � 6

�612

12

pq

�pq

� 1 �pq

�rr

�prqr

r � 0,q � 0 and

Section 7.3 Least Common Denominator

Concepts

1. Writing Equivalent Rational

Expressions

2. Least Common Denominator

3. Writing Rational Expressions

with the Least Common

Denominator


Section 7.3 Least Common Denominator 511

b. Multiply the numerator and denominator ofthe fraction by the missing factor of

c. Multiply numerator and denominator by themissing factor of

d.

Factor the denominator of the secondexpression. The factor missing from thedenominator of the first expression is

Multiply numerator and denominatorby

Convert each expression to an equivalent expression with theindicated denominator.

1. 2.

3. 4.

2. Least Common DenominatorRecall from Section R.2 that to add or subtract fractions, the fractions must havea common denominator. The same is true for rational expressions. In this sec-tion, we present a method to find the least common denominator of two ra-tional expressions.

The least common denominator (LCD) of two or more rational expressions isdefined as the least common multiple of the denominators. For example, con-sider the fractions and By inspection, you can probably see that the leastcommon denominator is 40. To understand why, find the prime factorization ofboth denominators:

A common multiple of 20 and 8 must be a multiple of 5, a multiple of anda multiple of However, any number that is a multiple of is automati-cally a multiple of Therefore, it is sufficient to construct the least com-mon denominator as the product of unique prime factors, in which each factoris raised to its highest power.

The LCD of 120

and 18

is 23 � 5 � 40.

22 � 4.23 � 823.

22,

20 � 22 � 5 and 8 � 23

18.

120

18

�8y � 16

xx � 4

�1x � 62 1x � 42

67y

�14y3

12a

�abc

Skill Practice

15

�1 � 1x � 42

5 � 1x � 42�

x � 451x � 42

1x � 42.

1x � 42.

15

�51x � 42

15

�5x � 20

ww � 5

�w � 1w � 22

1w � 52 � 1w � 22 or

w2 � 2w1w � 52 1w � 22

1w � 22.

ww � 5

�1w � 52 1w � 22

7 � 4p4

5p2 � 4p4 �28p4

20p6

4p4. 7

5p2 �20p6

TIP: Notice that in

Example 1(c) we

multiplied the polynomials

in the numerator but left

the denominator in

factored form. This

convention is followed

because when we add

and subtract rational

expressions in the next

section, the terms in the

numerators must be

combined.


1. 2.

3.

4.18

�y � 2

8 1y � 2 2

xx � 4

�x

2 � 6x1x � 4 2 1x � 6 2

67y

�12y 2

14y 3

12a

�12bcabc



Steps to Find the Least Common Denominator of Two or More Rational Expressions

1. Factor all denominators completely.

2. The LCD is the product of unique factors from the denominators, inwhich each factor is raised to the highest power to which it appears inany denominator.

Finding the Least Common Denominator of Rational Expressions

Find the LCD of the following sets of rational expressions.

a. b. c.

d.

Solution:

a.

Step 1: Factor the denominators.

Step 2: The LCD is the product of uniquefactors, each raised to its highestpower.

b.

Step 1: The denominators are alreadyfactored.

The LCD is Step 2: The LCD is the product of uniquefactors, each raised to its highestpower.

c.



d.



x1x � 222.

�x � 5

x1x � 22;

11x � 222

x � 5x2 � 2x

; 1

x2 � 4x � 4

21a � 52 1a � 52.

�a � b

1a � 52 1a � 52;

121a � 52

a � ba2 � 25

; 1

2a � 10

3x5y3z.

� 5

3x2z;

7x5y3

53x2z

; 7

x5y3

The LCD is 23 � 72 � 392.

� 5

2 � 7;

372;

123

514

; 349

; 18

x � 5x2 � 2x

; 1

x2 � 4x � 4

a � ba2 � 25

; 1

2a � 105

3x2z;

7x5y3

514

; 349

; 18

Example 2




5. 120 6.

7.

8. 1t � 7 2 1t � 2 2 1t � 1 23 1x � 4 2 1x � 4 2

10 a 4b

2

Find the LCD for each set of expressions.

5. 6.

7. 8.

3. Writing Rational Expressions with the LeastCommon Denominator

To add or subtract two rational expressions, the expressions must have the samedenominator. Therefore, we must first practice the skill of converting each ra-tional expression into an equivalent expression with the LCD as its denomina-tor. The process is as follows: Identify the LCD for the two expressions. Then,multiply the numerator and denominator of each fraction by the factors fromthe LCD that are missing from the original denominators.

Converting to the Least Common Denominator

Find the LCD of each pair of rational expressions. Then convert each expressionto an equivalent fraction with the denominator equal to the LCD.

a. b. c.

Solution:

a. The LCD is

The first expression ismissing the factor fromthe denominator.

The second expression ismissing the factor fromthe denominator.

2b

65a2 �

6 � 2b5a2 � 2b

�12b

10a2b

5a

32ab

�3 � 5a

2ab � 5a�

15a10a2b

10a2b.3

2ab;

65a2

w � 2w2 � w � 12

; 1

w2 � 94

x � 1;

7x � 4

32ab

; 6

5a2

Example 3

6t

2 � 5t � 14;

8t

2 � 3 t � 2x

x 2 � 16;

23x � 12

15a3b2;

110a4b

38

; 710

; 115

Skill Practice

b. The LCD is .

The first expression is missingthe factor from thedenominator.

The second expression ismissing the factor from the denominator.

1x � 12

7x � 4

�71x � 12

1x � 42 1x � 12�

7x � 71x � 42 1x � 12

1x � 42

4x � 1

�41x � 42

1x � 12 1x � 42�

4x � 161x � 12 1x � 42

1x � 12 1x � 424

x � 1;

7x � 4



TIP: In Example 4, the

expressions

have opposite factors in

the denominators. In such

a case, you do not need

to include both factors in

the LCD.

3

x � 7 and

1

7 � x


9.

10. ;

11.

�3z � 61z � 22 1z � 22 1z � 12

z 2 � z1z � 22 1z � 22 1z � 12

;

x 2 � 3x

1x � 1 2 1x � 3 2

5x � 51x � 3 2 1x � 1 2

2r 2

r 3s 2

; �s

r 3s 2

has the desired LCD.

c. To find the LCD, factor eachdenominator.

The LCD is

The first expression is missing the factor from the denominator.

The second expression is missing the factor from the denominator.

For each pair of expressions, find the LCD, and then converteach expression to an equivalent fraction with the denominatorequal to the LCD.

9. 10. 11.

Converting to the Least Common Denominator

Find the LCD of the expressions

Solution:

Notice that the expressions are opposites and differ by a fac-tor of Therefore, we may use either as a common denomi-nator. Each case is detailed in the following conversions.

Converting to the Denominator

Leave the first fraction unchanged because it

Multiply the second rational expression by theratio to change its denominator to

Apply the distributive property.

��1

x � 7

��1

�7 � x

x � 7.�1�1

17 � x

�1�121

1�12 17 � x2

3x � 7

; 1

7 � x

x � 7

x � 7 or 7 � x�1.x � 7 and 7 � x

3x � 7

and 1

7 � x.

Example 4

zz 2 � 4

; �3

z2 � z � 25

x � 3 ;

xx � 1

2

rs 2

; �1r

3s

Skill Practice

�w � 4

1w � 32 1w � 32 1w � 42

1w � 42

11w � 32 1w � 32

�11w � 42

1w � 32 1w � 32 1w � 42

�w2 � w � 6

1w � 42 1w � 32 1w � 32

1w � 32

w � 21w � 42 1w � 32

�1w � 22 1w � 32

1w � 42 1w � 32 1w � 32

1w � 42 1w � 32 1w � 32.

w � 21w � 42 1w � 32

; 1

1w � 32 1w � 32

w � 2w2 � w � 12

; 1

w2 � 9



Study Skills Exercise

1. Define the key term least common denominator.

Review Exercises

2. Evaluate the expression for the given values of x.

a. b. c.

For Exercises 3–4, write the domain in set-builder notation. Then reduce the expression to lowest terms.

3. 4.


5. 6.

7. 8.5b2 � 6b � 1b2 � 5b � 6

� 15b � 1216y2

9y � 36�

8y3

3y � 12

61a � 2b2

21a � 3b2�

41a � 3b2 1a � 3b2

91a � 2b2 1a � 2b2a � 3a � 7

�a2 � 3a � 10a2 � a � 6

x � 2x2 � 3x � 10

3x � 35x2 � 5

x � �5x � 5x � 1

2xx � 5

Converting to the Denominator

Leave the second fraction unchanged because

Multiply the first rational expression by theratio to change its denominator to


12a. Find the LCD of the expressions.b. Then convert each expression to an equivalent fraction with denominator

equal to the LCD.

9w � 2

; 11

2 � w

Skill Practice

��3

7 � x

��3

�x � 7

7 � x.�1�1

3x � 7

�1�123

1�12 1x � 72;

3x � 7

; 1

7 � x

7 � x

it has the desired LCD.


12a. The LCD is or.

b.

or

112 � w

�11

2 � w

9w � 2

��9

2 � w ;

11

2 � w�

�11w � 2

9w � 2

�9

w � 2 ;

12 � w 21w � 2 2




• NetTutor



Concept 1: Writing Equivalent Rational Expressions

For Exercises 9–24, convert the expressions into equivalent expressions with the indicated denominator.

9. 10. 11.

12. 13. 14.

15. 16. 17.

18. 19. 20.

21. 22. 23.

24.

25. Which of the expressions are equivalent to Circle all that apply.

a. b. c. d.

26. Which of the expressions are equivalent to Circle all that apply.

a. b. c. d.

Concept 2: Least Common Denominator

27. Explain why the least common denominator of

28. Explain why the least common denominator of 2y3, 9

y6, and 4y5 is y6.

1x3, 1

x5, and 1x4 is x5.

�a � 4

6

�14 � a2

�6a � 4

6a � 4

�6

4 � a6

?

5�1x � 32

53 � x

5�x � 3

�5x � 3

�5

x � 3?

2a � 9

�9 � a

6x � 3

�3 � x

3y

4y � 5�

16y2 � 255

x � 2�

x2 � 4

�83a � 2

�12a � 8

�4z � 3

�5z � 15

z � 1z � 1

�1z � 12 1z � 32

w � 6w � 7

�1w � 72 1w � 22

8ab2c

�2b4c5

2xyz

�6y2z4

23rs

�18rs3

3p2q

�5p3q

18

�64

213

�39

49

�72

67

�42


29. Explain why the least common denominator of 30. Explain why the least common denominator of

31. Explain why a common denominator of 32. Explain why a common denominator of

could be either could be either 16 � t2 or 1t � 62.1b � 12 or 11 � b2.

16 � t

and t

t � 6b � 1b � 1

and b

1 � b

is 1y � 82 1y � 12.is 1x � 32 1x � 22.

7y � 8

and 3

y � 11

x � 3 and

3x � 2



For Exercises 33–50, identify the LCD.

33. 34. 35.

36. 37. 38.

39. 40. 41.

42. 43. 44.

45. 46. 47.

48. 49. 50.

Concept 3: Writing Rational Expressions with the Least Common Denominator

For Exercises 51–74, find the LCD. Then convert each expression to an equivalent expression with thedenominator equal to the LCD.

51. 52. 53.

54. 55. 56.

57. 58. 59.

60. 61. 62.

63. 64. 65.1

a � 4;

a4 � a

5q2 � 6q � 9

; q

q2 � 9

6p

p2 � 4;

3p2 � 4p � 4

t1t � 22 1t � 122

; 18

1t � 22 1t � 226

1w � 32 1w � 82;

w1w � 82 1w � 12

4m � 3

; �3

5m � 1

62x � 5

; 1

x � 33

n � 5;

7n � 2

6m � 4

; 3

m � 1

x15y2;

y

5xy

56a2b

; a

12bc

3b2

315b2;

y

6x3

45x2;

79y2

3y

;1x

65x2;

96 � x

4x � 6

;7

x � 35

3 � x;

4w2 � 3w � 2

; w

w2 � 4

y

y2 � 4;

3y

y2 � 5y � 613x

151x � 122;

53x1x � 12

73t1t � 12

; 10t

91t � 122

61q � 42 1q � 42

; q2

1q � 12 1q � 42p

1p � 32 1p � 12;

21p � 32 1p � 22

2r;

3s2

6w2;

7y

52a4b2;

18ab3

13x2y

; 8

9xy3

23

; 58

17

; 29

12

; 1112

316

; 14

712

; 118

415

; 59

66. 67. 68.

69. 70. 71.

72. 73. 74.�1

4a � 8;

54a

35z

; 1

z � 4r

10r � 5;

216r � 8

�324y � 8

; 5

18y � 6p

�q � 8;

1q � 8

1a � b

; 6

�a � b

43x � 15

; z

5 � x

4x � 7

; y

14 � 2x

3b2b � 5

; 2b

5 � 2b



Addition and Subtraction of Rational Expressions

Let p, q, and r represent polynomials where Then,

1.

2.pq

�rq

�p � r

q

pq

�rq

�p � r

q

q � 0.

Adding and Subtracting Rational Expressions with a Common Denominator

Add or subtract as indicated.

a. b.

c. d.x2

x � 3�

�5x � 24x � 3

23d � 5

�7d

3d � 5

25p

�7

5p1

12�

712

Example 1

1. Addition and Subtraction of Rational Expressions with the Same Denominator

To add or subtract rational expressions, the expressions must have the samedenominator. As with fractions, we add or subtract rational expressions with thesame denominator by combining the terms in the numerator and then writing theresult over the common denominator. Then, if possible, we simplify the expressionto lowest terms.

Section 7.4 Addition and Subtraction of Rational Expressions

Concepts

1. Addition and Subtraction of

Rational Expressions with the

Same Denominator

2. Addition and Subtraction of

Rational Expressions with

Different Denominators

3. Using Rational Expressions

in Translations


For Exercises 75–78, find the LCD. Then convert each expression to an equivalent expression with thedenominator equal to the LCD.

75.

76.

77.

78.7

q3 � 125;

q

q2 � 25;

12q2 � 5q � 25

3p3 � 8

; p

p2 � 4;

5p

p2 � 2p � 4

6w2 � 3w � 4

; 1

w2 � 6w � 5;

�9ww2 � w � 20

zz2 � 9z � 14

; �3z

z2 � 10z � 21;

5z2 � 5z � 6


Solution:

a. The fractions have the same denominator.

Add the terms in the numerators, and write theresult over the common denominator.


b. The rational expressions have the samedenominator.

Subtract the terms in the numerators, and writethe result over the common denominator.


c. The rational expressions have the same denominator.


Because the numerator and denominator share nocommon factors, the expression is in lowest terms.

d. The rational expressions have the same denominator.

Subtract the terms in the numerators, and writethe result over the common denominator.

Simplify the numerator.

Factor the numerator and denominator to deter-mine if the rational expression can be simplified.


� x � 8

�1x � 82 1x � 32

1

1x � 32

�1x � 82 1x � 32

1x � 32

�x2 � 5x � 24

x � 3

x2

x � 3�

�5x � 24x � 3

�7d � 23d � 5

�2 � 7d3d � 5

23d � 5

�7d

3d � 5

� �1p

�1� 5

�12

5p

��55p

�2 � 7

5p

25p

�7

5p

�23

�812

�1 � 7

12

112

�7

12

Section 7.4 Addition and Subtraction of Rational Expressions 519

�x2 � 1�5x � 242

x � 3

Avoiding Mistakes:

When subtracting rational expres-sions, use parentheses to groupthe terms in the numerator thatfollow the subtraction sign. Thiswill help you remember to applythe distributive property.


Steps to Add or Subtract Rational Expressions

1. Factor the denominators of each rational expression.

2. Identify the LCD.

3. Rewrite each rational expression as an equivalent expression with theLCD as its denominator.

4. Add or subtract the numerators, and write the result over the commondenominator.

5. Simplify to lowest terms.



1. 2.

3. 4.

2. Addition and Subtraction of Rational Expressions with Different Denominators

To add or subtract two rational expressions with unlike denominators, we mustconvert the expressions to equivalent expressions with the same denominator. Forexample, consider adding

The LCD is 10y. For each expression, identify the factors from the LCD thatare missing from the denominator. Then multiply the numerator and denomi-nator of the expression by the missing factor(s).

missing missingy 2

The rational expressions now have the same denominators.

Add the numerators.

After successfully adding or subtracting two rational expressions, always checkto see if the final answer is simplified. If necessary, factor the numerator anddenominator, and reduce common factors. The expression

is in lowest terms because the numerator and denominator do not share anycommon factors.

y � 24

10y

�y � 24

10y

�y

10y�

2410y

�1 � y

10 � y�

12 � 25y � 2

110

� 125y

110

�125y

4t � 92t � 1

�t � 5

2t � 1x 2 � 2x � 3

�4x � 1x � 3

37d

�6

7d314

�4

14

Skill Practice

rr


1. 2.

3. 4.3t � 42t � 1

x � 1

�37d

12

Avoiding Mistakes:

In the expression , noticethat you cannot reduce the 24and 10 because 24 is not a factorin the numerator. It is a term. Onlyfactors can be reduced.

y � 2410y


Adding and Subtracting Rational Expressions with Unlike Denominators


a. b. c.

Solution:

a. Step 1: The denominators are alreadyfactored.

Step 2: The LCD is

Step 3: Write each expression with the LCD.

Step 4: Subtract the numerators, and writethe result over the LCD.

Step 5: The expression is in lowest termsbecause the numerator anddenominator share no commonfactors.

b. Step 1: The denominators are alreadyfactored.

Step 2: The LCD is 6.

Step 3: Write each expression with the LCD.

Step 4: Subtract the numerators, and writethe result over the LCD.

Step 5: The expression is in lowest termsbecause the numerator anddenominator share no commonfactors.

� q � 11

6

� 4q � 8 � 3q � 3

6

� 212q � 42 � 31q � 12

6

� 212q � 42

2 � 3�

31q � 12

3 � 2

2q � 43

�q � 1

2

�4k � 21

7k2

�4k7k2 �

217k2

�4 � k

7k � k�

3 � 7k2 � 7

7k2.

47k

�3k2

1x � 5

��10

x2 � 25

2q � 43

�q � 1

24

7k�

3k2

Example 2

Avoiding Mistakes:

Do not reduce after rewriting thefractions with the LCD. You willrevert back to the originalexpression.



c.


Step 2: The LCD is

Step 3: Write each expressionwith the LCD.

Step 4: Add the numerators, andwrite the result over theLCD.

Step 5: Simplify.


5. 6. 7.

Subtracting Rational Expressions with Different Denominators

Subtract the rational expressions.

Solution:


Step 2: The LCD is

Step 3: Write each expression withthe LCD.

Step 4: Combine the numerators,and write the result over the LCD.

� 1p � 22 1p � 62 � 21p � 12 � 14

1p � 12 1p � 62

� 1p � 22 1p � 62

1p � 12 1p � 62�

21p � 12

1p � 62 1p � 12�

141p � 12 1p � 62

1p � 12 1p � 62.

� p � 2p � 1

�2

p � 6�

141p � 12 1 p � 62

p � 2p � 1

�2

p � 6�

14p2 � 5p � 6

p � 2p � 1

�2

p � 6�

14p2 � 5p � 6

Example 3

1x � 4

��8

x 2 � 16

q

12�

q � 24

43x

�1

2x 2

Skill Practice

�1

x � 5

�x � 5

1

1x � 52 1x � 52

�x � 5 � 101x � 52 1x � 52

�11x � 52 � 1�102

1x � 52 1x � 52

�11x � 52

1x � 52 1x � 52�

�101x � 52 1x � 52

1x � 52 1x � 52.

�1

x � 5�

�101x � 52 1x � 52

1x � 5

��10

x2 � 25



5. 6.

7.1

x � 4

�q � 3

68x � 3

6x2


Step 5: Clear parentheses in thenumerator.

Combine like terms.

Factor the numerator todetermine if the expressionis in lowest terms.


Subtract.

8.

When the denominators of two rational expressions are opposites, we can pro-duce identical denominators by multiplying one of the expressions by the ratio This is demonstrated in Example 4.

Adding Rational Expressions with Different Denominators

Add the rational expressions.

Solution:

The expressions and are oppositesand differ by a factor of Therefore, multiplythe numerator and denominator of eitherexpression by to obtain a commondenominator.

Note that

Simplify.


� �4

d � 7

� 1 � 1�52

d � 7

� 1

d � 7�

�5d � 7

�117 � d2 � �7 � d or d � 7.� 1

d � 7�

1�125

1�12 17 � d2

�1

�1.7 � dd � 7

1d � 7

�5

7 � d

1d � 7

�5

7 � d

Example 4

�1�1.

2y

y � 1�

1y

�2y � 1

y2 � y

Skill Practice

� p

p � 1

� p1p � 6

12

1p � 12 1p � 62

� p1p � 62

1p � 12 1p � 62

� p2 � 6p

1p � 12 1p � 62

� p2 � 6p � 2p � 12 � 2p � 2 � 14

1p � 12 1p � 62


8. 9.2

p � 8 or

�28 � p

2y � 3

y � 1


Add.

9.3

p � 8�

18 � p

Skill Practice


3. Using Rational Expressions in Translations

Using Rational Expressions in Translations

Translate the English phrase into a mathematical expression. Then simplifyby combining the rational expressions.

The difference of the reciprocal of x and the quotient of x and 3

Solution:

The difference of the reciprocal of x and the quotient of x and 3

The difference of

the reciprocal the quotient of x of x and 3

The LCD is 3x.

Write each expression over the LCD.

Subtract the numerators.

Translate the English phrase into a mathematical expression.Then simplify by combining the rational expressions.

10. The sum of 1 and the quotient of 2 and a

Skill Practice

� 3 � x2

3x

� 3 � 13 � x

�x � x3 � x

1x

�x3

a1xb � a

x3b

Example 5


Review Exercises

1. Write the domain of the expression.

2. For the rational expression

a. Find the value of the expression (if possible) when and 5.

b. Factor the denominator and identify the domain. Write the domain in set-builder notation.

c. Reduce the expression to lowest terms.

x � 0, 1, �1, 2,

x2 � 4x � 5x2 � 7x � 10

x � 4x2 � 36




• NetTutor



10. 1 �2a

; a � 2

a



a. Find the value of the expression (if possible) when and 6.

b. Factor the denominator, and identify the domain. Write the domain in set-builder notation.

c. Reduce the expression to lowest terms.


4. 5.

Concept 1: Addition and Subtraction of Rational Expressions with the Same Denominator

For Exercises 6–27, add or subtract the expressions with like denominators as indicated.

6. 7. 8.

9. 10. 11.

12. 13. 14.

15. 16. 17.

18. 19. 20.

21. 22. 23.

24. 25. 26.

27.

For Exercises 28–29, find an expression that represents the perimeter of the figure (assume that and

28. 29.

12t � 3

6t � 3

6xy

2xy

7xy

t 7 02.x 7 0, y 7 0,

52x2 � 13x � 20

�2x

2x2 � 13x � 20

13y2 � 22y � 7

��3y

3y2 � 22y � 7x

x2 � x � 12�

4x2 � x � 12

rr 2 � 3r � 2

�2

r 2 � 3r � 2

y2

y � 7�

49y � 7

x2

x � 5�

25x � 5

5bb � 4

�20

b � 4

2aa � 2

�4

a � 2k2

k � 3�

6k � 9k � 3

m2

m � 5�

10m � 25m � 5

4w2

2w � 1�

12w � 1

9x2

3x � 7�

493x � 7

7p � 12p � 1

�p � 4

2p � 1

5t � 8

�2t � 1t � 8

122 � d

�6d

2 � d5c

c � 6�

30c � 6

2bb � 3

�b � 9b � 3

5aa � 2

�3a � 4a � 2

1415

�4

15

916

�3

1613

�73

78

�38

6t � 15t � 30

�10t � 25

2t2 � 3t � 52b2 � b � 3

2b2 � 3b � 9�

b2 � 14b � 6

a � 0, 1, �2, 2,

a2 � a � 2a2 � 4a � 12



Concept 2: Addition and Subtraction of Rational Expressions with Different Denominators

For Exercises 30–71, add or subtract the expressions with unlike denominators as indicated.

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.

57. 58. 59.

60. 61. 62.

63. 64. 65.

66. 67. 68.

69. 70. 71.

For Exercises 72–73, find an expression that represents the perimeter of the figure (assume that and ).

72. 73.

32t

1t2

5t

1x � 2

2x � 3

t 7 0x 7 0

�2xx2 � y2 �

1x � y

�1

x � y2

a � b�

2a � b

�4a

a2 � b2

xx � y

�2xy

x2 � y2 �y

x � y

mm � n

�m

m � n�

1m2 � n2

13q � 2

�6q � 4

9q2 � 43

2p � 1�

4p � 4

4p2 � 1

56y2 � 7y � 3

�4y

3y2 � 4y � 1

3y

2y2 � y � 1�

4y

2y2 � 7y � 4x

x2 � 5x � 4�

2xx2 � 2x � 3

3xx2 � x � 6

�x

x2 � 5x � 63b � 5

b2 � 4b � 3�

�b � 5b2 � 2b � 3

3a � 8a2 � 5a � 6

�a � 2

a2 � 6a � 8

z � 23z2 � z � 20

�2

5 � z4w

w2 � 2w � 3�

21 � w

4q2 � 2q

�5

3q � 6

y

4y � 2�

3y

6y � 31

7x�

52y2

5p � 3

�2

p � 1

6y � 1

�9y

5x

�3

x � 2m

m � 2�

3m � 12 � m

4nn � 8

�2n � 18 � n

r7

�r � 5�7

p

3�

4p � 1�3

7xx2 � 2xy � y2 �

3xx2 � xy

6aa2 � b2 �

2aa2 � ab

82w � 1

�4

1 � 2w

103x � 7

�5

7 � 3xk � 2

8k�

3 � k12k

3a � 76a � 10

�10

3a2 � 5a

7h � 5

�2h � 3h2 � 25

kk2 � 9

�4

k � 32

c � 4�

15c � 20

5a � 1

�4

3a � 33w � 82w � 4

�w � 3w � 2

z3z � 9

�z � 2z � 3

1p2q

�2

pq3

2s3t3 �

3s4t

53a2b

��76b2

45xy3 �

2x15y2

116p

��74p

54

�3

2a



Concept 3: Using Rational Expressions in Translations

74. Let a number be represented by n. Write the reciprocal of n.

75. Write the reciprocal of the sum of a number and 6.

76. Write the quotient of 5 and the sum of a number and 2.

77. Let a number be represented by p. Write the quotient of 12 and p.

For Exercises 78–81, translate the English phrases into algebraic expressions. Then simplify by combining therational expressions.

78. The sum of a number and the quantity seven times the reciprocal of the number.

79. The sum of a number and the quantity five times the reciprocal of the number.

80. The difference of the reciprocal of n and the quotient of 2 and n.

81. The difference of the reciprocal of m and the quotient of 3m and 7.


For Exercises 82–87, perform the indicated operations.

82. 83.

84. 85.

86. 87.

For Exercises 88–91, simplify by applying the order of operations.

88. 89.

90. 91. a1

2m�

n2m2b � a

14

�n

4mba

110a

�b

10a2b � a110

�b

10ab

ap � 1

3p � 4b a

1p � 1

� 2ba2

k � 1� 3b a

k � 14k � 7

b

2n3n2 � 8n � 3

�1

6 � 2n�

33n � 1

3mm2 � 3m � 10

�5

4 � 2m�

1m � 5

3t8t2 � 2t � 1

�5t

2t2 � 9t � 5�

24t2 � 21t � 5

2p

p2 � 5p � 6�

p � 1

p2 � 2p � 3�

3p2 � p � 2

mm3 � 1

�1

1m � 122�3

w3 � 27�

1w2 � 9



11. Divide.

12. Add.

13. Simplify.

14. Multiply.

15. Divide.

16. Simplify.

17. Subtract.

18. Add.

19. Multiply.

20. Simplify.

For Exercises 21–22, determine the domain.

21. 22.x � 2x2 � 9

x � 3x � 1

6b2 � 7b � 10b � 2

1t2 � 5t � 242 at � 8t � 3

b

4y2 � 36

�2

y2 � 4y � 12

aa2 � 9

�3

6a � 18

xy � 7x � 5y � 35

x2 � ax � 5x � 5a

8x2 � 18x � 54x2 � 25

�4x2 � 11x � 3

3x � 9

w2 � 81w2 � 10w � 9

�w2 � w � 2zw � 2zw2 � 9w � zw � 9z

20x2 � 10x4x3 � 4x2 � x

3k � 8k � 5

�k � 12k � 5

p2 � 10pq � 25q2

p2 � 6pq � 5q2 �10p � 50q

2p2 � 2q2


Chapter 7 Problem Recognition Exercises—Operations on Rational Expressions

In Sections 7.1–7.4, we learned how to simplify, add,subtract, multiply, and divide rational expressions. Theprocedure for each operation is different, and it takesconsiderable practice to determine the correct methodto apply for a given problem. The following reviewexercises give you the opportunity to practice the spe-cific techniques for simplifying rational expressions.

1. Subtract.

2. Divide.

3. Multiply.

4. Add.

5. Simplify.

6. Add and subtract.

7. Divide.

8. Multiply.

9. Simplify.

10. Subtract.2a

a � b�

ba � b

��4ab

a2 � b2

6a2b3

72ab7c

2x2 � 5x � 3x2 � 9

�x2 � 6x � 9

10x � 5

c2 � 5c � 6c2 � c � 2

�c

c � 1

1p

�3

p2 � 3p�

p

3p � 9

x � 99x � x2

�1x � 3

�2

2x � 1

3y

�y2 � 5y

6y � 9

w � 1w2 � 16w � 1w � 4

53x � 1

�2x � 43x � 1


Section 7.5Complex Fractions

Concepts

1. Simplifying Complex Fractions

(Method I)

2. Simplifying Complex Fractions

(Method II)

1. Simplifying Complex Fractions (Method I)A complex fraction is a fraction whose numerator or denominator contains oneor more rational expressions. For example,

are complex fractions.Two methods will be presented to simplify complex fractions. The first method

(Method I) follows the order of operations to simplify the numerator and denom-inator separately before dividing. The process is summarized as follows.

1ab2b

and 1 �

34

�16

12

�13

Simplifying Complex Fractions (Method I)

Simplify the expression.

Solution:

Step 1: The numerator and denominator of the complexfraction are already single fractions.

This fraction bar denotes division

Step 2: Multiply the numerator of the complex fraction bythe reciprocal of

1

Step 3: Reduce common factors and simplify.

�12a

�1

ab�

b2

2b, which is b2.

�1

ab�

b2

�1

ab�

2b

1�2.

1ab2b

1ab2b

Example 1

Steps to Simplify a Complex Fraction (Method I)

1. Add or subtract expressions in the numerator to form a single fraction.Add or subtract expressions in the denominator to form a single fraction.

2. Divide the rational expressions from step 1 by multiplying the numera-tor of the complex fraction by the reciprocal of the denominator of thecomplex fraction.

3. Simplify to lowest terms if possible.

Section 7.5 Complex Fractions 529



1.

Sometimes it is necessary to simplify the numerator and denominator of acomplex fraction before the division can be performed. This is illustrated in thenext example.



Solution:

Step 1: Combine fractions in the numera-tor and denominator separately.

Form a single fraction in thenumerator and in thedenominator.

Step 2: Multiply by the reciprocal of

Step 3: Simplify.


2.

34

�16

� 2

13

�12

Skill Practice

�1910

56, which is 65.

�1912

2

�61

5

�

191256

�

1212

�9

12�

212

36

�26

�

1 �1212

�34

�33

�16

�22

12

�33

�13

�22

1 �34

�16

12

�13

1 �34

�16

12

�13

Example 2

Skill Practice


The LCD in the numerator is 12.The LCD in the denominator is 6.


1.

2.3110

4x3

6xy

92y





Solution:

The LCD in the numerator is xy. The LCD inthe denominator is x.

Rewrite the expressions using commondenominators.

Form single fractions in the numerator anddenominator.

Multiply by the reciprocal of the denominator.

Factor and reduce. Note that

Simplify.


3.

2. Simplifying Complex Fractions (Method II)We will now simplify the expressions from Examples 2 and 3 again using a sec-ond method to simplify complex fractions (Method II). Recall that multiplyingthe numerator and denominator of a rational expression by the same quantitydoes not change the value of the expression because we are multiplying by anumber equivalent to 1. This is the basis for Method II.

1 �qp

pq

�qp

Skill Practice

�1

y1x � y2

1y � x2� 1x � y2.�y � x

1

xy�

x1

1x � y2 1x � y2

�y � x

xy�

xx2 � y2

�

y � xxy

x2 � y2

x

�

yxy

�x

xy

x2

x�

y2

x

�

1 � yx � y

�1 � xy � x

x � x1 � x

� y2

x

1x

�1y

x �y2

x

1x

�1y

x �y2

x

Example 3


3.q

p � q


Simplifying Complex Fractions (Method II)


Solution:

The LCD of the expressions

Step 1: Multiply the numerator and denominator of the complexfraction by 12.

Step 2: Apply the distributive property.

Simplify each term.

Step 3: Simplify.


4.1 �

35

14

�7

10� 1

Skill Practice

�1910

�12 � 9 � 2

6 � 4

�

12 � 1 � 123

�34

� 122

�16

612 �

12

�4

12 �13

�

12 � 1 � 12 �34

� 12 �16

12 �12

� 12 �13

�

12 a1 �34

�16b

12 a12

�13b

and 13 is 12.1, 34, 16,

12,

1 �34

�16

12

�13

1 �34

�16

12

�13

Example 4


Steps to Simplifying a Complex Fraction (Method II)

1. Multiply the numerator and denominator of the complex fraction by theLCD of all individual fractions within the expression.

2. Apply the distributive property, and simplify the numerator anddenominator.

3. Simplify to lowest terms if possible.

TIP: In step 1, we are

multiplying the original

expression by , which

equals 1.

1212


4.811



Simplifying a Complex Fraction (Method II)


Solution:

The LCD of the expressions and

Step 1: Multiply numerator and denominator ofthe complex fraction by xy.

Step 2: Apply the distributive property, and simplify each term.

Step 3: Factor completely, and reduce commonfactors.

Note that


5.

Simplifying a Complex Fraction (Method II)


1k � 1

� 1

1k � 1

� 1

Example 6

z3

�3z

1 �3z

Skill Practice

�1

y1x � y2

1y � x2 � 1x � y2.�y � x

1

y1x � y2 1x � y2

�y � x

y1x2 � y22

�y � x

x2y � y3

�

xy �1x

� xy �1y

xy � x � xy �y2

x

�

xy a1x

�1yb

xy ax �y2

xb

y2

x is xy.

1x,

1y, x,

1x

�1y

x �y2

x

1x

�1y

x �y2

x

Example 5


5.z � 3

3



Solution:

The LCD of and 1 is

Step 1: Multiply numerator anddenominator of the com-plex fraction by

Step 2: Apply the distributive property.

Simplify.

Step 3: The expression is already inlowest terms.


6.

4p � 3

� 1

1 �2

p � 3

Skill Practice

��k

k � 2

�1 � k � 11 � k � 1

�1 � 1k � 12

1 � 1k � 12

�

1k � 11

2 �1

1k � 12� 1k � 12 � 1

1k � 11

2 �1

1k � 12� 1k � 12 � 1

1k � 12.

�

1k � 12a1

k � 1� 1b

1k � 12a1

k � 1� 1b

1k � 12.

1k � 1

1k � 1

� 1

1k � 1

� 1


6.p � 1

p � 1




• NetTutor



1. Define the key term complex fraction.

Review Exercises

For Exercises 2–3, write the domain in set-builder notation, and simplify the expression.

2. 3.


4. 5. 6.p2 � 2p

2p � 1�

10p2 � 5p

12p3 � 24p2

65

�3

5k � 102

w � 2�

3w

a � 52a2 � 7a � 15

y12y � 92

y212y � 92



7. 8. 9.

Concepts 1–2: Simplifying Complex Fractions (Methods I and II)

For Exercises 10–35, simplify the complex fractions.

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.

For Exercises 36–39, translate the English phrases into algebraic expressions. Then simplify the expressions.

36. The sum of one-half and two-thirds, divided by five.

37. The quotient of ten and the difference of two-fifths and one-fourth.

38. The quotient of three and the sum of two-thirds and three-fourths.

39. The difference of three-fifths and one-half, divided by four.

94m

�9

2m2

32

�3m

t � 4 �3t

t � 4 �5t

1 �9p2

1 �1p

�6p2

1 �4t 2

1 �2t

�8t 2

2w � 1

� 3

3w � 1

� 4

8a � 4

� 2

12a � 4

� 2

1m2 �

23

1m

�56

15

�1y

710

�1y2

2p

�p

2p

3�

3p

m7

�7m

17

�1m

6 �6k

1 �1k

2 �1x

4 �1x

5k � 5k � 1

k2 � 25

n � 1n2 � 9

2n � 3

1b

� 1

1b

1h

�1k

1hk

89

�13

76

�19

18

�43

12

�5

12

5p4q

w4

10p2

qw2

4r 3s

t5

2s7

r 2t 9

12x2

5y8x6

9y2

8a4b3

3ca7b2

9c

2x � 104

x2 � 5x3x

3x � 2y

2y6x � 4y

2

a2

2a � 35a

8a � 12

718y29

a2

3a2 �3bb � a

5ab

� 4ba1z

�1

2zb � a

12

�1

2zb

x2 � 2xy � y2

x4 � y4 �3x2y � 3xy2

x2 � y2


40. In electronics, resistors oppose the flow of current. For two resistors in parallel, the total resistance is given by

a. Find the total resistance if (ohms) and .

b. Find the total resistance if and .

41. Suppose that Joëlle makes a round trip to a location that is d miles away. If theaverage rate going to the location is and the average rate on the return trip isgiven by the average rate of the entire trip, R, is given by

a. Find the average rate of a trip to a destination 30 miles away when the average rate going there was 60 mphand the average rate returning home was 45 mph. (Round to the nearest tenth of a mile per hour.)

b. Find the average rate of a trip to a destination that is 50 miles away if the driver travels at the same ratesas in part (a). (Round to the nearest tenth of a mile per hour.)

c. Compare your answers from parts (a) and (b) and explain the results in the context of the problem.


For Exercises 42–45, simplify the complex fractions using either method.

42. 43.

44. 45.

For Exercises 46–48, simplify the complex fractions. (Hint: Use the order of operations and begin with the fractionon the lower right.)

46. 47. 48. 1 �1

1 �1

1 �1

1 � 1

1 �1

1 �1

1 � 1

1 �1

1 � 1

1y � 3

� 1

2y � 3

� 1

2x � 1

� 2

2x � 1

� 2

5w2 � 25

�3

w � 54

w � 5

1z2 � 9

�2

z � 33

z � 3

R �2d

dr1

�dr2

r2,r1

R2 � 15 R1 � 10

R2 � 3 R1 � 2

R �1

1R1

�1

R2



Section 7.6 Rational Equations 537

1. Introduction to Rational EquationsThus far we have studied two specific types of equations in one variable: linearequations and quadratic equations. Recall,

.

We will now study another type of equation called a rational equation.

ax2 � bx � c � 0, where a � 0, is a quadratic equation.

ax � b � 0, where a � 0, is a linear equation

Section 7.6Rational Equations

Concepts

1. Introduction to Rational

Equations

2. Solving Rational Equations

3. Solving Formulas Involving

Rational Equations

Definition of a Rational Equation

An equation with one or more rational expressions is called a rationalequation.

The following equations are rational equations:

To understand the process of solving a rational equation, first review the processof clearing fractions from Section 2.3.

Solving a Rational Equation

Solve.

Solution:

The LCD of all terms in the equation is 4.

Multiply both sides of the equation by 4 to clearfractions.


Clear fractions.

Solve the resulting equation (linear).

Check:

✔ 6 � 6

4 � 2 � 6

182

2�182

4� 6

y

2�

y

4� 6

y � 8

3y � 24

2y � y � 24

4 �y

2� 4 �

y

4� 4162

4 ay

2�

y

4b � 4162

y

2�

y

4� 6

y

2�

y

4� 6

Example 1

y

2�

y

4� 6

1x

�13

�56

6t2 � 7t � 12

�2t

t � 3�

3tt � 4



Solve the equation.

1.

2. Solving Rational EquationsThe same process of clearing fractions is used to solve rational equations whenvariables are present in the denominator.


Solve the equation.

Solution:

Check:

✔

Solve the equation.

2.34

�5 � a

a�

12

Skill Practice

36

�26

�56

12

�13

� 56

�1�2

�13

� 56

1�22 � 1

1�22�

13

� 56

x � 1x

�13

�56

x � �2

3x � �6

8x � 6 � 5x

6x � 6 � 2x � 5x

61x � 12 � 2x � 5x

6x � ax � 1

xb � 6x � a

13b � 6x � a

56b

6x � ax � 1

x�

13b � 6x � a

56b

x � 1

x�

13

�56

x � 1x

�13

�56

Example 2

t5

�t4

� 2

Skill Practice


1. 2. a � �4t � �40

The LCD of all the expressionsis 6x.

Multiply by the LCD.


Clear fractions.

Solve the resulting equation.




Solve the equation.

Solution:

The LCD of all theexpressions is


Check:

The denominator is 0 when a � 2.

Because the value makes the denominator zero in one (or more) of therational expressions within the equation, the equation is undefined for That is, is not in the domain of the equation, therefore, it is an extrane-ous solution. No other potential solutions exist for the equation.

The equation has no solution.

Solve the equation.

3.

Examples 1–3 show that the steps to solve a rational equation mirror theprocess of clearing fractions from Section 2.3. However, there is one significant dif-ference. The solutions of a rational equation must not make the denominator equalto zero for any expression within the equation. When a � 2 is substituted into theexpression

3aa � 2

or 6

a � 2

xx � 1

� 2 ��1

x � 1

Skill Practice

1 �3a

a � 2�

6a � 2

a � 2a � 2.

a � 2

1 �60

� 60

1 �3122122 � 2

� 6122 � 2

1 �3a

a � 2�

6a � 2

a � 2

4a � 8

4a � 2 � 6

a � 2 � 3a � 6

1a � 221 � 1a � 22a3a

a � 2b � 1a � 22a

6a � 2

b

1a � 22a1 �3a

a � 2b � 1a � 22a

6a � 2

b

a � 2. 1 �

3aa � 2

�6

a � 2

1 �3a

a � 2�

6a � 2

Example 3

Apply the distributiveproperty.

Solve the resultingequation (linear).


3. No solution; (x � �1 does not check.)



the denominator is zero and the expression is undefined. Hence, cannotbe a solution to the equation

The steps to solve a rational equation are summarized as follows.

1 �3a

a � 2�

6a � 2

a � 2

Steps to Solve a Rational Equation

1. Factor the denominators of all rational expressions.

2. Identify the LCD of all expressions in the equation.

3. Multiply both sides of the equation by the LCD.

4. Solve the resulting equation.

5. Check potential solutions in the original equation.

Solving Rational Equations

Solve the equations.

a. b.

Solution:

a. Step 1: The denominators are already factored.

Step 2: The LCD of all expressions is

Step 3: Multiply by the LCD.


Step 4: Solve the resulting equation(quadratic).

Set the equation equal to zeroand factor.


or Step 5: Check: Check:

Both solutions and check. ✔ ✔�3 � �3�13

� �13

p � 1p � 3

1 � 4 � �333

�43

� �39

1 �4112

� �31122

1 �4132

� �31322

1 �4p

� �3p21 �

4p

� �3p2

p � 1p � 3p � 1 p � 3

p � 3 � 0 or p � 1 � 0

1p � 32 1p � 12 � 0

p2 � 4p � 3 � 0

p2 � 4p � �3

p2112 � p2 a

4pb � p2

a�3p2b

p2 a1 �

4pb � p2

a�3p2b

p2.

1 �4p

� �3p2

6t2 � 7t � 12

�2t

t � 3�

3tt � 4

1 �4p

� �3p2

Example 4



b.


Step 2: The LCD is .

Step 3: Multiply by the LCD on bothsides.

Step 4: Solve the resulting equation.

Because the resultingequation is quadratic, set the equation equal to zero and factor.


Step 5: Check the potential solutionsin the original equation.

Check: Check:

cannot be a solution to the equation because it will make the denominator zero in the original equation.

✔

is a solution.

The only solution is

Solve the equations.

4. 5. �2

x � 2�8

x 2 � 6x � 8�

xx � 4

z2

�1

2z�

12z

Skill Practice

t � �2.

t � �2

15

�45

� 160

�60

� 9�1

630

�45

� 16

1322 � 7132 � 12�

2132

132 � 3�

3132

132 � 4

64 � 14 � 12

��4�5

� �6�6

6t2 � 7t � 12

�2t

t � 3�

3tt � 4

6

1�222 � 71�22 � 12�

21�22

1�22 � 3�

31�22

1�22 � 4

6t2 � 7t � 12

�2t

t � 3�

3tt � 4

t � 3

t � �2t � 3

t � 3 or t � �2

t � 3 � 0 or t � 2 � 0

0 � 1t � 32 1t � 22

0 � t2 � t � 6

0 � 3t2 � 2t2 � 9t � 8t � 6

6 � 2t2 � 8t � 3t2 � 9t

6 � 2t1t � 42 � 3t1t � 32

1t � 32 1t � 42a6

1t � 32 1t � 42b � 1t � 32 1t � 42a

2tt � 3

b � 1t � 32 1t � 42a3t

t � 4b

1t � 32 1t � 42a6

1t � 32 1t � 42�

2tt � 3

b � 1t � 32 1t � 42a3t

t � 4b

1t � 421t � 32

61t � 32 1t � 42

�2t

t � 3�

3tt � 4

6

t2 � 7t � 12�

2tt � 3

�3t

t � 4


4.

5. does not check.)x � 4; 1x � �4z � 5 or z � �5

zero in the denominator

TIP: t � 3 is not a

solution because it is not

in the domain of the

equation.

TIP: Note that t � 3

and t � 4 are not defined

in the original expressions.

Therefore, they cannot be

solutions to the original

equation.



Translating to a Rational Equation

Ten times the reciprocal of a number is added to four. The result is equal to thequotient of twenty-two and the number. Find the number.

Solution:

Let x represent the number.

10 the reciprocal the quotient oftimes of a number 22 and the number

is added the resultto four is equal to

Step 1: The denominators arealready factored.

Step 2: The LCD is x.

Step 3: Multiply both sides bythe LCD.

Apply the distributiveproperty.

Step 4: Solve the resultingequation (linear).

is a potential solution. Step 5: Substituting intothe original equationverifies that it is a solution.

The number is 3.

6. The quotient of ten and a number is two less than four times the reciprocalof the number. Find the number.

3. Solving Formulas Involving Rational EquationsA rational equation may have more than one variable. To solve for a specificvariable within a rational equation, we can still apply principles of clearingfractions.

Solving a Formula Involving a Rational Equation

Solve for k. F �mak

Example 6

Skill Practice

x � 3x � 3

4x � 12

4x � 10 � 22

x a4 �10xb � x a

22xb

4 �10x

�22x

4 � 10 a1xb �

22x

Example 5


6. The number is �3.



Solution:

To solve for k, we must clear fractions so that k appears in the numerator.

The LCD is k.

Multiply both sides of the equation by the LCD.

Clear fractions.

Divide both sides by F.

7. Solve for t.

Solving a Formula Involving a Rational Equation

Solve for b.

Solution:

To solve for b, we must clear fractions so that b appears in the numerator.

The LCD is

Multiply both sides of the equation bythe LCD.


Subtract hB from both sides to isolatethe b term.

Divide by h.

8. Solve the formula for x. y �3

x � 2

Skill Practice

b �2A � hB

h

hbh

�2A � hB

h

hb � 2A � hB

hB � hb � 2A

h1B � b2 � a2A

B � bb � 1B � b2

1B � b2. h �2A

B � b

h �2A

B � b

Example 7

C �rtd

Skill Practice

k �maF

kFF

�maF

kF � ma

k � 1F 2 � k � amakb

F �mak

Avoiding Mistakes:

Algebra is case-sensitive. Thevariables B and b representdifferent values.


7.

8. x �3 � 2y

y or x �

3y

� 2

t �Cdr



TIP: The solution to Example 7 can be written in several forms. The quantity

can be left as a single rational expression or can be split into two fractions

and simplified.

b �2A � hB

h�

2A

h�

hB

h�

2A

h� B

2A � hB

h

Solving a Formula Involving a Rational Expression

Solve for z.

Solution:

To solve for z, we must clear fractions so that z appears in the numerator only.

LCD is

Multiply both sides of the equation bythe LCD.


Collect z terms on one side of the equation.

Factor out a z.

Divide by to solve for z.

9. Solve for h.bx

�ah

� 1

Skill Practice

y � 1z �x � yx

y � 1

z1y � 12 � x � yx

yz � z � x � yx

yx � yz � x � z

y1x � z2 � ax � zx � z

b1x � z21

1x � z2.y �x � zx � z

y �x � zx � z

Example 8



a. linear equation b. quadratic equation c. rational equation


9. or�ax

x � bh �

xab � x




• NetTutor




14. For the equation

a. Identify the domain of the equation.

b. Identify the LCD of all the denominators ofthe equation.

c. Solve the equation.

1w

�12

� �14





3z

�45

� �15

Review Exercises


2. 3. 4.

5. 6. 7.

Concept 1: Introduction to Rational Equations

For Exercises 8–13, solve the equations by first clearing the fractions.

8. 9. 10.

11. 12. 13.4y � 2

3�

76

� �y

62x � 3

4�

910

�x5

53

�16

k �3k � 5

4

32

p �13

�2p � 3

452

�12

b � 5 �13

b13

z �23

� �2z � 10

1 �1x

�12x 2

w � 4w2 � 9

�w � 3

w2 � 8w � 16

h �1h

15

�1

5h

2y

y � 3�

4y2 � 9

2x � 64x2 � 7x � 2

�x2 � 5x � 6

x2 � 42

x � 3�

3x2 � x � 6





x � 1x2 � 2x � 3

�1

x � 3�

1x � 1





10x � 2

�40

x2 � x � 6�

12x � 3

Concept 2: Solving Rational Equations

For Exercises 18–47, solve the equations.

18. 19. 20.

21. 22. 23.

24. 25. 26.a � 1

a� 1 �

a � 22a

1 �2m

�8

m21 �2y

�3y2

143x

�5x

� 25

6x�

7x

� 19b

�8b

�14

4t

�3t

�18

27

�1x

�23

18

�35

�5y



27. 28. 29.

30. 31. 32.

33. 34. 35.

36. 37. 38.

39. 40. 41.

42. 43.

44. 45.

46. 47.

For Exercises 48–51, translate to a rational equation and solve.

48. The reciprocal of a number is added to three. The result is the quotient of 25 and the number. Find thenumber.

49. The difference of three and the reciprocal of a number is equal to the quotient of 20 and the number. Findthe number.

50. If a number added to five is divided by the difference of the number and two, the result is three-fourths.Find the number.

51. If twice a number added to three is divided by the number plus one, the result is three-halves. Find thenumber.

Concept 3: Solving Formulas Involving Rational Equations

For Exercises 52–69, solve for the indicated variable.

52. 53. 54.

55. 56. 57.

58. 59. 60.

61. 62. for t 63. for mT � mf

m� gx �

at � bt

Vlw

� h for w

Vph

� r2 for hCpr

� 2 for rh �2A

B � b for B

I �E

R � r for rI �

ER � r

for RK �IRE for R

K �IRE for EK �

maF

for aK �maF

for m

6w � 1

�w

w � 5�

16w2 � 6w � 5

y � 2

y � 3�

�1y2 � 7y � 12

�3y

y � 4

a5a � 10

�1

a � 5�

7a2 � 3a � 10

x3x � 3

�3

x � 2�

7x2 � 3x � 2

y

y � 4�

32y2 � 16

� 3x

x � 6�

72x2 � 36

� 4

4xx � 3

�12

x � 3�

4x2 � 36x2 � 9

2xx � 4

�8

x � 4�

2x2 � 32x2 � 16

2x2 � 212x � 3

��11x

2x � 3

x2 � 3xx � 1

�4

x � 1x2 � 9x � 4

��10xx � 4

x2 � xx � 2

�12

x � 2

4ww � 3

� 3 �3w � 1w � 3

2tt � 2

� 2 �t � 8t � 2

�5q � 5

�q

q � 5� 2

p

p � 4� 5 �

4p � 4

24n � 4

�74

��3

n � 12

m � 3�

54m � 12

�38

t12

�t � 3

3t�

1t

w5

�w � 3

w� �

3w

7b � 45b

�95

�4b

B&ImiL52817_ch07_493-570 11/8/06 11:42 PM 6CONFIRMING PAGES

Problem Recognition Exercises—Comparing Rational Equations and Rational Expressions 547

64. for x 65. for w 66. for h

67. for P 68. for R 69. for bb � a

ab�

1f

1R

�1

R1�

1R2

1 � rt �AP

a � b �2Ah

w � nwn

� Px � y

xy� z

Often adding or subtracting rational expressions is confused with solving rational equations. When adding rationalexpressions, we combine the terms to simplify the expression. When solving an equation, we clear the fractions andfind numerical solutions, if possible. Both processes begin with finding the LCD, but the LCD is used differentlyin each process. Compare these two examples.

Chapter 7 Problem Recognition Exercises—ComparingRational Equations and Rational Expressions

2.

4.

6.

8.

10.

12.10

2t � 1� 1 �

t2t � 1

3m

�65

� �3m

2w � 1

�3

1w � 122

3bb � 1

�2b

b � 1

3 �2

a � 5

1x � 2

� 2 �x � 11x � 2

1.

3.

5.

7.

9.

11.7

2x � 2�

3x4x � 4

1x � 6

�3

x2 � 6x�

4x

4 �2

h � 3� 5

76p2 �

29p

�1

3p2

5t2

�t � 2

3� 5

y

2y � 4�

2y2 � 2y

For Exercises 1–12, solve the equation or simplify the expression by combining the terms.

Example 1:

Add.

�12 � x2

3x

�123x

�x2

3x

�33

� a4xb � a

x3b �

xx

4x

�x3

1The LCD is 3x.2

Example 2:

Solve.

The final answers are numbers.x � �2 or x � 2

x � 2 � 0 or x � 2 � 0

1x � 22 1x � 62 � 0

x2 � 8x � 12 � 0

12 � x2 � �8x

3x1

a4x

�x3b �

3x1a�

83b

4x

�x3

� �83

1The LCD is 3x.2

The final answer is arational expression.



1. Solving ProportionsIn this section, we look at how rational equations can be used to solve a vari-ety of applications. The first type of rational equation that will be applied iscalled a proportion.

Definition of a Proportion

An equation that equates two ratios or rates is called a proportion. Thus,for b � 0 and d � 0, is a proportion.

A proportion can be solved by multiplying both sides of the equation by theLCD and clearing fractions.

Solving a Proportion

Solve the proportion.

Solution:

The LCD is 11w.

Multiply by the LCD and clear fractions.


Check: w � 451

✔ Simplify to lowest terms.

Solve the proportion.

1.10b

�2

33

Skill Practice

311

�3

11

311

� 12314512

311

�123w

w � 451

3w3

�1353

3

3w � 1353

3w � 11 � 123

11w a311b � 11w a

123wb

311

�123w

311

�123w

Example 1

ab � c

d

Section 7.7 Applications of Rational Equations and Proportions

Concepts

1. Solving Proportions

2. Applications of Proportions

and Similar Triangles

3. Distance, Rate, and Time

Applications

4. Work Applications


1. b � 165


Section 7.7 Applications of Rational Equations and Proportions 549

2. Applications of Proportions and Similar Triangles

Using a Proportion in an Application

For a recent year, the population of Alabama was approximately 4.2 million. Atthat time,Alabama had seven representatives in the U.S. House of Representatives.In the same year, North Carolina had a population of approximately 7.2 million.If representation in the House is based on population in equal proportions foreach state, how many representatives did North Carolina have?

Solution:

Let x represent the number of representatives for North Carolina.

Set up a proportion by writing two equivalent ratios.

Multiply by the LCD,


North Carolina had 12 representatives.

2. A college keeps the ratio of students to faculty at 105 to 2. If the studentpopulation at the college is 1575, how many faculty members are needed?

Skill Practice

x � 12

4.2x4.2

�50.44.2

4.2x � 50.4

4.2x � 17.22 172

7x. 7x �4.27

� 7x �7.2x

4.27

�7.2x

Population of North Carolina

number of representatives4.27

�7.2x

Population of Alabama

number of representatives

Example 2


2. 30 faculty members are needed.

TIP: The cross products of any proportion are equal. That is, for and

the proportion is equivalent to Some rational equations

are proportions and can be solved by equating the cross products. Consider

the proportion from Example 1:

Equate the cross products.

Solve the resulting equation.

w � 451

3w

3�

1353

3

3w � 1353

3 � w � 11 � 123

3

11�

123

w

ad � bc.ab � c

dd � 0,

b � 0

TIP: The equation

from Example 2 could

have been solved by first

equating the cross

products:

x � 12

4.2x � 50.4

4.2x � 17.22 172

4.2

7�

7.2

x


Using Similar Triangles to Find an Unknown Side in a Triangle

The triangles in Figure 7-2 are similar.

a. Solve for x. b. Solve for y.

Solution:

a. The lengths of the upper right sides of the triangles are given. These forma known ratio of . Because the triangles are similar, the ratio of the othercorresponding sides must be equal to . To solve for x, we have:

The LCD is 18.


Clear fractions.

Divide by 2.

The length of side x is 15 in.

x � 15

2x � 30

18 � ax9b � 18 � a

106b

x9

�106

Right side from large triangle

Right side from small triangle

x9 in.

�10 in.6 in.

Bottom side from large triangle

Bottom side from small triangle

106

106

Example 3

8 in. 10 in.

x 9 in.

y 6 in.

Figure 7-2

Proportions are used in geometry with similar triangles. Two triangles are saidto be similar if their corresponding angles have equal measures. In such a case, thelengths of the corresponding sides are proportional. The triangles in Figure 7-1are similar. Therefore, the following ratios are equivalent.

ax

�by

�cz

z x

yb

c a

Figure 7-1



Figure 7-3

x

11 ft2 ft

1 yd � 3 ft

Using Similar Triangles in an Application

The shadow cast by a yardstick is 2 ft long. The shadow cast by a tree is 11 ft long.Find the height of the tree.

Solution:

Let x represent the height of the tree. Label the variable.

We will assume that the measurements were taken at the same time of day. There-fore, the angle of the sun is the same on both objects, and we can set up similartriangles (Figure 7-3).

Example 4

b. To solve for y, the ratio of the upper left sides of the triangles must equal .

The LCD is 6y.


Clear fractions.

The length of side y is 4.8 in.

3. The two triangles shown are similar triangles. Solve for the lengths of themissing sides.

Skill Practice

4.8 � y

4810

�10y

10

48 � 10y

6y � a8yb � 6y � a

106b

8y

�106

Right side from large triangle

Right side from small triangle

8 in.y

�10 in.6 in.

Left side from large triangle

Left side from small triangle

106

12 in. 16 in.

4 in.

6 in.

x

y



3. x � 1.5 in., and y � 4.5 in.


Create a verbal model.

Write a mathematical equation.


Solve the equation.

Interpret the results and write theanswer in words.

The tree is 16.5 ft high.

4. The sun casts a 3.2-ft shadow of a 6-ft man. At the same time, the sun castsan 80-ft shadow of a building. How tall is the building?

Skill Practice

16.5 � x

332

�2x2

33 � 2x

2211

� a32b � a

x11b � 22

2

32

�x11

Height of tree

Length of tree’s shadow�

x11 ft

3 ft2 ft

Height of yardstick

Length of yardstick’s shadow

Memphis

LittleRock

Oklahoma City

Figure 7-4

3. Distance, Rate, and Time Applications In Sections 2.4 and 4.4, we presented applications involving the relationship amongthe variables distance, rate, and time. Recall that

Using a Rational Equation in a Distance, Rate,and Time Application

A small plane flies 440 miles with the wind from Memphis, Tennessee, to Okla-homa City, Oklahoma. In the same amount of time, the plane flies 340 miles againstthe wind from Oklahoma City to Little Rock, Arkansas (see Figure 7-4). If thewind speed is 30 mph, find the speed of the plane in still air.

Example 5

d � rt.



4. The building is 150 ft tall.

Solution:

Let x represent the speed of the plane in still air.

Organize the given information in a chart.

Because .d � rt, then t �dr

Distance Rate Time

With the wind 440

Against the wind 340340

x � 30x � 30

440x � 30

x � 30


The plane travels with the wind for the same amount of time as it travels againstthe wind, so we can equate the two expressions for time.

The LCD is

Solve the resultinglinear equation.

The plane’s speed in still air is 234 mph.

5. Alison paddles her kayak in a river where the current of the water is 2 mph.She can paddle 20 miles with the current in the same time that she can pad-dle 10 miles against the current. Find the speed of the kayak in still water.

Using a Rational Equation in a Distance, Rate,and Time Application

A motorist drives 100 miles between two cities in a bad rainstorm. For the return trip in sunny weather, she averages 10 mph faster and takes hour lesstime. Find the average speed of the motorist in the rainstorm and in sunnyweather.

Solution:

Let x represent the motorist’s speed during the rain.

Then represents the speed in sunny weather.

Because .d � rt, then t �dr

x � 10

12

Example 6

Skill Practice

x � 234

100x � 23,400

440x � 13,200 � 340x � 10,200

4401x � 302 � 3401x � 302

1x � 302 1x � 302 �440

x � 30� 1x � 302 1x � 302 �

340x � 30

1x � 302 1x � 302.

440x � 30

�340

x � 30

aTime withthe wind b � a

time againstthe wind b


Distance Rate Time

Trip during rainstorm 100 x

Trip during sunny weather 100100

x � 10x � 10

100x


5. The speed is 6 mph.

TIP: The equation

is a proportion. The

fractions can also be

cleared by equating the

cross products.

4401x � 302 � 3401x � 302

440

x � 30�

340

x � 30

440

x � 30�

340

x � 30


Because the same distance is traveled in hr less time, the difference between thetime of the trip during the rainstorm and the time during sunny weather is hr.

Verbal model

Mathematicalequation

Multiply bythe LCD.

Apply thedistributiveproperty.

Clear fractions.

Solve theresultingequation (quadratic).

Set theequation equalto zero.

Factor.

Because a rate of speed cannot be negative, reject Therefore, the speedof the motorist in the rainstorm is 40 mph. Because theaverage speed for the return trip in sunny weather is 50 mph.

6. Harley rode his mountain bike 12 miles to the top of the mountain and the same distance back down. His speed going up was 8 mph slower thancoming down. The ride up took 2 hours longer than coming down. Find his speeds.

4. Work ApplicationsSuppose Winston can paint a room in 2 hours. Then he paints room per hour.Suppose Clyde can paint the room in 4 hours. Then he paints room per hour. Ingeneral, we can define a work rate as follows:

Work rate: jobs per hour, where t is the total time requiredto complete the job.

Furthermore, if we multiply a work rate by the time an individual works, we com-pute the portion of the job completed. That is,

Portion of job completed � (Work rate)(Time)

Therefore, in 3 hours, Clyde would paint ( room/hr)(3 hr) � room.We use this basic principle to solve equations involving “work.”

34

14

1t

14

12

Skill Practice

x � 10 � 40 � 10 � 50,x � �50.

x � 40 or x � �50

0 � 1x � 402 1x � 502

0 � x2 � 10x � 2000

2000 � x2 � 10x

200x � 2000 � 200x � x2 � 10x

2001x � 102 � 200x � x1x � 102

2x1x � 102a100xb � 2x1x � 102a

100x � 10

b � 2x1x � 102a12b

2x1x � 102a100x

�100

x � 10b � 2x1x � 102a

12b

100x

�100

x � 10�

12

aTime duringthe rainstorm

b � atime during

sunny weatherb � a

12

hrb

12

12


Avoiding Mistakes:

The equation

is not a proportion because theleft-hand side has more than onefraction. Do not try to multiply thecross products. Instead, multiplyby the LCD to clear fractions.

100x

�100

x � 10�

12


6. Uphill speed was 4 mph; downhillspeed was 12 mph.



Using a Rational Equation in a “Work”Application

A new printing press can print the morning edition in 2 hours, whereas the oldprinter required 4 hours. How long would it take to print the morning edition ifboth printers were working together?

Solution:

Let x represent the time required for both printers working together to completethe job.

One method to approach this problem is to determine the portion of the job thateach printer can complete in 1 hour and extend that rate to the portion of the jobcompleted in x hours.

• The old printer can perform the job in 4 hours. Therefore, it completes of thejob in 1 hour and jobs in x hours.

• The new printer can perform the job in 2 hours. Therefore, it completes ofthe job in 1 hour and jobs in x hours.

The sum of the portions of the job completed by each printer must equal onewhole job.

The LCD is 4.



Solve the resulting linear equation.

The time required to print the morning editionusing both printers is

7. The computer at a bank can process and prepare the bank statements in 30hours. A new faster computer can do the job in 20 hours. If the bank usesboth computers together, how long will it take to process the statements?

Skill Practice

1 13 hr.

x �43 or x � 1

13

3x � 4

x � 2x � 4

4 �14

x � 4 �12

x � 4 � 1

4a14

x �12

xb � 4112

14

x �12

x � 1

°

Portion of jobcompleted by

old printer¢ � °

portion of jobcompleted bynew printer

¢ � °

1whole

job¢

12 x

12

14 x

14

Example 7

Work Rate Time Portion of Job Completed

Old printer x hours

New printer x hours12

x1 job

2 hr

14

x1 job

4 hr


7. 12 hours




a. proportion b. similar triangles

Review Exercises

For Exercises 2–8, determine whether each of the following is an equation or an expression. If it is an equation, solveit. If it is an expression, perform the indicated operation.

2. 3. 4.

5. 6. 7.

8.

Concept 1: Solving Proportions

For Exercises 9–22, solve the proportions.

9. 10. 11.

12. 13. 14.

15. 16. 17.

18. 19. 20.

21. 22.

23. Charles’ law describes the relationship between the initial and final temperature and volume of a gas heldat a constant pressure.

a. Solve the equation for . b. Solve the equation for .TfVf

Vi

Vf�

Ti

Tf

4p � 13

�2p � 5

68

9a � 1�

53a � 2

1t

�1

4 � t9

2z � 1�

3z

w � 24w

�16

y � 1

2y�

23

161.3

�30p

21.9

�x

38

b14

�38

53

�a8

15135

�w9

1976

�z4

67

�96y

85

�152p

1t2 �

23

1t

�56

3p � 3

�12p � 19

p2 � 7p � 12�

5p � 4

z2 � z24

�8

z � 1

3y � 6

20�

4y � 8

8

2a � 5

�5

a2 � 25m

m � 1�

2m � 3

b5

� 3 � 9





• NetTutor



24. The relationship between the area, height, and base of a triangle is given by the proportion

, where A is area, b is the base, and h is the height.

a. Solve the equation for A. b. Solve the equation for b.

Concept 2: Applications of Proportions and Similar Triangles

For Exercises 25–32, solve using proportions.

25. Toni drives her Honda Civic 132 miles on the highway on 4 gallons of gas. At this rate how many miles canshe drive on 9 gallons of gas?

26. Tim takes his pulse for 10 seconds and counts 12 beats. How many beats per minute is this?

27. Suppose a household of 4 people produces 128 lb of garbage in one week. At this rate, how many poundswill 48 people produce in one week?

28. Property tax on a $180,000 house is $4000. At this rate, how much tax would be paid on a $216,000 home?

29. Martin won an election by a ratio 5 to 4. If he received 5420 votes, how many votes did his opponentreceive?

30. Cooking oatmeal requires 1 cup of water for every cup of oats. How many cups of water will be requiredfor cup of oats?

31. A map has a scale of 75 miles/in. If two cities measure 3.5 in. apart, how many miles does this represent?

32. A map has a scale of 50 miles/in. If two cities measure 6.5 in. apart, how many miles does this represent?

33. is similar to

a. Find the length of

b. Find the length of

34. Figure ABCD is similar to Figure EFGH.

a. Find the length of

b. Find the length of

c. Find the length of

35. Solve for x and y. 36. Solve for x and y.

BC.

AB.

EH.

DF.

EF.

^DEF.^ABC

34

12

Ab

�h2


25 cm?

?20 cm 3 cm15 cm

BE

DA

C

F

8 in.

27 in.12 in.

22.5 in.

6 in.

B

E

D

A

C

FG

H

x

y3 cm 12 cm 18 cm

15 cm x

y9 ft 3 ft

5 ft

7 ft


40. For a science project at school, a student must measure the height of a tree. The student measures thelength of the shadow of the tree and then measures the length of the shadow cast by a yardstick. Usesimilar triangles to find the height of the tree.

Concept 3: Distance, Rate, and Time Applications

41. A boat travels 54 miles upstream against the current in the same amount of time it takes to travel 66 milesdownstream with the current. If the boat has the speed of 20 mph in still water, what is the speed of thecurrent? (Use to complete the table.)t � d

r

2.4 m

16.8 m

1 m

x

37. To estimate the height of a light pole, a mathematics student measures the length of a shadow cast by ameterstick and the length of the shadow cast by the light pole. Find the height of the light pole (see figure).


38. To estimate the height of a building, a student measures the length of ashadow cast by a yardstick and the length of the shadow cast by thebuilding (see figure). Find the height of the building.

39. A 6-ft-tall man standing 54 ft from a light post casts an 18-ft shadow.What is the height of the light post?

4 ft

48 ft

3 ft

x

54 ft 18 ft

6 ft

4.25 ft 51 ft3 ft

Distance Rate Time

With the current(downstream)

Against the current(upstream)


42. A fisherman travels 9 miles downstream with the current in the same time that he travels 3 miles upstreamagainst the current. If the speed of the current is 6 mph, what is the speed at which the fisherman travelsin still water?

43. A plane flies 630 miles with the wind in the same time that it takes to fly 455 miles against the wind. Ifthis plane flies at the rate of 217 mph in still air, what is the speed of the wind? (Use to complete thetable.)

t � dr


Distance Rate Time

With the wind

Against the wind

44. A plane flies 370 miles with the wind in the same time that it takes to fly 290 miles against the wind. If thespeed of the wind is 20 mph, what is the speed of the plane in still air?

45. Devon can cross-country ski 5 km/hr faster than his sister Shanelle. Devon skis 45 km in the same timeShanelle skis 30 km. Find their speeds.

46. Brooke walks 2 km/hr slower than her older sister Adrianna. Brooke can walk 12 km in the same amountof time that Adriana can walk 18 km. Find their speeds.

47. One motorist travels 15 mph faster than another. The slower driver takes 2 hours longer to travel 360 mithan the faster driver. What are the speeds of the two motorists?

48. A train travels 180 miles in 1 hour less time than a bus traveling the same distance. If the speed of the busis 15 mph slower than the speed of the train, find the speed of the train and the bus.

49. Kendra flew 900 miles to Cincinnati, Ohio. When she returned she traveled 30 mph slower. If it took her1 hour longer on her return flight, what were her speeds going to and returning from Cincinnati?

50. A plane flew 500 km from Atlanta, Georgia, to Louisville, Kentucky. When returning to Atlanta, the flighttook hr less time. If the rate to Louisville was 50 km/hr slower than the rate returning, find the two ratesin km/hr.

51. Sergio rode his bike 4 miles. Then he got a flat tire and had to walk back 4 miles. It took him 1 hr longerto walk than it did to ride. If his rate walking was 9 mph less than his rate riding, find the two rates.

52. Amber jogs 10 km in hr less than she can walk the same distance. If her walking rate is 3 km/hr less thanher jogging rate, find her rates jogging and walking (in km/hr).

Concept 4: Work Applications

53. If it takes a person 2 hr to paint a room, what fraction of the room would be painted in 1 hr?

54. If it takes a copier 3 hr to complete a job, what fraction of the job would be completed in 1 hr?

55. If the cold-water faucet is left on, the sink will fill in 10 min. If the hot-water faucet is left on, the sink willfill in 12 min. How long would it take the sink to fill if both faucets are left on?

56. The CUT-IT-OUT lawn mowing company consists of two people: Tina and Bill. If Tina cuts a lawn byherself, she can do it in 4 hr. If Bill cuts the same lawn himself, it takes him an hour longer than Tina. Howlong would it take them if they worked together?

34

12


57. A manuscript needs to be printed. One printer can do the job in 50 min, and another printer can do thejob in 40 min. How long would it take if both printers were used?

58. A pump can empty a small pond in 4 hr. Another more efficient pump can do the job in 3 hr. How longwould it take to empty the pond if both pumps were used?

59. Tim and Al are bricklayers. Tim can construct an outdoor grill in 5 days. If Al helps Tim, they can build itin only 2 days. How long would it take Al to build the grill alone?

60. Norma is a new and inexperienced secretary. It takes her 3 hr to prepare a mailing. If her boss helps her,the mailing can be completed in 1 hr. How long would it take the boss to do the job by herself?

61. A pipe can fill a reservoir in 16 hr. A drainage pipe can drain the reservoir in 24 hr. How long would ittake to fill the reservoir if the drainage pipe were left open by mistake? (Hint: The rate at which waterdrains should be negative.)

62. A hole in the bottom of a child’s plastic swimming pool can drain the pool in 60 min. If the pool had no hole,a hose could fill the pool in 40 min. How long would it take the hose to fill the pool with the hole?

63. A new copy machine works three times faster than the older model. It takes 12 minutes to complete a jobwhen both machines are working together. Find the time required for the new copy machine to completethe job by itself.

64. A cold-water faucet can fill a tub twice as fast as a hot-water faucet. Together it takes 12 min to fill thetub. How long will it take to fill the tub using only the cold-water faucet?


For Exercises 65–68, solve using proportions.

65. The ratio of smokers to nonsmokers in a restaurant is 2 to 7. There are 100 more nonsmokers thansmokers. How many smokers and nonsmokers are in the restaurant?

66. The ratio of fiction to nonfiction books sold in a bookstore is 5 to 3. One week there are 180 more fictionbooks sold than nonfiction. Find the number of fiction and nonfiction books sold during that week.

67. There are 440 students attending a biology lecture. The ratio of male to female students at the lecture is6 to 5. How many men and women are attending the lecture?

68. The ratio of dogs to cats at the humane society is 5 to 8. There are a total of 650 dogs and cats. How manydogs and how many cats are at the humane society?



Summary 561

B&I

Key Concepts

A rational expression is a ratio of the form wherep and q are polynomials and

The domain of an algebraic expression is the setof real numbers that when substituted for the variablemakes the expression result in a real number. For arational expression, the domain is all real numbersexcept those that make the denominator zero.

Simplifying a Rational Expression to Lowest Terms

Factor the numerator and denominator completely, andreduce factors whose ratio is equal to 1 or to Arational expression written in lowest terms will stillhave the same restrictions on the domain as theoriginal expression.

�1.

q � 0.

pq

Examples

Example 1

is a rational expression.

Example 2

To find the domain of factor the

denominator:

The domain is

.

Example 3


1

Simplify.

(provided .x � 7, x ��22� 1

x � 7

x � 2

1x � 22 1x � 72

x � 2x2 � 5x � 14

x � �2, x � 76

5x 0 x is a real number and

x � 2

1x � 22 1x � 72

x � 2x2 � 5x � 14

x � 2x2 � 5x � 14

Introduction to Rational ExpressionsSection 7.1

Chapter 7 SUMMARY

Key Concepts

Multiplying Rational Expressions

Factor the numerator and denominator completely.Then reduce factors whose ratio is 1 or �1.

Examples

Example 1

Multiply.

�1 1 1

� �a � 2b

2 or

2b � a

2

� 1b � a2 1b � a2

1a � b2 1a � b2�1a � 2b2 1a � b2

21a � b2

b2 � a2

a2 � 2ab � b2 �a2 � 3ab � 2b2

2a � 2b

Multiplication and Division of Rational Expressions

Section 7.2


Dividing Rational Expressions

Multiply the first expression by the reciprocal of thesecond expression. That is, for and

pq

�rs

�pq

�sr

s � 0,r � 0,q � 0,

Example 2

Divide.

�4d 2

9c2e3

�40c2d5e90c4d3e4

�2c2d5

15e4 �20e

6c4d3

2c2d5

15e4 �6c4d3

20e

Key Concepts

Converting a Rational Expression to an Equivalent

Expression with a Different Denominator

Multiply numerator and denominator of the rationalexpression by the missing factors necessary to createthe desired denominator.

Finding the Least Common Denominator (LCD) of Two

or More Rational Expressions

1. Factor all denominators completely.2. The LCD is the product of unique factors from the

denominators, where each factor is raised to itshighest power.

Examples

Example 1

Convert to an equivalent expression with the

indicated denominator:

Multiply numerator and denominator by the missingfactors from the denominator.

Example 2

Identify the LCD.

1. Write the denominators as a product of primefactors:

2. The LCD is or 24x3y4z233x3y4z

123x3y2z

; 5

2 � 3xy4

18x3y2z

; 5

6xy4

�3 � 51x � 221x � 22 � 51x � 22

��15x � 30

51x � 22 1x � 22

�3

x � 2�

51x � 22 1x � 22

�3

x � 2�

51x2 � 42 Factor.

�3

x � 2�

5x2 � 20

�3x � 2

Least Common DenominatorSection 7.3



Summary 563

B&I

Key Concepts

To add or subtract rational expressions, the expres-sions must have the same denominator.

Steps to Add or Subtract Rational Expressions

1. Factor the denominators of each rationalexpression.

2. Identify the LCD.3. Rewrite each rational expression as an equivalent

expression with the LCD as its denominator.4. Add or subtract the numerators, and write the

result over the common denominator.5. Simplify.

Examples

Example 1

Subtract.

The LCD is

�

2c � c � 321c � 42 1c � 32

�c � 3

21c � 42 1c � 32

�

2c � 1c � 32

21c � 42 1c � 32

�

2c21c � 42 1c � 32

�11c � 32

21c � 42 1c � 32

21c � 42 1c � 32.

�c

1c � 42 1c � 32�

121c � 42

cc2 � c � 12

�1

2c � 8

Addition and Subtraction of Rational Expressions

Section 7.4

Key Concepts

Complex fractions can be simplified by using Method Ior Method II.

Method I

1. Simplify the numerator and denominator of thecomplex fraction separately to form a singlefraction in the numerator and a single fraction inthe denominator.

2. Perform the division represented by the complexfraction. (Multiply the numerator of the complexfraction by the reciprocal of the denominator ofthe complex fraction.)

3. Simplify to lowest terms, if possible.

Examples

Example 1

Simplify.

�w � 2w � 3

�1w � 22 1w � 22

w2 �w2

1w � 32 1w � 22

�

w2 � 4w2

w2 � w � 6w2

�w2 � 4

w2 �w2

w2 � w � 6

1 �4

w2

1 �1w

�6

w2

�

w2

w2 �4

w2

w2

w2 �ww2 �

6w2

Complex FractionsSection 7.5


Method II

1. Multiply the numerator and denominator of thecomplex fraction by the LCD of all individualfractions within the expression.

2. Apply the distributive property, and simplify theresult.

3. Simplify to lowest terms, if possible.

Example 2

Simplify.

�w � 2w � 3

�w2 � 4

w2 � w � 6�1w � 22 1w � 22

1w � 32 1w � 22

1 �4

w2

1 �1w

�6

w2

�

w2a1 �4

w2b

w2a1 �1w

�6

w2b

Key Concepts

An equation with one or more rational expressionsis called a rational equation.

Steps to Solve a Rational Equation

1. Factor the denominators of all rationalexpressions.

2. Identify the LCD of all expressions in theequation.

3. Multiply both sides of the equation by the LCD.4. Solve the resulting equation.5. Check each potential solution in the original

equation.

Examples

Example 1

Solve.

The LCD is

Quadratic equation

orDoes not check. Checks.

Example 2

Solve for I.

I �VQq

Iq � VQ

I � q �VQ

I� I

q �VQ

I

w � �1w � 12

12w � 12 1w � 12 � 0

2w2 � w � 1 � 0

2w � 1 � w � �2w2

12w � 12 112 � w112 � w1�2w2

� w12w � 12 �2w

2w � 1

w12w � 12 1w

� w12w � 12 1

2w � 1

w12w � 12.

1w

�1

2w � 1�

�2w2w � 1

Rational EquationsSection 7.6



Summary 565

B&I

Key Concepts

Solving Proportions

An equation that equates two rates or ratios is calleda proportion:

To solve a proportion, multiply both sides of the equa-tion by the LCD.

Examples 2 and 3 give applications of rational equations.

1. Applications involving and work.

Example 2

Two cars travel from Los Angeles to Las Vegas. One cartravels an average of 8 mph faster than the other car. Ifthe faster car travels 189 miles in the same time as theslower car travels 165 miles, what is the average speedof each car?

Let r represent the speed of the slower car.Let represent the speed of the faster car.r � 8

1distance � rate � time2d � rt

ab

�cd 1b � 0, d � 02

Examples

Example 1

A 90-g serving of a particular ice cream contains 10 gof fat. How much fat does 400 g of the same ice creamcontain?

Example 3

Beth and Cecelia have a house cleaning business. Bethcan clean a particular house in 5 hr by herself. Ceceliacan clean the same house in 4 hr. How long would ittake if they cleaned the house together?

Let x be the number of hours it takes for bothBeth and Cecelia to clean the house.

Beth can clean of the house in an hour and ofthe house in x hours.

Cecelia can clean of the house in an hour and of the house in x hours.

Together they clean one whole house.

working together. x �209

, or 229 hr

9x � 20

4x � 5x � 20

20a15

x �14

xb � 11220

15

x �14

x � 1

14x

14

15x

15

x �4009

� 44.4 g

400 � 9x

360040

� a1090b � a

x400b � 3600

9

1090

�x

400

10 g fat

90 g ice cream�

x grams fat

400 g ice cream

Distance Rate Time

Slower car 165 r

Faster car 189189

r � 8r � 8

165r

Applications of Rational Equations and Proportions

Section 7.7

The slower car travels 55 mph, and the faster car travels55 � 8 � 63 mph.

55 � r

1320 � 24r

165r � 1320 � 189r

1651r � 82 � 189r

165

r�

189r � 8


Section 7.1


a. Evaluate the expression (if possible) for

b. Write the domain of the expression in set-builder notation.


a. Evaluate the expression for

b. Write the domain of the expression in set-builder notation.

3. Which of the rational expressions are equal tofor all values of x for which the expressions

are defined?

a. b.

c. d.

For Exercises 4–13, write the domain in set-buildernotation. Then simplify the expressions to lowest terms.

4. 5.

6. 7.

8. 9.

10. 11.

12. 13.p � 7

p2 � 14p � 49n � 3

n2 � 6n � 9

3m2 � 12m � 159m � 9

2b2 � 4b � 64b � 12

15 � 3k2k2 � 10k

z2 � 4z8 � 2z

2w2 � 11w � 12w2 � 16

4a2 � 7a � 2a2 � 4

h � 713h � 12 1h � 72

x � 312x � 52 1x � 32

x2 � 44 � x2

�x � 7x � 7

x � 5x � 5

2 � xx � 2

�1

�1, �2k � 0, 1, 5,

k � 1k � 5

t � 0, 1, 2, �3, �9

t � 2t � 9

Section 7.2


14. 15.

16. 17.

18. 19.

20. 21.

22. 23.

24.

25.

26.

27.

Section 7.3

For Exercises 28–33, convert each expression into anequivalent expression with the indicated denominator.

28. 29.

30. 31.

32. 33.u � 1u � 6

�u2 � 36

s � 2s � 4

�s2 � 16

2r

�r2 � 3r

6w

�w2 � 4w

y � 2

y � 3�

2y � 6x � 1x � 2

�5x � 10

4y2 � 1

1 � 2y�

y2 � 4y � 5

5 � y

x � 2x2 � 3x � 18

�6 � xx2 � 4

3m � 36m2 � 18m � 12

�2m2 � 8

m2 � 3m � 2�

m � 3m � 1

5h2 � 6h � 1h2 � 1

�16h2 � 9

4h2 � 7h � 3�

3 � 4h30h � 6

n2 � n � 1n2 � 4

n2 � n � 1n � 2

a2 � 5a � 17a � 7

a2 � 5a � 1a � 1

1s2 � 6s � 82 a4s

s � 2ba

�2tt � 1

b 1t2 � 4t � 52

q2 � 5q � 62q � 4

�2q � 6q � 2

4c2 � 4cc2 � 25

�8c

c2 � 5c

8x2 � 25

�3x � 15

1611

v � 2�

2v2 � 822

2u � 10u

�u3

4u � 20

3y3

3y � 6�

y � 2y

Chapter 7 Review Exercises



Review Exercises 567

B&I

For Exercises 34–41, identify the LCD.

34. 35.

36. 37.

38.

39.

40. 41.

42. State two possible LCDs that could be used to addthe fractions.

43. State two possible LCDs that could be used to sub-tract the fractions.

Section 7.4

For Exercises 44–55, add or subtract as indicated.

44. 45.

46. 47.

48. 49.

50. 51.

52.

53.3q

q2 � 7q � 10�

2q

q2 � 6q � 8

4p

p2 � 6p � 5�

3p

p2 � 5p � 4

52r � 12

�1r

43m

�1

m � 2

34 � t2 �

t2 � t

y

y2 � 81�

29 � y

x2

x � 7�

49x � 7

a2

a � 5�

25a � 5

b � 6b � 2

�b � 2b � 2

h � 3h � 1

�h � 1h � 1

103 � x

�5

x � 3

7c � 2

�4

2 � c

�23k � 1

; 6

1 � 3k4

2t � 5;

55 � 2t

6n2 � 9

; 5

n2 � n � 6

8m2 � 16

; 7

m2 � m � 12

6q

; 1

q � 85

p � 2;

p

p � 4

6xy2z

; 3

xy2z4

2a2bc2;

5ab3

54.

55.

Section 7.5

For Exercises 56–63, simplify the complex fractions.

56. 57.

58. 59.

60. 61.

62. 63.

Section 7.6

For Exercises 64–71, solve the equations.

64. 65.

66. 67.

68.

69.

70.y � 1

y � 3�

y2 � 11y

y2 � y � 6�

y � 3

y � 2

4p � 4

p2 � 5p � 14�

2p � 7

�1

p � 2

t � 13

�t � 1

6�

16

ww � 1

�3

w � 1� 1

2h � 2

� 1 �h

h � 2

1y

�34

�14

2x

�12

�14

25k � 5

� 5

5k � 5

� 5

6p � 2

� 4

8p � 2

� 4

x3x � 9

�3

x2 � 3x�

1x

1h

�h

2h � 4�

2h2 � 2h

a � 43

a � 23

z � 56z

2z � 103

2 � 3w2

2w

� 3

yx

�xy

1x

�1y

2y

� 6

3y � 1

4

ba

�ab

1b

�1a


71.

72. Four times a number is added to 5. The sum isthen divided by 6. The result is Find the number.

73. Solve the formula for h.

74. Solve the formula for b.

Section 7.7

For Exercises 75–76, solve the proportions.

75. 76.

77. A bag of popcorn states that it contains 4 g offat per serving. If a serving is 2 oz, how manygrams of fat are in a 5-oz bag?

78. Bud goes 10 mph faster on his Harley Davidsonmotorcycle than Ed goes on his Honda motorcycle.If Bud travels 105 miles in the same time that

12a

�58

m � 28

�m3

Ab

�h2

Vh

�pr2

3

72.

1z � 2

�4

z2 � 4�

1z � 2

Ed travels 90 miles, what are the rates of the twobikers?

79. There are two pumps set up to fill a smallswimming pool. One pump takes 24 min byitself to fill the pool, but the other takes 56 minby itself. How long would it take if both pumpswork together?

80. Consider the similar triangles shown here. Findthe values of x and y.

105

28.6

13

x

y


B&I

For Exercises 1–2,

a. Write the domain in set-builder notation.

b. Reduce the rational expression to lowest terms.

1. 2.

3. Identify the rational expressions that are equalto for all values of x for which theexpression is defined.

a. b.

c. d. �x � 5x � 5

9x2 � 16�9x2 � 16

7 � 2x2x � 7

x � 4x � 4

�1

7a2 � 42aa3 � 4a2 � 12a

51x � 22 1x � 12

3012 � x2

For Exercises 4–9, perform the indicated operation.

4.

5.

6.

7.

8. 9.1 �

4m

m �16m

13x � 4

�2

3x2 � 2x � 8�

xx � 2

tt � 2

�8

t2 � 4

w2 � 4ww2 � 8w � 16

�w � 4

w2 � w

9 � b2

5b � 15�

b � 3b � 3

� 19b2 � 28b � 32

2y2 � 4y � 3

�1

3y � 9

Chapter 7 Test


Cumulative Review Exercises 569

B&I

18. A motorboat can travel 28 miles downstream inthe same amount of time as it can travel 18 milesupstream. Find the speed of the current if theboat can travel 23 mph in still water.

19. One printer requires 3 hr to do a job and asecond printer requires 6 hr to do the same job. Ifthey worked together, how long would it take tocomplete the task?

20. Consider the similar triangles shown here. Findthe values of a and b.

21. Find the LCD of the following pairs of rationalexpressions.

a. b.�23x2y

; 4

xy2

x31x � 32

; 7

51x � 32

1215

7 a

9.6b

For Exercises 10–13, solve the equation.

10.

11.

12.

13.

14. Solve the formula for r.

15. If is added to the reciprocal of a number theresult is times the reciprocal of that number.Find the number.

16. Solve the proportion.

17. A recipe for vegetable soup calls for cup ofcarrots for six servings. How many cups of carrotsare needed to prepare 15 servings?

12

y � 7

�4�

14

25

32

C2

�Ar

4x

x � 4� 3 �

16x � 4

3c � 2

�1

c � 1�

7c2 � c � 2

p

p � 1�

1p

�2p2 � 1

p2 � p

3a

�52

�7a

For Exercises 1–2, simplify completely.

1. 2.

3. Solve for y:

4. Complete the table.

12

�34

1y � 12 �5

12

03 � 5 0 � 0�2 � 7 0a12b

�4

� 24

6. The height of a triangle is 2 in. less than thebase. The area is 40 in.2 Find the base and heightof the triangle.

7. Simplify.

8. The length and width of a rectangle are given interms of x.

a. Write a polynomial that represents theperimeter of the rectangle.

b. Write a polynomial that represents the areaof the rectangle.

2x � 3

x � 1

a4x�1y�2

z4 b�2

12y�1z323

Chapters 1–7 Cumulative Review Exercises

Set-Builder IntervalNotation Graph Notation

1�, 52

5x 0 x � �16

5. The perimeter of a rectangular swimming pool is104 m. The length is 1 m more than twice thewidth. Find the length and width.



B&I

16. Solve the proportion.

17. Determine the x- and y-intercepts.

a. b.

18. Determine the slope

a. of the line containing the points and

b. of the line

c. of a line parallel to a line having a slope of 4.

d. of a line perpendicular to a line having aslope of 4.

19. Find an equation of a line passing through thepoint (1, 2) and having a slope of 5. Write theanswer in slope-intercept form.

20. A group of teenagers buys 2 large popcorns and6 drinks at the movie theater for $16. A couplebuys 1 large popcorn and 2 drinks for $6.50.Find the price for 1 large popcorn and the pricefor 1 drink.

y � �23

x � 6

1�5, 1210, �62

y � 5x�2x � 4y � 8

2b � 56

�4b7

9. Factor completely:

10. Factor.

11. Determine the domain of the expression.

12. Simplify to lowest terms.

13. Divide.

14. Simplify.

15. Solve.

7y2 � 4

�3

y � 2�

2y � 2

34

�1x

13x

�14

2x � 6x2 � 16

�10x2 � 90

x2 � x � 12

x2 � 9x2 � 8x � 15

x � 31x � 52 12x � 12

10cd � 5d � 6c � 3

25x2 � 30x � 9


mil52817 ch07 493-570...the denominator is a perfect square trinomial. skill practice answers 7. 8....

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