Chapter 7 Radicals, Radical Functions, and Rational …lhoang.wikispaces.com/file/view/Chapter 7...

Chapter 7 Radicals, Radical Functions, and Rational Exponents

393

7.1 Check Points

1. a. 64 8= because 28 64=

b. 49 7− = − because 2( 7) 49− =

c. 16 425 5

= because 24 16

5 25⎛ ⎞ =⎜ ⎟⎝ ⎠

d. 0.0081 0.09= because 20.09 0.0081=

e. 9 16 25 5+ = =

f. 9 16 3 4 7+ = + =

2. a. ( ) 12 20

(3) 12(3) 20

36 20

164

f x x

f

= −

= −

= −

==

b. ( ) 9 3

( 5) 9 3( 5)

9 15

244.90

g x x

g

= − −

− = − − −

= − +

= −≈ −

3. ( ) 9 27f x x= − 9 27 0

9 273

xxx

− ≥≥≥

Domain of f is { }3x x ≥ or [3, )∞ .

4. ( ) 0.7 12.5M x x= + Because 2010 is 14 years after 1996, substitute 14 for x.

(14) 0.7 14 12.5 15.1M = + ≈ The model indicates that there will be about 15.1 non-program minutes in an hour of prime-time cable in 2010.

5. a. 2( 7) 49 7− = =

b. 2( 8) 8x x+ = +

c. 10 549 7x x=

d. 2 26 9 ( 3) 3x x x x− + = − = −

6. a. 3

3

3

( ) 6

(33) 33 6

273

f x x

f

= −

= −

==

b. 3

3

3

( ) 2 2

( 5) 2( 5) 2

82

g x x

g

= +

− = − +

= −= −

7. 3 3 3327 ( 3 ) 3x x x− = − = −

8. a. 4 16 2=

b. 4 16 2− = −

c. 4 16− is not a real number.

d. 5 1 1− = −

9. a. 44 ( 6) 6x x+ = +

b. 55 (3 2) 3 2x x− = −

c. 66 ( 8) 8− =

7.1 Exercise Set

2. 16 4= because 24 16=

4. 16 4− = − because 2( 4) 16− =

6. 16− not a real number

Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall

Chapter 7: Radicals, Radical Functions, and Radical Exponents

394

8. 1 149 7

= because 21 1

7 49⎛ ⎞ =⎜ ⎟⎝ ⎠

10. 4 225 5

− = − because 22 4

5 25⎛ ⎞ =⎜ ⎟⎝ ⎠

12. 0.49 0.7= because 20.7 0.49=

14. 0.64 0.8− = − because 20.8 0.64=

16. 144 25 169 13+ = =

18. 144 25 12 5 17+ = + =

20. 25 144 119− = − not a real number

22. ( )28 28 3 25 5f = − = =

( )4 4 3 1 1f = − = =

( )3 3 3 0 0f = − = =

( )1 1 3 4f − = − − = − not a real number

24. ( ) ( )4 2 4 1 8 1 9 3g = − + = − + = − = −

( ) ( )1 2 1 1 2 1 3 1.73g = − + = − + = − ≈ −

1 12 1 1 12 2

0 0 0

g ⎛ ⎞ ⎛ ⎞− = − − + = − − +⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

= − = − =

( ) ( )1 2 1 1 2 1 1g − = − − + = − − + = − − not a real number

26. ( ) ( ) ( )2 25 5 2 3 3 3h = − = = =

( ) ( ) ( )2 23 3 2 1 1 1h = − = = =

( ) ( ) ( )2 20 0 2 2 2 2h = − = − = − =

( ) ( ) ( )2 25 5 2 7 7 7h − = − − = − = − =

28. 2 0

2x

x+ ≥

≥ −

The domain of f is { }2x x ≥ − or [ )2, .− ∞ This corresponds to graph (f).

30. 3 15 03 15

5

xxx

− ≥≥≥

The domain of f is { }5x x ≥ or [ )5, .∞ This corresponds to graph (a).

32. 8 2 02 8

4

xxx

− ≥− ≥ −

≤

The domain of f is { }4x x ≤ or ( ], 4 .−∞ This corresponds to graph (b).

34. 27 7 7= =

36. ( )210 10 10− = − =

38. ( )22 2x x− = −

40. ( )24 2 2 281 9 9 9x x x x= = =

42. ( )26 3 3 349 7 7 7x x x x− = − = − = −

44. ( )22 14 49 7 7x x x x+ + = + = +

46. ( )22 10 25 5 5x x x x− − + = − − = − −

48. 33 64 4 because 4 64= =

50. ( )33 64 4 because 4 64− = − − = −

52. 3

3 1 1 1 1 because 1000 10 10 1000

⎛ ⎞= =⎜ ⎟⎝ ⎠

54. 3

3 8 2 2 8 because 125 5 5 125− −⎛ ⎞= − − =⎜ ⎟

⎝ ⎠

56. ( ) 3 330 30 3 27 3f = − = =

( ) 3 311 11 3 8 2f = − = =

( ) 332 2 3 1 1f = − = − = −

( ) 3 3122 122 3 125 5f − = − − = − = −


Intermediate Algebra for College Students 5E Section 7.1

395

58. ( ) ( )

( ) ( )

3313 2 13 1 27 3

0 2 0

g

g

= − + = − = −

= −

( ) ( ) ( )

3 33

33

1 1 1 1

63 2 63 1 125 5 5g

+ = − = − = −

− = − − + = − − = − − =

60. 55 1 1 because 1 1= =

62. 44 81 3 because 3 81= =

64. 44 81 3 because 3 81− = − =

66. 4 81− not a real number

68. ( )77 1 1 because 1 1− = − − = −

70. 8 1− not a real number

72. 44 10, 000 10 because 10 10, 000− = − =

74. 55 32 2 because 2 32= =

76. 66 64 2 because 2 64− = − =

78. 5 5x x=

80. 66 y y=

82. 3 3125 5x x− = −

84. ( )33 6 6− = −

86. ( )66 6 6 6− = − =

88. ( )44 5 5x x+ = +

90. ( ) ( )55 32 2 2 2x x− − = − −

92. ( )( )( )( )( )

2

0 0 0 2 0 2 2

1 1 1 2 1 2 1

4 4 4 2 2 2 0

9 9 9 2 3 2 1

x f x x

f

f

f

f

= −

= − = − = −

= − = − = −

= − = − =

= − = − =

Domain: { }| 0x x ≥ or [ )0,∞

Range: { }| 2y y ≥ − or [ )2,− ∞

94. ( )( ) ( )( )( )( )

4

5 5 4 5 9 3

0 0 4 0 4 2

3 3 4 3 1 1

4 4 4 4 0 0

x f x x

f

f

f

f

= −

− − = − − = =

= − = =

= − = =

= − = =

Domain: { }| 4x x ≤ or ( ], 4−∞

Range: { }| 0y y ≥ or [ )0,∞

96. The domain of the cube root function is all real numbers, so we only need to worry about the square root in the denominator. We need the radicand of the square root to be 0≥ , but we also cannot divide by 0. Therefore, we have

80 5 05 80

16

xxx

− >− > −

<

The domain of f is { }| 16x x < or ( ),16−∞ .



396

98. 2 0x − ≥ . From the denominator, we need 7 0x− > . We need to solve the two inequalities. The domain of the function is the overlap of the two solution sets.

2 02

xx

− ≥≥

and 7 07

7

xxx

− >− > −

<

We need 2x ≥ and 7x < . Therefore, the domain of f is { }| 2 7x x≤ < or [ )2, 7 .

100. 3 3 3

3

3

3 3

169 9 1000 216

13 3 10 6

16 16

4 4 8 2

+ + +

= + + +

= +

= + = =

102. a. ( ) 3.1 19

(48) 3.1 48 19 40.5

f x x

f

= +

= + ≈

The model estimates the median height of girls who are 48 months to be 40.5 inches. This overestimates the actual median height by 0.3 inches.

b. Find (0)f and (10).f

( ) 3.1 19

(0) 3.1 0 19 19

(10) 3.1 10 19 28.8

f x x

f

f

= +

= + =

= + ≈

Average rate of change is (10) (0) 28.8 19 1.0

10 0 10f fm − −

= = ≈−

inches per

month.

c. Find (50)f and (60).f

( ) 3.1 19

(50) 3.1 50 19 40.9

(60) 3.1 60 19 43.0

f x x

f

f

= +

= + ≈

= + ≈

Average rate of change is (60) (50) 43.0 40.9 0.2

60 50 10f fm − −

= = ≈−

inches

per month. This is a much smaller rate of change. This is shown on the graph because the graph is not as steep between 50 and 60 as it is between 0 and 10.

104. ( ) ( )45 20 45 900 30f = = =

The officer should believe the motorist. The model predicts that the motorist’s speed was 30 miles per hour. This is below the 35 mile per hour speed limit.

106. – 114. Answers will vary.

116. y x=

4y x= +

3y x= −

The graphs start at the y-axis, are increasing, and have the same shape. They differ in where they begin on the y-axis (i.e. different y-intercepts).

118. 21

2

y xy x

== −

a. 2x x= − for { }0 .x x ≤

b. 2x x≠ − for { }0 .x x >

120. does not make sense; Explanations will vary. Sample explanation: Zero is a real number with only 1 even root.

122. make sense

124. false; Changes to make the statement true will vary. A sample change is: If b is negative and n is even, the expression does not represent a real number. However, this does not hold when n is odd.



397

126. false; Changes to make the statement true will vary. A sample change is: As the index n gets larger, the value of the radical will get smaller.

128. The domain of f g+ is the set of values of x for which both radicands are greater than or equal to zero.

3 0 and 1 03 1

x xx x

− ≥ + ≥≥ ≥ −

The domain of f g+ is the intersection of the two

sets,{ }3 .x x ≥ In interval notation, the domain of

f g+ is [ )3, .∞

( ) ( )( )

31

f xf xxg g x x

⎛ ⎞ −= =⎜ ⎟

+⎝ ⎠

The domain of fg

is the set of values of x for which

the radicands are greater than zero and the denominator does not equal zero.

3 0 and 1 03 1

x xx x

− ≥ + >≥ > −

The domain of fg

is { }3x x ≥ or [ )3, .∞

130. ( ) 2g x x= +

The graph of g is the graph of f shifted up two units.

132. ( ) [ ][ ]

3 2 3 5 3 2 3 15

3 2 2 15

3 4 307 30

x x x x x x

x x

x xx

⎡ ⎤− − + = − − −⎣ ⎦= − − −

= + += +

133. ( ) ( ) ( ) ( )

( )

( )

2 2 224 3 4 3

8 62

8 8

2 66

3 3

1

3

93

x y x y

x y

x xyy

− − −−− −

−

− = −

=−

= =−

134. 3 4 113 4 11 or 3 4 11

3 7 3 157 53

xx x

x xxx

− >

− < − − >< − >

>< −

The solution set is

{ }| or 573x x x< − >

or ( )7, 5, .3

⎛ ⎞−∞ − ∞⎜ ⎟⎝ ⎠

∪

135. 7

3 5 4 6 3 4 5 ( 6) 7 1 2 128(2 )(2 ) 2 2x x x xx x

− + + − −= = = =

136. 2

2 5 35 3

32 22 216

x x xx x

− −= = =

137. 12

2 3 4 2 4 3 4 8 128( ) yx y x y x y

x− − ⋅ ⋅ −= = =

7.2 Check Points

1. a. 1/ 225 25 5= =

b. ( )1/ 3 38 8 2− = − = −

c. ( )1/ 42 245 5xy xy=

2. a. ( )1/ 44 5 5xy xy=

b. 1/ 53 3

52 2

a b a b⎛ ⎞= ⎜ ⎟⎜ ⎟⎝ ⎠

3. a. ( )44 / 3 438 8 2 16= = =

b. ( )33/ 2 325 25 5 125= = =

c. ( )33/ 4 3481 81 3 27− = − = − = −

4. a. 3 4 4 / 36 6=

b. ( ) ( )7 7 / 55 2 2xy xy=



398

5. a. 1/ 21/ 2

1 1 110010100100

− = = =

b. 1/ 31/ 3 31 1 18

288− = = =

c. ( )

3/ 53/ 5 3 35

1 1 1 132832 232

− = = = =

d. ( )( )

5 / 95 / 9

133

xyxy

− =

6. a. 1 1 1 513 2 3 627 7 7 7

+⋅ = =

b.

11 43 13 3

43

50 55 5

10

x x xx

x

− −= ⋅ = =

c.

32 2 3 345 5 4 109.1 9.1 9.1

⋅⎛ ⎞⎜ ⎟ = =⎜ ⎟⎝ ⎠

d.

1 13 3 1 1 1 11 13 125 5 3 4 3 54 12

15

yx y x y x y

x

− − ⋅ ⋅ −⎛ ⎞⎜ ⎟ = = =⎜ ⎟⎝ ⎠

7. a. 3 1

6 3 6 2x x x x= = =

b. ( )1 11 123 12 12 43 338 8 8 2a a a a

⋅= = =

c. ( ) ( ) ( )

( )

1 1 14 2 4 2 4 28 8 8 8

2 1 12 244 4 4

x y x y x y

x y x y x y

= =

= = =

d.

11 1 3 2 12 62 3 6 6 6

3 13

x x x x x xx

x

− −= = = = =

e.

11 1 1 12

3 63 3 2 6x x x x x⋅⎛ ⎞

⎜ ⎟= = = =⎜ ⎟⎝ ⎠

7.2 Exercise Set

2. 1/ 2100 100 10= =

4. ( )1/ 3 364 64 4− = − = −

6. 1/ 4 481 81 3− = − = −

8. ( )1/ 4 4xy xy=

10. ( )1/ 54 453 3xy xy=

12. ( )33/ 2 325 25 5 125= = =

14. ( )22 / 3 231000 1000 10 100= = =

16. ( ) ( ) ( )22 / 3 2327 27 3 9− = − = − =

18. ( ) ( ) ( ) ( )5 2 5 25 / 2 2 / 3 34 8 4 8 2 2

32 4 28

− = − = −

= − =

20. ( ) ( ) ( )44 / 9 499 or xy xy xy=

22. 1/ 213 13=

24. 1/ 33 6 6=

26. ( )1/ 55 13 13x x=

28. 5 5 / 2x x=

30. 7 4 4 / 7x x=

32. ( )1

3 37 7xy xy=

34. ( ) ( )33211 11xy xy=

36. ( )55

2 26 69 9x y x y⎛ ⎞ =⎜ ⎟⎝ ⎠

38. 2 2 / 554 4x y xy=



399

40. 1/ 21/ 21 1 19

399− = = =

42. 1/ 31/ 3 3

1 1 11255125125

− = = =

44. ( )

5 / 45 / 4 5 54

1 1 1 18124381 381

− = = = =

46. ( )

4 / 54 / 5 4 45

1 1 1 1321632 232

− = = = =

48. 1/ 3 1/ 3

38 125 125 5125 8 8 2

−⎛ ⎞ ⎛ ⎞= = =⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

50. ( )22 / 3 2

2 / 3 31 1 1 188 8 2 4

− ⎛ ⎞⎛ ⎞ ⎛ ⎞− = = = =⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟− − −⎝ ⎠ ⎝ ⎠⎝ ⎠

52. ( )( )

4 / 74 / 7

144

xyxy

− =

54. 1/ 4

1/ 41/ 4

7 771

xz xxzz

−− = =

56. ( ) ( )2 / 3 1/ 32 / 3 1/ 3 3 / 3 15 5 5 5 5 5+⋅ = = = =

58. ( ) ( )3/ 43 / 4 1/ 4 2 / 4

1/ 4

1/ 2

100 100 100100

100 100 10

−= =

= = =

60. ( ) ( ) ( ) ( )1/ 2 2 / 3 3/ 6 4 / 61/ 2 2 / 3 7 / 6x x x x x+ +⋅ = = =

62. ( ) ( )3/ 73/ 7 1/ 7 2 / 7

1/ 7x x xx

−= =

64. ( ) ( ) ( ) ( )1/ 41/ 4 3/ 5 5 / 20 12 / 20

3 / 5

7 / 207 / 201

x x xx

xx

− −

−

= =

= =

66. ( )54 / 5 43 3 81= =

68. ( )1/ 63/ 4 3 / 24 1/ 81/ 81y y y

y− − −= = =

70. ( ) ( )4 1/ 4 41/ 4 4 12 2 16 16x x x x⋅= = =

72. ( ) ( ) ( )1/ 3 1/ 3 1/ 39 6 1/ 3 9 6

3 2 3 23

125 125

125 5

x y x y

x y x y

=

= =

74. ( )1/ 31/ 4 1/121/ 31/ 4 2 / 5

2 / 5 2 /15x xx yy y

− ⎛ ⎞= =⎜ ⎟⎜ ⎟⎝ ⎠

76. ( ) ( ) ( ) ( ) ( ) ( )3/ 4 1/ 23/ 4 1/ 2 1/ 4 3/ 4 2 / 4 1/ 4

1/ 4

4 / 4 1

5 5 5 55

5 5 5

+ − + −⋅= =

= = =

78. ( ) ( )

( ) ( ) ( ) ( )

4 41/ 5 4 1/ 5 4 / 5

3/10 3/10 3/10

4 / 5 3/10 8 /10 3 /10

5 /10 1/ 2

2 2 16

16 16

16 16

y y y

y y y

y y

y y

− −

= =

= =

= =

80. 10 2 2 /10 1/ 5 5x x x x= = =

82. 3 12 1/ 3 12 / 3 427 27 3a a a= =

84. 15 20 15 / 5 20 / 5 3 45 x y x y x y= =

86. ( ) ( ) ( )21 21/ 3 7 7 73 xy xy xy x y= = =

88. ( ) ( ) ( )2 2 /12 1/ 612 63 3 3 3y y y y= = =

90. ( ) ( ) ( )

( )

6 6 / 8 3/ 48

3 4 34

2 2 2

2 8

a a a

a a

= =

= =

92. 2 6 2 / 4 6 / 4 1/ 2 3/ 2 34 x y x y x y xy= = =

94. ( ) ( )

( ) ( )

1/ 2 1/ 31/ 2 1/ 33

3/ 6 2 / 6 5 / 6

6 5 6

3 3 3 3 3

3 3

3 or 243

+

+

⋅ = ⋅ =

= =

=



400

96. ( ) ( ) ( ) ( )72 1 42 17 2 7 2 14 147 2

11 14 1114

x x x x x x

x x

+ +⋅ = ⋅ = =

= =

98. ( ) ( )1/ 6 1/ 36 32 2 2 2ab a b ab a b⋅ = ⋅

( ) ( )

( ) ( ) ( )

1/ 6 1/ 31/ 6 2 2 1/ 3

1/ 6 2(1/ 6) 2(1/ 3) 1/ 3

1/ 6 1/ 3 2 / 3 1/ 3

1/ 6 2 / 3 (1/ 3) 1/ 3

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

65 / 6 4 / 6 5 4

a b a b

a b a b

a b a b

a b

a b a b

a b a b

+ +

+

= ⋅

= ⋅

= ⋅

= ⋅

= ⋅ =

= =

100. ( ) ( ) ( ) ( )1/ 33 1/ 3 1/ 4 4 /12 3/124 1/ 4

1/12 12

x x x xx x

x x

− −= = =

= =

102. ( ) ( ) ( ) ( )25 2 / 5

2 / 5 3/10 4 /10 3/103/10310

1/10 10

y y y yyy

y y

− −= = =

= =

104. ( ) ( ) ( )1/ 5 1/ 2 1/ 55 1/ 2 1/ 25

1/10 10

x x x x

x x

⋅= = =

= =

106. ( ) ( )

( )( ) ( ) ( )

1/ 21/ 2 1/ 22 2 2

1/ 2 1/ 2 1/ 42 2 24

xy xy xy

xy xy xy⋅

⎛ ⎞= = ⎜ ⎟

⎝ ⎠

= = =

108. ( ) ( )( )

( )( ) ( ) ( )

1/ 51/ 3 1/ 35 3 5

1/ 3 1/ 5 1/15 15

2 2 2

2 2 2

x x x

x x x⋅

= =

= = =

110. ( ) ( )( )

( ) ( )

2020 1/ 5 1/ 5 204 2 4 2 4 25

20 / 5 44 2 4 2

4 4 2 4 16 8

x y x y x y

x y x y

x y x y

⋅

⋅ ⋅

⎛ ⎞⎛ ⎞ = =⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

= =

= =

112.

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

( )

4 3 3 3/ 4 3/ 4

1/ 2 1/ 2

3/ 4 1/ 2 3/ 4 1/ 2

3/ 4 2 / 4 3/ 4 2 / 4

1/ 41/ 4 1/ 4

4

a b a bab a b

a b

a b

a b ab

ab

− −

− −

=

=

=

= =

=

114. ( )

( ) ( ) ( ) ( )

1/ 4 9 / 4 1/ 4

1/ 4 9 / 4 1/ 4 1/ 4

1/ 4 9 / 4 1/ 4 1/ 4

8 / 4 0

2 1

x x x

x x x x

x x

x x

x

−

− −

− + − +

−

= ⋅ − ⋅

= −

= −

= −

116. ( ) ( )

( ) ( )

1/ 3 1/ 3

1/ 3 1/ 3 1/ 3 1/ 3

1/ 3 1/ 3 1/ 3 1/ 3

2 / 3 1/ 3

2 6

6 2 2 6

6 2 12

4 12

x x

x x x x

x x x

x x

+

− +

= ⋅ + ⋅ − ⋅ − ⋅

= + − −

= + −

118. ( ) ( )

( )

1/ 4 4 / 41/ 4 5 / 4 1/ 4

1/ 4 1/ 4 4 / 4

1/ 4 1/ 4 1

1/ 4

8 4 2 4 4

2 4 4

2 4 4

4 2

x x x x

x x x

x x x

x x

++ = ⋅ +

= ⋅ + ⋅

= ⋅ + ⋅

= +

120. ( ) ( )

( )

1/ 3 1/ 3 3/ 3

1/ 3 2 / 31/ 3

1/ 3 1/ 3 2 / 3

1/ 3 1/ 3 2 / 3

1/ 3 2 / 3

7 70 7 70

7 70

7 70

7 1 7 10

7 1 10

x x x x

x x

x x x

x x x

x x

+

− = −

= −

= − ⋅

= ⋅ − ⋅

= −

122. ( ) ( )( )( ) ( )( ) ( )( ) ( )( )

( )

1/ 3 66 3 5 / 6 1/ 3

6 1/ 3 3 1/ 3 5 / 6 6 1/ 3 61/ 3

1 22 1 5 2 2 5

33 1

8

8

2 2

22

x y x y

x y x y

x y x y x y

xx yy

− −

− −

+ −− − − +

−

=

= =

= =



401

124. ( ) ( )( )( ) ( )( ) ( )( )

41/ 2 7 / 4 47 / 4 5 / 41/ 25 / 4

4 1/ 2 4 2 / 4 41/ 2 2 / 4

22 2

2

x y x yy

x y x y

yx yx

−− −− − −−

− − − −−

−

⎛ ⎞=⎜ ⎟⎜ ⎟

⎝ ⎠

= =

= =

126. ( ) ( ) ( )1/ 3 327 29 27 29 27 29 3 87f = = = = There are 87 species of plants on an island with an area of 27 square miles.

128. ( )( ) ( )

3/ 4

3/ 4

70

80 70 70 1694

f x x

f

=

= ≈

A person who weighs 70 kilograms needs about 1694 calories per day to maintain life.

130. a. 4 / 25 4 / 2535.74 0.6215 35.74 0.4275C t v tv= + − +

b. 4 / 25 4 / 2535.74 0.6215(35) 35.74(15) 0.4275(35)(15) 25 FC = + − + ≈ °

132. 4 / 25 4 / 2535.74 0.6215 35.74 0.4275C t v t v= + − + ⋅

a. For 30t = , we get

4 / 25 4 / 25 4 / 25 4 / 25

4 / 25

( ) 35.74 0.6215(30) 35.74 0.4275(30) 35.74 18.645 35.74 12.825

54.385 22.915

C v v v v v

v

= + − + ⋅ = + − + ⋅

= −

b. 4 / 25(40) 54.385 22.915(40) 13C = − ≈ When the air temperature is 30 F° and the wind speed is 40 miles per hour, the windchill temperature is13 F° .

c. The solution to part (b) is represented by the point ( )40, 13 on the graph.

134. 31.25 9.8 16.296L S D+ − ≤

a. 1/ 2 1/ 31.25 9.8 16.296L S D+ − ≤

b.

( ) ( )

( ) ( )

1/ 2 1/ 3

1/ 2 1/ 3

3

1.25 9.8 16.296

22.85 1.25 312.5 9.8 22.34 16.296

22.85 1.25 312.5 9.8 22.34 16.29622.85 1.25 17.678 9.8 2.816 16.296

22.85 22.098 27.597 16.29617.351 16.296

L S D+ − ≤

+ − ≤

+ − ≤

+ − ≤

+ − ≤≤

The yacht is not eligible to participate in the America’s Cup.




402

146.

The simplification is not correct.

( ) ( ) ( )2 2 21/ 2 3/ 4 1/ 2 3/ 4

( 1/ 2)( 2) (3 / 4)( 2)

1 3/ 2 1 ( 3 / 2)

1/ 21/ 21

x x x x

x x

x x x

xx

− − −− −

− − −

− + −

−

⋅ =

=

= =

= =

The new simplification is correct. The right side should be 1/ 21

x.

148. does not make sense; Explanations will vary. Sample explanation: 1 1 1 1

12 2 2 27 7 7 7 7.+

⋅ = = =

150. does not make sense; Explanations will vary. Sample explanation: Since the 12

power is equivalent to an even root,

1/ 2 1/ 2( 64) 64 .− ≠ −

152. true

154. false; Changes to make the statement true will vary. A sample change is: ( )

2 / 32 / 3 2 23

1 1 1 1848 28

− = = = =

156. ( ) ( ) ( )

( ) ( ) ( )

5 / 2 3 / 4 1/ 45 / 2 3/ 4 1/ 4

10 / 4 3/ 4 1/ 4

12 / 4 3

2 2 2 2

2

2 2 8

+ −

+ −

⋅ ÷ =

=

= = =

The son is 8 years old.



403

158. ( ) 3/ 22 / 3 2 / 5 1/ 23 27 32 9⎡ ⎤+ + −⎢ ⎥⎣ ⎦

( ) ( )

( )( )

( ) [ ]

( ) ( )

3/ 22 23 5

3/ 22 2

3/ 2

3/ 2 3/ 2

3 3

3 27 32 9

3 3 2 3

3 9 4 3

3 13 3 16 3

16 3 4 3

64 3 61

⎡ ⎤⎛ ⎞= + + −⎢ ⎥⎜ ⎟⎝ ⎠⎣ ⎦

⎡ ⎤= + + −⎢ ⎥⎣ ⎦

⎡ ⎤= + + −⎣ ⎦

⎡ ⎤= + − = −⎣ ⎦

= − = −

= − =

160. 2 1

2 1

3 1 2 24 5 1

y ym

x x− −

= = = = −− − −

( )3 2 4y x− = − − Solve for y to write the equation in slope–intercept form.

3 2 82 11

or( ) 2 11

y xy x

f x x

− = − += − +

= − +

161. 3 32

y x≤ − +

162. ( ) ( )5 35 1 7 3 5 21 26

7 1D

−= = − − = + =

( ) ( )3 33 1 25 3 3 75 78

25 1xD−

= = − − = + =

( ) ( )5 35 25 7 3 125 21 104

7 25yD = = − = − =

78 1043 426 26

yx DDx y

D D= = = = = =

The solution set is ( ){ }3, 4 .

163. a. 16 4 4 2 8⋅ = ⋅ =

b. 16 4 64 8⋅ = =

c. 16 4 16 4⋅ = ⋅

164. a. 300 17.32≈

b. 10 3 17.32≈

c. 300 10 3=

165. a. 3 21 21/ 3 7x x x= =

b. 24 24 / 6 46 y y y= =

7.3 Check Points

1. a. 5 11 5 11 55⋅ = ⋅ =

b. 24 4 ( 4)( 4) 16x x x x x+ ⋅ − = + − = −

c. 3 3 3 36 10 6 10 60⋅ = ⋅ =

d. 7 73 47 2 6 12x x x⋅ =

2. a. 80 16 5 16 5 4 5= ⋅ = ⋅ =

b. 3 3 3 3 340 8 5 8 5 2 5= ⋅ = ⋅ =

c. 4 44 4 432 16 2 16 2 2 2= ⋅ = ⋅ =

d. 2 2200 100 2 10 2x y x y x y= ⋅ =

3. 2

2

2

2

( ) 3 12 12

3( 4 4)

3( 2)

3 ( 2)

3 2

f x x x

x x

x

x

x

= − +

= − +

= −

= ⋅ −

= ⋅ −

4. 9 11 3 8 10 2 4 5x y z x y z xyz x y z xyz= ⋅ =



404

5. 10 14 9 12 23 3

9 12 2 3 4 23 3 3

40 8 5

8 5 2 5

x y x x y y

x y xy x y xy

= ⋅ ⋅ ⋅ ⋅ ⋅

= =

6. 12 2 8 10 2 2 5 35 5

5 10 5 2 2 35

2 2 2 35

32 32

32

2

x y z x x y z z

x z x y z

x z x y z

= ⋅ ⋅ ⋅ ⋅ ⋅

=

=

7. a. 6 2 12 4 3 4 3 2 3⋅ = = ⋅ = ⋅ =

b. 33 3 3

33 3

3 3

10 16 5 2 50 16 2 50 32

50 8 4 50 8 4

50 2 4 100 4

⋅ = ⋅ =

= ⋅ = ⋅

= ⋅ ⋅ =

c. 2 6 3 8 44 4 4

8 4 24 44

4 8 32

16 2 2 2

x y x y x y

x y x y

⋅ =

= ⋅ =

7.3 Exercise Set

2. 7 5 35⋅ =

4. 33 35 4 20⋅ =

6. 5 5 59 3 27⋅ =

8. 5 11 55x y xy⋅ =

10. 4 42 346 3 18x x x⋅ =

12. 26 6 36x x x+ ⋅ − = −

14. ( ) ( )4 56 6 65 5 5x x x− ⋅ − = −

16. 2 5 2 55 2 5 2x x x⋅ = ⋅ =

18. 4 4 47 7

3 3x x

y y⋅ =

20. 2 3 3 4 2 3 3 49 9 9

5 79

12 3 12 3

36

x y x y x x y y

x y

⋅ = ⋅

=

22. 27 9 3 3 3= ⋅ =

24. 28 4 7 2 7= ⋅ =

26. 40 4 10 2 10x x x= ⋅ =

28. 33 354 27 2 3 2= ⋅ =

30. 3 33 3 3250 125 2 5 2x x x= ⋅ ⋅ =

32. 32 3 2 3 23 332 8 4 2 4x y x y y x− = − ⋅ = −

34. ( ) ( )281 2 9 2f x x x= − = −

36. ( ) ( ) ( )( )

3 33 3

3

48 2 8 6 2

2 2 6

f x x x

x

= − = ⋅ −

= −

38. ( ) ( )( )

2 2

2

5 10 5 5 2 1

5 1 1 5

f x x x x x

x x

= − + = − +

= − = −

40. 5 4 2x x x x x= ⋅ =

42. 6 7 6 6 3 3x y x y y x y y= =

44. 3 240 4 10 2 10x x x x x= ⋅ =

46. 11 9 2 3 23 3 3y y y y y= ⋅ =

48. 3 17 2 2 15 2 2 5 2 23 3 3x y z x y y z xy y z= =

50. 9 17 9 15 2 3 5 23 3 332 8 4 2 4x y x y y x y y= ⋅ =

52. ( ) ( ) ( )

( )( )

4 33 3

33 3

3

x y x y x y

x y x y

x y x y

+ = + ⋅ +

= + ⋅ +

= + +

54. 18 15 3 3 35 5 5y y y y y= ⋅ =

56. 7 16 5 2 15 3 25 5 564 32 2 2 2x y x x y y xy x y= ⋅ =

58. 4 4 411 8 3 2 396 16 6 2 6x x x x x= ⋅ =



405

60. ( ) ( ) ( )

( ) ( )

( )

14 12 24 4

3 24

3

2 2 2

2 2

or

2 2

x x x

x x

x x

− = − −

= − −

= − −

62. 3 6 18 9 2 3 2⋅ = = ⋅ =

64. 8 10 80 16 5 4 5x y xy xy xy⋅ = = ⋅ =

66. 220 5 20 5 100 10x x x x x x⋅ = ⋅ = =

68. 2 2 3 2 25 10 50 25 2

5 2

xy xy x y x y y

xy y

⋅ = = ⋅

=

70. 3 15 5 6 15 90 15 9 10

15 3 10 45 10

⋅ = = ⋅

= ⋅ =

72. 4 44 4 44 8 32 16 2 2 2⋅ = = ⋅ =

74. 7 4 11 10 52 12 24 4 6 2 6x x x x x x x⋅ = = ⋅ =

76. 7 4 12 11 133 3 3

9 2 123

3 4 23

6 9 54

27 2

3 2

x y x y x y

x x y y

x y x y

⋅ =

= ⋅

=

78. 2 3 3 4 6 6 4 94 4 44 8 32x y z x yz x y z⋅ = 44 2 4 8 2 24 16 2 2 2x x y z z xyz x z= ⋅ =

80. 4 3 3 9 8 5 12 115 5 5

5 10 2 10 2 2 25 5

8 8 64

32 2 2 2

x y z xy z x y z

x y y z z xy z y z

⋅ =

= ⋅ =

82. ( ) ( )

( ) ( )

( ) ( )

7 83 3 3

6 23

2 23

6 6 6

6 6

6 6

x x x

x x

x x

− ⋅ − = −

= − −

= − −

84.

( )

2 74 9 73

2 73 9 63

2 73 9 6 33

2 73 2 3

3 10 2 3

322

8 42

8 42

2 42

4

x y x y z

x y x xy z z

x y x y z xz

x y xy z xz

x y z xz

− ⎛ ⎞−⎜ ⎟⎝ ⎠

−= − ⋅

−= − ⋅

−= ⋅ − ⋅

=

86.

( )

2 7 11 175

2 7 10 15 25

2 7 10 15 25 5

2 7 2 3 25

4 10 25

4 32

4 32

4 32

4 2

8

x y x y

x y x xy y

x y x y xy

x y x y xy

x y xy

⎛ ⎞− −⎜ ⎟⎝ ⎠

= − −

= − − ⋅

= − −

=

88. ( ) ( )2 3 45 2 10x y z xyz x z xz− −

( )

6 3 2

6 3 2 2 2

6 3 2 2 2

6 3 2 2 2

6 3 2

7 3 3

5 2 10

5 20

5 4 5

5 4 5

5 2 5

10 5

x y z xyz xz

x y z x yz

x y z x z y

x y z x z y

x y z xz y

x y z y

= ⋅

=

= ⋅

= ⋅

=

=

90. 4 42 2 3 2

43 2 2 3 2

5 8 4 4

20 8 4

a b a b ab a b

a b a b a b

⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

= ⋅

( )

43 2 5 3

43 2 4 3

43 2 3

44 2 3

20 32

20 16 2

20 2 2

40 2

a b a b

a b a ab

a b a ab

a b ab

=

= ⋅

=

=

92. ( ) 3(120)120 180

236 5 36 5 6 5 miles

r = =

= ⋅ = ⋅ =

The captain can see 6 5 13.4 miles≈ .



406

94. ( ) 4 2W x x=

( ) ( )10 4 2 10 4 20

4 4 5

4 4 5

4 2 5

8 5 18 feet per second

W = =

= ⋅

= ⋅

= ⋅

= ≈

A dinosaur with a leg length of 6 feet has a walking speed of about 18 feet per second.

96. a. ( ) 4

4

4

4

7.6448080

7.64416 5

7.6442 5

3.8225

C =

=⋅

=

=

The cardiac index of an 80-year-old is4

3.8225

.

b. 43.822 3.822

1.49552.56

=

≈

The cardiac index of a 80-year-old is 2.56 liters per minute per square meter. This is represented on the graph as the point (80, 2.56) .


104. 28 4 2x x=

The graphs do not coincide. Correct the simplification.

2 28 4 2 2 2x x x= ⋅ = .

106. 3 23 2 4 4x x x⋅ =


3 3 32 2 33 2 4 2 4 8 2x x x x x x⋅ = ⋅ = =

108. makes sense

110. does not make sense; Explanations will vary. Sample explanation: A single radical of an nth root is still not simplified if it contains a factor that is a perfect nth power.

112. false; Changes to make the statement true will vary. A sample change is: 3 3 3 34 4 4 4 16⋅ = ⋅ =

114. true

116. Assume the cube root of a number is x. Triple the cube root and work in reverse to find the number.

( )3 3 333 3 27x x x= = The number must be multiplied by 27 for the cube root to be tripled.



407

118. ( ) ( )21f x x= −

( ) ( )

( ) ( )

( ) ( )

( ) ( )

( ) ( )

( ) ( )

( ) ( )

( ) ( )

2

2

2

2

2

2

2

2

1

3 3 3 1 16 4

2 2 2 1 9 3

1 1 1 1 4 2

0 0 0 1 1 1

1 1 1 1 0 0

2 2 2 1 1 1

3 3 3 1 4 2

x f x x

f

f

f

f

f

f

f

= −

− − = − − = =

− − = − − = =

− − = − − = =

= − = =

= − = =

= − = =

= − = =

119. 2 1 21 and 2 2 122 22 2 10

11 5

x xx xx x

− ≤ + ≥≤ ≥≤ ≥

{ }5 11x x≤ ≤ or [ ]5,11

120. Multiply the first equation by –3, the second equation by 2 and solve by addition.

15 6 6 8 6 8

7 142

x yx y

xx

− − = −+ = −

− = −=

Back-substitute 2 for x to find y.

( )5 2 2

5 2 2 210 2 2

2 84

x yyyyy

+ =

+ =

+ == −= −

The solution is ( )2, 4 .−

The solution set is ( ){ }2, 4 .−

121. ( ) ( )3 264 27 4 3 16 12 9x x x x− = − + +

122. a. 21 10 31x x x+ =

b. 21 2 10 2 31 2+ =

123. a. 4 12 8x x x− = −

b. 3 3 34 2 12 2 8 2− = −

124. 5 5 44 4 4 4 4

3 3 34 12

7 7 7 7y y y y y y

x x xx

⋅= = =

7.4 Check Points

1. a. 8 13 2 13 10 13+ =

b. ( )( )

3 3 3 3

3

9 7 6 7 12 7 9 6 12 7

21 6 7

x x

x

− + = − +

= −

c. 3 34 4 4

34

7 3 2 3 2 3 (7 2) 3 2 3

5 3 2 3

x x x x x

x x

− + = − +

= +

2. a. 3 20 5 45 3 4 5 5 9 5

3 2 5 5 3 5

6 5 15 5

21 5

+ = ⋅ + ⋅

= ⋅ + ⋅

= +

=

b. 3 12 6 27 3 4 3 6 9 3

3 2 3 6 3 3

6 3 18 3

12 3

x x x x

x x

x x

x

− = ⋅ − ⋅

= ⋅ − ⋅

= −

= −

c. 8 5 6 2− cannot be simplified.

3. a. 3 3 3 3

3 3

3 3

3

3 24 5 81 3 8 3 5 27 3

3 2 3 5 3 3

6 3 15 3

9 3

− = ⋅ − ⋅

= ⋅ − ⋅

= −

= −



408

b. 2 5 4 2 3 3 23 3 3 3

2 3 3 23 3 3

2 23 3

23

5 27 5 27

5 27

5 3

(5 3 )

x y x y x y x y x y

x y x y x y

x y xy x y

xy x y

+ = +

= +

= +

= +

4. a. 3 3 3 3 3

33 3 3

24 24 8 3 8 3 2 3125 5125 125 125

⋅ ⋅= = = =

b. 3 3 2 2

10 510 10 10

9 9 9 9 3x x x x x x x xy yy y y

⋅= = = =

c. 7 6 23 37 3

312 43 312 12

8 8 28 y y y y yyx xx x

⋅= = =

5. a. 5 5

4 4 4 240 40 20 4 5 4 5 2 522

x x x x x xxx

= = = ⋅ = =

b. 50 51 50 1 125 5

2 2 2 2 22 2xy xyxy xy xy= ⋅ = ⋅ = ⋅ =

c. 73 7

7 1 1 ( 2) 6 3 2333223

48 48 48 8 2666

x y x y x y x y x yxyxy

− − −−−

= = ⋅ ⋅ = ⋅ =

7.4 Exercise Set

2. 7 3 2 3 9 3+ =

4. ( )3 3 3 39 7 4 7 9 4 7 5 7− = − =

6. 5 5 5 56 3 2 3 3 3 5 3+ − =

8. 8 17 5 19 6 17 4 19 2 17 19− − + = −

10. 3 3 36 7 2 7 5 8 7 4x x x− + + = +

12. 5 20 5 4 5 5 2 5 3 5+ = + ⋅ = + =

14. 5 12 75 5 4 3 25 3

5 2 3 5 3

10 3 5 3 15 3

+ = ⋅ + ⋅

= ⋅ +

= + =



409

16. 5 45 2 20 5 9 5 2 4 5

5 3 5 2 2 5

15 5 4 5 11 5

x x x x

x x

x x x

− = ⋅ − ⋅

= ⋅ − ⋅

= − =

18. 3 3 3 3 3 3

3 3 3

3 24 81 3 8 3 27 3 3 2 3 3 3

6 3 3 3 9 3

+ = ⋅ + ⋅ = ⋅ +

= + =

20. 3 28 45 5 8 9 5 5

8 3 5 5

24 5 5

(24 1) 5

x x x x x

x x x

x x x

x x

+ = ⋅ +

= ⋅ +

= +

= +

22. 3 33 33 324 81 8 3 27 3xy y x xy y x+ = ⋅ + ⋅ 3 3 32 3 3 3 5 3y x y x y x= + =

24.

( )

3 34 33 3

3 3

3

81 24 27 3 8 33 3 2 33 2 3

x x x x xx x xx x

− = ⋅ − ⋅= −= −

26. ( )4 12 3 4 3 3

2 3 33 3

x x x x

x xx

− + − = − + −

= − + −= −

28. 4 2 2 3 2 23 3 3 3

2 23 3

23

4 5 4 5

4 5

9

x y x xy x xy x xy

x xy x xy

x xy

+ = +

= +

=

30. 19 1925 5

=

32. 3

3 11 1164 4

=

34. 2

12 6144 12x x

y y=

36. 3 2

8 4 450 25 2 5 281 9 9

x x x x xy y y

⋅= =

38. 3 35 3 2 2

33 5 5125

x x x x xy yy

= =

40. 3 38 6 2 2 2

315 5 5

81 27 3 3 38 2 2

x x x x xy y y

⋅= =

42. 4 3 34 47

412 3 3

13 1313 y y y yy

x x x= =

44. 5 514 10 4 2 4

515 3 3

64 32 2 2 2x x x x xy y y

⋅= =

46. 200 20 4 5 2 510

= = ⋅ =

48. 3

3354 27 32

= =

50. 3

272 36 62

x x xx

= =

52. 7 6

4 4 2 23 2

x yx y x y

x y= =

54. ( )33 1 4

1

4 2

500 50 5010

25 2 5 2

x x xx

x x

− −

−= =

= ⋅ =

56. 7 11 7 11

11 134 24 2

10 12 5 6

54 54 1833

9 2 3 2

a b a b a ba ba b

a a b b a b ab

− −− −= =

= ⋅ ⋅ =

58. 50 1 125 5

2 22 255 or

2 2

xyxy xy

xyxy

= = ⋅

=

60. 5 33

32 3 233 3

250125 5

2

x yx y y x

x= =

62. ( )12 115

11 25 10 10 1352 25

10 10 3 2 2 35 5

9632 32

3

32 2

x yx y x y

x y

x y y x y y

− −

−= =

= =



410

64.

( ) ( )

3 2 23

3

33

7 12 7 1233

4 34

3

x x x xxx

x xx

x

+ + + +=

+++ +

= = ++

66.

( ) ( )

3 3 3 3 33

3

2 23

3 2 2

a b a ba ba b

a b a ab b

a b

a ab b

− −=

−−

− + +=

−

= + +

68.

( )

27 75 9 3 25 32 7 2 7

9 3 25 32 7

3 3 5 3 21 3 10 32 7 14 1421 10 3 31 3

14 14

⋅ ⋅+ = +

⋅ ⋅= +

= + = +

+= =

70.

( ) ( ) ( )

( )

2 2 5

2 2 4

2 2

2 2

2

2

6 3 4 27 5 75

6 3 4 9 3 5 25 3

6 3 4 3 3 5 5 3

6 3 12 3 25 3

6 3 12 25 3

6 3 37 3

x xy x xy x y

x y x x xy x xy

x y x x xy x xy

xy x x xy x xy

xy x x xy

xy x x xy

− −

= ⋅ − ⋅ − ⋅

= − −

= − −

= − +

= −

72. 3 2

32

40 1507 25 3x xxx x

+

( )

( )

23

3

2

1507 2 83

7 2 8 50

7 2 8 25 2

7 2 8 5 2

7 2 40 2

7 40 2

47 2

xx xx

x x x

x x x x

x x x x

x x x x

x x

x x

= +

= +

= ⋅ + ⋅

= +

= +

= +

=

74. 2

2 32

9 645 8

3 2

x yx y

x y−−

( ) ( )

( )

2 32

2 3 3

2 2 2

645 8 3

2

5 8 3 32

5 4 2 3 16 2

5 2 2 3 4 2

10 2 12 2

10 12 2

2 2

yx y x

y

x y x y

x y y x y y

xy y x y y

xy y xy y

xy y

xy y

−= −

= −

= ⋅ − ⋅

= −

= −

= −

= −

76. 4 3 2 33 3

33 2 13

16 48 20 2

8 3 4

x x y x y

x x y x−−

( )

( )

3 2 333

2 1

433

333

3 3

3 3

3 3

48 22 5

3

2 16 5 2

2 8 2 5 2

2 2 2 5 2

4 2 5 2

4 5 2 2

x y x yx

x y x

x xy x y

x xy x xy

x xy x xy

x xy x xy

x xy x xy

−= −

= −

= ⋅ − ⋅

= −

= −

= − = −

78. ( )

( ) ( )

2 225 2555

5 55

5

f x xxg xx

x xx

x

⎛ ⎞ − −= =⎜ ⎟ ++⎝ ⎠

− += = −

+

To get the domain, we need 5 0

5x

x+ >

> − and 5 0

5x

x− ≥

≥

Therefore, we need 5x ≥ . Domain: { }| 5x x ≥ or [ )5,∞



411

80. ( )3 6 6 5

3 33

33 23 2

3

2 216 816

28

f x x xxg xx

x xx x

⎛ ⎞= = =⎜ ⎟

⎝ ⎠

⋅= =

Our only restriction here is that we cannot divide by 0. Thus, we need 0x ≠ .

Domain: { }| 0x x ≠ or ( ) ( ), 0 0,−∞ ∪ ∞

82. Perimeter: 2 2

2 4 20 2 80

8 4 5 2 16 5

8 2 5 2 4 5

16 5 8 5

24 5 feet

P l w= +

= ⋅ + ⋅

= ⋅ + ⋅

= ⋅ + ⋅

= +

=

Area:

4 20 80

4 20 80

4 16004 40160 square feet

A lw=

= ⋅

= ⋅

== ⋅=

84. Area: 1

1 221212121212323232

( )

6 ( 8 2)

6 ( 4 2 2)

6 (2 2 2)

6 (3 2)

3 6 2)

12)

4 3)

2 3)

3 3 square feet

A h b b= +

= ⋅ ⋅ +

= ⋅ ⋅ ⋅ +

= ⋅ ⋅ +

= ⋅ ⋅

= ⋅ ⋅ ⋅

= ⋅

= ⋅ ⋅

= ⋅

=

86. a. 2 2 2 2

22

2 2

2 2

21

f f

f

f

c v c vR Rcc

c vRc c

vRc

− −=

= −

⎛ ⎞= − ⎜ ⎟⎝ ⎠

b.

( )

2 2

2

1 1

1 1

1 1

0

0 0

f f

f

f

f

f

v cR Rc c

R

R

R

R

⎛ ⎞ ⎛ ⎞− = −⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

= −

= −

=

= ⋅ =

Your aging rate is zero. This means that a person moving close to the speed of light does not age relative to a friend on Earth.


94. 4 9 5x x x+ =

The graphs coincide, so the simplification is correct.

96. 8 2 10x x x+ =


( )

8 2 4 2 2 2 2 2

2 2 3 2

x x x x x x

x x x

+ = ⋅ + = +

= + =



412

98. makes sense

100. makes sense

102. false; Changes to make the statement true will vary. A sample change is: 4 3 5 3 9 3+ =

104. false; Changes to make the statement true will vary. A sample change is: The square root of a negative number is not a real number, 4 2.− ≠ −

106. 20 45 803 4

4 5 9 5 16 53 4

+ −

⋅ ⋅= + − ⋅

2 5 3 5 4 53 4

= + −

2 35 5 4 53 42 3 4 53 42 4 3 3 4 12 53 4 4 3 1 128 9 48 5

12 12 12

31 31 55 or 12 12

= + −

⎛ ⎞= + −⎜ ⎟⎝ ⎠⎛ ⎞= ⋅ + ⋅ − ⋅⎜ ⎟⎝ ⎠⎛ ⎞= + −⎜ ⎟⎝ ⎠

= − −

108. ( ) ( )2 3 1 4 2 66 2 4 2 6

6 6 3 63 0

0

x x xx x x

x xxx

− − = − −

− − = − +− = −

==

The solution set is { }0 .

109. ( ) ( )2 28 12 6 2x xy y x y x y− + = − −

110. 2 2

2 35 6 6 9

xx x x x

++ + + +

( ) ( ) ( )

( ) ( )( )( ) ( )

( )( )

( ) ( )( ) ( )

( ) ( )

( ) ( )

2

2

2

2

2

2

2

2 33 2 3

3 22 33 2 3 23

2 3 3 2

3 2

2 6 3 6

3 2

3 8 6

3 2

xx x x

x xxx x x xx

x x x

x x

x x x

x x

x x

x x

= ++ + +

+ += ⋅ + ⋅

+ + + ++

+ + +=

+ +

+ + +=

+ +

+ +=

+ +

111. a. 7( 5) 7 35x x+ = +

b. ( )7 5 7 35x x+ = +

112. a. ( ) ( ) 25 6 3 6 33 15x x x x+ + = + +

b. ( )( ) 22 5 6 2 3 6 2 33 2 15

6 2 33 2 15

12 33 2 15

27 33 2

+ + = + +

= ⋅ + +

= + +

= +

113. 2 4 2 45 5

3 2 4 3 2 45 5 5 5

2 45

5 55

2 45

2 45

8 10 810

4 8 4 8

10 8

32

10 82

5 8

x y y x yy

x y x y x y x y

y x y

x y

y x yxy

x yx

⋅ =⋅

=

=

=


Intermediate Algebra for College Students 5E Mid-Chapter Check Point

413

Mid-Chapter Check Point – Chapter 7

1. ( )3100 27 10 3 10 3 13− − = − − = + =

2. 5 7 4 6 2 38 4 2 2 2x y x y xy x y xy= ⋅ =

3. ( )3 3 3 32 2 2 23 4 2 4 3 2 4 5 4x x x x+ = + =

4.

( )

3 3 32 2 2 2

3 4

3 3

3

3

3 4 2 4 6 4 4

6 16

6 8 2

6 2 2

12 2

x x x x

x

x x

x x

x x

⎛ ⎞ ⎛ ⎞ = ⋅⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

=

= ⋅

=

=

5. ( ) ( ) ( )( ) ( )

( )

2 33/ 52 / 3 3 5

2 3

27 32 27 32

3 2

9 81

+ − = + −

= + −

= + −

=

6. ( ) ( ) ( ) ( )( ) ( )( )

1/ 3 1/ 3 1/ 31/ 33 1/ 4 3 1/ 4

3 1/ 3 1/ 4 1/ 33

1/12

64 64

64

4

x y x y

x y

xy

=

= ⋅ ⋅

=

7.

( ) ( )

( )

5 27 4 48 5 9 3 4 16 3

5 3 3 4 4 3

15 3 16 3

15 16 3

3

− = ⋅ − ⋅

= −

= −

= −

= −

8. 3 3 2

4 4 4

2 2

500 500 100 54 4 4

10 5 5 5

2

x x x xy y y

x x x x

y y

⋅= =

= =

9. ( )

( ) ( )

1 1/ 44 1/ 4

4 / 4 1/ 4 3/ 4

4 3

x x xx x

x x

x

−

−

= =

= =

=

10. 3 3 35 3 2 254 27 2 3 2x x x x x= ⋅ =

11. 3

3 3 333160 160 80 8 10 2 10

22= = = ⋅ =

12.

( )( )

( )

( )

1/ 510 105

20 20

1/ 510

1/ 520

10 1/ 5 2

420 1/ 5

x xy y

x

y

x xyy

⎛ ⎞= ⎜ ⎟⎜ ⎟⎝ ⎠

=

= =

13. ( )( )

( )

( )

( ) ( )

( ) ( )

22 / 3 2 / 3 2 4 / 3

3 3/ 41/ 4 31/ 4

4 / 3 3/ 4

16 /12 9 /12

7 /12

x x xxxx

x

x

x

⋅

⋅

−

−

= =

=

=

=

14. ( ) ( ) ( )( ) ( )

1/ 6 / 6 1/ 66 4 6 4 6 46

6 1/ 6 4 1/ 6 1 2 / 3

23

x y x y x y

x y x y

x y

1= =

= =

=

15. ( ) ( )

( ) ( )

( )

( ) ( )

( ) ( )

3 67 7

3 67

92

7 27

27

2 2

2 2

2

2 2

2 2

x x

x x

x

x x

x x

− ⋅ −

= − ⋅ −

= −

= − ⋅ −

= − −



414

16. 11 17 8 16 34 4

8 16 34 4

2 4 34

32 16 2

16 2

2 2

x y x y x y

x y x y

x y x y

= ⋅

= ⋅

=

17.

( ) ( )

( )

3 3 3 3

3 33 3

3 3

3 3

3

3

4 16 2 54 4 8 2 2 27 2

4 8 2 2 27 2

4 2 2 2 3 2

8 2 6 2

8 6 2

14 2

+ = ⋅ + ⋅

= ⋅ + ⋅

= +

= +

= +

=

18. 4 97 4 9

9 2775 75 77

7 2 27

2 27

x y x y x yx yx y

x x y

x x y

−−= =

= ⋅

=

19. ( )( ) ( )

( )

2 / 32 / 3 23

2

1 1125125 125

1 1255

−− = =− −

= =−

20. ( ) ( )

( ) ( )

1/ 2 1/ 31/ 2 1/ 33

3/ 6 2 / 6 5 / 6

6 5 6

2 2 2 2 2

2 2

2 32

+

+

⋅ = ⋅ =

= =

= =

21. 2 2

3 33 4 2 4 2

33 33

6 63

2

32 2 32 2

64 64

4

x x x xy y y y

x xy y

xy

⋅ = ⋅

= =

=

22. 2 3 5 2 3 5

4 7

4 6

2 3

32 2 32 2

64

64

8

xy x y xy x y

x y

x y y

x y y

⋅ = ⋅

=

= ⋅

=

23.

( ) ( )

( )

4 3 6

4 2 6

2 3

3 3

3

3

4 6 7 24

4 6 7 4 6

4 6 7 2 6

4 6 14 6

4 14 6

10 6

x x y y x y

x x y y y x y

x x y y y x y

x y y x y y

x y y

x y y

−

= ⋅ − ⋅

= −

= −

= −

= −

24. ( ) 30 5f x x= − To find the domain, we set the radicand greater than or equal to 0.

30 5 05 30

6

xxx

− ≥− ≥ −

≤

Domain: { }| 6x x ≤ or ( ], 6−∞

25. ( ) 3 3 15g x x= − The domain of a cube root is all real numbers. Since there are no other restrictions, we have Domain: { }| is a real numberx x or ( ),−∞ ∞



415

7.5 Check Points

1. a. ( )6 10 6 6 10

6 60

6 4 15

6 2 15

x x

x

x

x

+ = ⋅ +

= +

= + ⋅

= +

b. 2 23 33 33 3 3

33 3

3

7 7

7

7

y y y y y

y y

y y

⎛ ⎞− = ⋅ − ⋅⎜ ⎟⎝ ⎠

= −

= −

c. ( )( ) ( )( ) ( )( ) ( )( ) ( )( )6 5 3 2 2 5 4 2 6 5 2 5 6 5 4 2 3 2 2 5 3 2 4 2

12 5 24 10 6 10 12 2

60 24 10 6 10 24

36 18 10

+ − = + − + + −

= ⋅ − + − ⋅

= − + −

= −

2. a. ( ) ( ) ( )2 2 25 6 5 2 5 6 6

5 2 30 6

11 2 30

+ = + ⋅ ⋅ +

= + +

= +

b. ( )( ) ( ) ( )2 26 5 6 5 6 5

6 51

+ − = −

= −=

c. ( )( ) ( ) ( )2 27 7 7

7

a a a

a

− + = −

= −

3. a. 3 3 7 21 2177 7 7 49

= ⋅ = =

b. 3 3 3 3 3 3

33 33 3 32 2 3

2 2 2 2 3 6 69 39 33 3 3= = = ⋅ = =

4. a. 7 142 2 2

7 77 7 7y xyx x x

y yy y y= = ⋅ =

b. 2 2 23 3 33 3 3

3 2 2 2 3 33 3 3 3

3 3 339 3 3 3 3

y xy xyx x xyy y y y y

= = ⋅ = =



416

c. 2 4 3 2 45 5

6 6

8 2

x x

x y x y=

2 35

3 2 4 2 35 5

2 35

5 5 55

2 35

35

26

2 2

6 2

2

6 22

3 4

x yx

x y x y

x x y

x y

x x yxy

x yy

= ⋅

=

=

=

5. 18 18 2 3 32 3 3 2 3 3 2 3 3

−= ⋅

+ + −

2 236 3 542 3 336 3 54

12 936 3 54

33(12 3 18)

312 3 18

−=

⋅ −−

=−−

=

−=

= −

6. 3 7 3 7 5 25 2 5 2 5 2+ + +

= ⋅− − +

3 5 3 2 35 145 2

3 5 3 2 35 143

+ + +=

−+ + +

=

7. 3 3 33 3 3

x x x x x xx x

+ − + − + += ⋅

+ +

( ) ( )

( )

2 23

3 3 33

3 3 33

3 3

13

x x

x xx xx x

x x

x x

+ −=

+ ++ −

=+ +

=+ +

=+ +

7.5 Exercise Set

2. ( )5 3 5 15x x+ = +

4. ( )3 5 3 5 3 9 5 3 3− = − = −

6. ( )6 4 6 3 2

4 36 3 12 4 6 3 4 3

24 3 2 3 24 6 3

−

= − = ⋅ − ⋅

= − ⋅ = −

8. ( )3 33 3 33 6 7 4 18 7 12+ = +

10. 3 3 32 2 23 3 3

3 3 33 2 23

24 8 3

8 3 2 3

x x x x x x

x x x x

⎛ ⎞− = ⋅ ⋅ −⎜ ⎟⎝ ⎠

= ⋅ − = −

12. ( ) ( ) ( )7 2 8 2 56 7 8 2 2

58 15 2

+ + = + + +

= +

14. ( ) ( ) ( )4 5 10 3 5 40 12 10 5 15

25 2 5

+ − = + − + −

= −

16. ( ) ( )( )

( )

2

7 2 7 5 3 7

35 21 7 10 7 6 7

35 21 10 7 6 7

35 31 7 42 77 31 7

− −

= − − +

= + − − + ⋅

= − + = −

18. ( ) ( )3 2 10 11

30 33 20 22

30 33 4 5 22

30 33 2 5 22

+ +

= + + +

= + + ⋅ +

= + + +

20. ( ) ( )3 2 10 11

30 33 20 22

30 33 4 5 22

30 33 2 5 22

− −

= − − +

= − − ⋅ +

= − − +

22. ( ) ( )3 5 2 3 4 5 5 3

12 5 15 15 8 15 10 3

− +

= ⋅ + − − ⋅

60 7 15 30 30 7 15= + − = +



417

24. ( )22 7 2 2 14 7 9 2 14+ = + + = +

26. ( )22 2 2 2x y x xy y− = − +

28. ( ) ( )6 2 6 2 6 4 2+ − = − =

30. ( ) ( ) 23 5 2 3 5 2 9 5 2 9 25 2

9 50 41

− + = − ⋅ = − ⋅

= − = −

32. ( ) ( )4 3 3 2 4 3 3 2

16 3 12 6

+ −

= ⋅ − 12 6+ 9 2 48 18 30− ⋅ = − =

34. ( ) ( )4 3 12 7x x x x− − = − +

36. ( ) ( ) 3 23 3 33 7 4 21x x x x− + = + −

38. 3 3 2 3 65 5 5 5

2 3 55 5

2 35 5

2 2

2

2

x y x y x x y y

x x y y y

x x y y y

⎛ ⎞ ⎛ ⎞− + = − −⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

= − −

= − −

40. 7 7 3 2133 3 3

= ⋅ =

42. 6 6 6x xx xx x= ⋅ =

44. 3 12 312 12

3 3 3 3 3

4 3

y y

y y y y y= ⋅ =

⋅

⋅=

33

y 4 3yyy

=

46. 3 32 2 3

3 3 3 32 3

1 1 3 3 933 3 3 3

= ⋅ = =

48. 3 2 3 3

3 3 3 32 3

10 10 5 10 25 10 2555 5 5 5

5

= ⋅ = =

=32 25

5⋅ 32 25=

50. 3 32 23 3 3

33 3 3 32 3

3 3

3 3 3 4 3 4 3 164 44 4 4 4

48 8 6 24 4

⋅ ⋅= = ⋅ = =

⋅= = =

3 62

3 622

=⋅

52. 3 3 3 32 2 2 2

3 3 3 3 32 2 33

7 7 7 7 7x x x xxx x x x x x

= ⋅ = = =

54. 3 3 3 3 3

3 2 32 2 33 3 3

5 55 5 5 y y yyyy y y y

= = ⋅ = =

56. 3 2 3 3

3 3 3 32 2 2 3 3

3 3

10 10 4 10 16 10 8 244 4 4 4

10 2 2 20 2 5 44 4

x x xxx x x x

x xx x

⋅= ⋅ = =

⋅ ⋅= = =

3 24

x

35 2

x

xx

=

58. 233 3

3 2 2 2 23 3 3

2 23 3

3 33

3 3 3

3 3

x y

xy xy xy x y

x y x yxyx y

= = ⋅

= =

60. 4 4 43 3 3

4 4 4 43 4

5 5 5 5x x xxx x x x

= ⋅ = =

62. 5 53 3

5 5 5 52 4 2 3 5 5

5 53 3

10 10 2 10 2

16 2 2 2

10 2 5 22

x x

x x x x

x xx x

= ⋅ =

= =

64. 2 4 252 2 2

3 3 3 3 3 2 4 25 5 5 5

2 4 2 2 4 25 5

5 5 55

4 25

23 3 3

8 2 2 2

3 4 3 422

3 42

x yxy xy xy

xy xy xy x y

xy x y xy x yxyx y

y x y

= = ⋅

= =

=



418

66.

( ) ( )

2 2

2

25 25 2555 5

5255 5

25 5 25 555

5

x yx y x y

y

x y y

y yx yx y

= =⋅

= ⋅

= =

=5 55

y⋅ 5 5yxyxy

=

68.

( )

3 3 2

5 5 4

2 2

22 2

3

150 150 25 6

5 6 5 6

5 6 5 6

5 6

a a a ab b b b

a a a a bbb b b b

a ab a abb bb b

a abb

⋅− = − = −

⋅

= − = − ⋅

= − = −

= −

70. 4 6 2 2

3 4

2

53 315

3 3 333 3 (3)

m n mn mnm n

n m n m n m

= =

= ⋅ = =

72.

( )

2 7 4 2 24 4 4

2 34

2 2 34 4

2 3 2 3 24 4 4

24 44

5 5 5

5

5 5 5

x y y x y y x y

x y

y x y x y

x y x y x yy xy xyy x y

= =⋅

= ⋅

= = =

74. 4 11 3 9 23 3

15 15

27 27x y x y xy=

− − ⋅

( )

3 23

23

3 2 23 3

2 23 3

33 3 33

2 23 3

2 4 2 4

15

3

15

3

15 15

33

15 5

3

xy xy

x y

xy xy x y

x y x y

xy xyxy x y

x y x y

x y x y

=−

= ⋅−

= =−−

= = −−

76. ( )

( ) ( )

15 6 115 15 6 16 16 1 6 1 6 1

15 6 13 6 1 3 6 3

5

−−= ⋅ =

−+ + −

−= = − = −

78. 17 17 10 210 2 10 2 10 2

17 10 34 17 10 3410 4 6

+= ⋅

− − +

+ += =

−

80.

( ) ( )

12 12 7 37 3 7 3 7 3

12 7 3 12 7 3

7 3 43 7 3 3

−= ⋅

+ + −

− −= =

−= −

82. b b a b ab ba ba b a b a b

+ += ⋅ =

−− − +

84.

( ) ( )

( ) ( )

35 35 5 2 3 55 2 3 5 5 2 3 5 5 2 3 5

35 5 2 3 5 35 5 2 3 5

25 2 9 5 50 4535 5 2 3 5

7 5 2 3 55

35 2 21 5

+= ⋅

− − +

+ += =

⋅ − ⋅ −

+= = +

= +



419

86.

( )

11 5 11 5 11 511 5 11 5 11 5

11 2 55 5 16 2 5511 5 6

2 8 55 8 556 3

− − −= ⋅

+ + −

− + −= =

−

− −= =

88. 2 2 5 3 10255 5 5

x x x x xxx x x

− − + + −= ⋅ =

−− − +

90. 2 6 5 2 6 5 3 6 53 6 5 3 6 5 3 6 5

36 5 30 5 41 5 309 6 5 54 5

41 5 3049

+ + += ⋅

− − +

+ + += =

⋅ − −+

=

92. 3 3 3

3 3 3

6 99

x y x y y x

y x y x y x

xy x yy x

+ + += ⋅

− − +

+ +=

−

94. 5 5 5 53 3 5 15= ⋅ =

96. 3 32 2 3 33 3

3 3 3 2 2 2 23 3

2 2 2 2 2

2 4 4

x x x x xy y x x y x y

= ⋅ = =

98. 4 4 4 164 4

x x x xx x x x x+ + − −

= ⋅ =− −

100.

2

a b a b a ba b a b a b

a ba ab b

− − += ⋅

+ + +−

=+ +

102.

( )

7 7 77 7 7

7 7

7 7

x x x x x xx x

x x

x x

+ − + − + += ⋅

+ +

+ −= =

+ + 7 ( )7

17

x x

x x

+ +

=+ +

104. 2 2 2 2x y x y x y

x yx y x y

− − += ⋅

+− −

( ) ( )2 2x y

x y x y

x y

−=

− +

−=( ) ( )x y x y+ − ( )

( ) ( )1

x y

x y x y

+

=+ +

106. 1 1 55 55 5 5

5 5 5 555 5 5

5 5 5 6 55 5

+ = + ⋅

= + = +

+= =

108. 4 3

4 44 4 4 3

44

4 4 4

20 20 28 82 2 2

20 882

8 10 8 9 8

− = − ⋅

= −

= − = −

110. 5 3 5 315 153 5 3 5

− + = − +

5 3 3 5153 3 5 515 15153 5

15 15 5 15 3 1515 15 15

15 15 5 15 3 1515

13 1515

= − ⋅ + ⋅

= − +

= − +

− +=

=



420

112. 5 2 32 282 7

− ++

( ) ( )( )2 2

5 2 7 2 16 2 4 72 7 2 75 2 5 7 2 4 2 2 72 7

5 2 5 7 8 2 2 72 7

5 2 5 7 8 2 2 75

7 2 8 2 2 7

3 7 9 2

−= ⋅ − ⋅ + ⋅

+ −

−= − +

−

−= − +

−−

= − +−

= − − +

= −

114. ( ) 2 4 2f x x x= + −

( ) ( ) ( )22 6 2 6 4 2 6 2

4 4 6 6 8 4 6 20

f − + = − + + − + −

= − + − + −=

116. ( ) 2f x x=

( )( )( ) ( )

( ) ( )

2

2 2

2

1 1

1 1

1 2 1 1 1

1 2 1 1 1

2 2 1

f a a

a a

a a a a

a a a a

a a

+ − −

= + − −

= + − + ⋅ − + −

= + − + − + −

= − −

118. 7 2 2 3 7 2 36 6

7 2

⋅ ⋅ ⋅=

⋅=

32

7 333

=⋅



421

120. ( )Perimeter 4 4 2 3 2

8 3 4 2

s= = +

= +

The perimeter is 8 3 4 2+ inches.

( )( ) ( )

22

2 2

Area 2 3 2

2 3 2 2 3 2 2

4 3 4 6 2

12 4 6 2

14 4 6

s= = +

= + ⋅ +

= ⋅ + +

= + +

= +

The area is 14 4 6+ square inches.

122. 2 2 2c a b= +

( ) ( )( ) ( ) ( ) ( )

2 22

2 2 2 22

2

2

20 6 20 6

20 2 20 6 6 20 2 20 6 6

20 2 120 6 20 2 120 6

52

52

2 13 inches

c

c

c

c

c

c

= + + −

= + + + − +

= + + + − +

=

=

=


132. ( ) ( ) 22 2 4 for 0x x x x+ − = − ≥


( ) ( )2 2 4 for 0x x x x+ − = − ≥



422

134. 3 33

x xx x

= + ++ −

The graphs coincide, so the simplification is correct.

136. does not make sense; Explanations will vary. Sample explanation: 2( )a b+ does not equal 2 2.a b+ Actually, 2 2 2( ) 2 .a b a ab b+ = + + Thus, 2 2 2( 2 5) ( 2) 2 2 5 ( 5)

2 2 10 5

7 2 10

+ = + ⋅ ⋅ +

= + +

= +

138. makes sense

140. false; Changes to make the statement true will vary. A sample change is: 44 4 x y x y

x yx y x y x y

+ += ⋅ =

++ + +

142. false; Changes to make the statement true will vary. A sample change is: ( ) ( ) ( ) ( )2 2 27 2 7 7

14 49

x x x

x x

− = + ⋅ ⋅ − + −

= − +

144. ( )22 3 2 3+ + −

( ) ( )( ) ( )

22 3 2 3

22 3 2 2 3

2 3 2 2 3 2 3 2 3

2 3

+ −+= + + −

= + + + − + −

= + 2 4 2 3+ − 2 3+ 3 2 3− + −

4 2 4 3

4 2 14 2 6

= + −

= += + =

146. 2

2 32 4x x+

− −

( ) ( )( )

( ) ( ) ( ) ( )( )

( ) ( ) ( ) ( )

( ) ( ) 2

2 32 2 2

2 2 32 2 2 2

2 2 3 2 4 32 2 2 22 7 2 7 or 2 2 4

x x x

xx x x x

x xx x x x

x xx x x

= +− + −

+= +

− + + −

+ + + += =

− + − +

+ +=

− + −



423

147. Using the results from Exercise 146, we know that the left side of the equation simplifies

to( ) ( )

2 72 2x

x x+

− +. Setting this equal to zero, we

have ( ) ( )

2 7 02 2x

x x+

=− +

.

To solve, set the numerator equal to zero. 2 7 0

2 772

xx

x

+ == −

= −

The solution set is 7 .2

⎧ ⎫−⎨ ⎬⎩ ⎭

148. 2 1 0 3 2 52 4 2 8

1 2 1 4 13

− − −− −− −

The remainder is 13. This means that ( )2 13.f − =

149. ( ) ( )2 2 24 1 4 2 4 1

4 2 4 1

5 2 4

x x x

x x

x x

+ + = + + + +

= + + + +

= + + +

150. 2

2

2

4 16 16 4( 4)

4 16 16 4 16

4 20 04 ( 5) 0

x x x

x x x

x xx x

− + = +

− + = +

− =− =

0 or 5 05

x xx

= − ==

The solution set is { }0,5 .

151. 2

2

2

26 11 16 8

0 10 3

0 3 100 ( 5)( 2)

x x x

x x

x xx x

− = − +

= − + +

= + −= + −

5 0 or 2 05 2

x xx x

+ = − == − =

The solution set is { }5,2 .−

7.6 Check Points

1.

( )2 2

3 4 8

3 4 8

3 4 643 60

20

x

x

xxx

+ =

+ =

+ ===

Check: 3 4 8

3 20 4 8

60 4 8

64 88 8

x + =

⋅ + =

+ =

==


2. 1 7 2

1 5

x

x

− + =

− = −

A principal square root cannot be negative. Thus, this equation has no solution. The solution set is { }.

3.

( )2 2

2

2

6 7 2

6 7 2

6 7 ( 2)

6 7 4 4

0 2 30 ( 1)( 3)

x x

x x

x x

x x x

x xx x

+ − =

+ = +

+ = +

+ = + +

= − −= + −

1 0 or 3 01 3

x xx x

+ = − == − =

Check: 6 7 2

6( 1) 7 ( 1) 2

1 1 21 1 2

2 2

x x+ − =

− + − − =

+ =+ =

=

6 7 2

6(3) 7 (3) 2

25 3 25 3 2

2 2

x x+ − =

+ − =

− =− =

=




424

4. 5 3 2x x+ − − =

( ) ( )( )

( )

2 2

2 2

22

5 3 2

5 3 2

5 3 2 2 3 2

5 3 4 3 4

5 1 4 3

4 4 3

4 4 34 41 3

1 3

1 34

x x

x x

x x x

x x x

x x x

x

x

x

x

xx

+ = − +

+ = − +

+ = − + ⋅ − +

+ = − + − +

+ = + + −

= −

−=

= −

= −

= −=

Check: 5 3 2

4 5 4 3 2

9 1 23 1 2

2 2

x x+ − − =

+ − − =

− =− =

=


5.

( )

13

13

13

33

(2 3) 3 0

(2 3) 3

(2 3) 3

2 3 272 24

12

x

x

x

xxx

− + =

− = −

⎛ ⎞− = −⎜ ⎟⎝ ⎠

− = −= −= −

Check:

13

13

13

3

(2 3) 3 0

(2( 12) 3) 3 0

( 27) 3 0

27 3 03 3 0

0 0

x − + =

− − + =

− + =

− + =− + =

=

The solution set is { }12 .−

6.

( )22

( ) 3.5 38

73 3.5 38

35 3.5

35 3.53.5 3.510

10

100

f x x

x

x

x

x

x

x

= +

= +

=

=

=

=

=

The model projects that 73% of U.S. women will participate in the work force 100 years after 1960, or 2060.

7.6 Exercise Set

2.

( )2 2

5 1 8

5 1 8

5 1 645 65

13

x

x

xxx

− =

− =

− ===


4.

( )2 2

3 2 5 0

3 2 5

3 2 5

3 2 253 27

9

x

x

x

xxx

− − =

− =

− =

− ===




425

6. 2 5 11 6

2 5 5

x

x

+ + =

+ = −

Since the square root of a number is always positive, the solution set is { } or .∅

8.

( )

( ) ( )

22

2

2

6 7

6 7

6 7

6 7 07 1 0

x x

x x

x x

x xx x

= +

= +

= +

− − =

− + =

7 0 or 1 07 1

x xx x

− = + == = −

We disregard –1 because square roots are always positive. The solution set is { }7 .

10.

( ) ( )

( ) ( )

2 2

2

2

2 1 7

2 1 7

2 1 14 49

0 16 480 12 4

x x

x x

x x x

x xx x

+ = −

+ = −

+ = − +

= − +

= − −

12 0 or 4 012 4

x xx x

− = − == =

We disregard 4, because it does not check. The solution set is { }12 .

12.

( ) ( )

( ) ( )

22

2

2

3 7 3

3 3 7

3 3 7

6 9 3 7

3 2 02 1 0

x x

x x

x x

x x x

x xx x

= + −

+ = +

+ = +

+ + = +

+ + =

+ + =

2 0 or 1 02 1

x xx x

+ = + == − = −

Both solutions check. The solution set is { }2, 1 .− −

14.

( ) ( )

( )( ) ( )

22

2

2

2

3 3 7 5

3 5 3 7

3 5 3 7

9 30 25 3 7

9 27 18 0

9 3 2 0

9 2 1 0

x x

x x

x x

x x x

x x

x x

x x

− + = −

+ = +

+ = +

+ + = +

+ + =

+ + =

+ + =

( )9 2 02 0

2

xx

x

+ =

+ =

= −

or 1 01

xx

+ == −

We disregard –2 because the solution does not check. The solution set is { }1 .−

16.

( ) ( )2 2

6 2 5 3

6 2 5 3

6 2 5 32 3

1

x x

x x

x xx

x

+ = +

+ = +

+ = ++ =

=

The solution checks. The solution set is { }1 .

18.

( )3

3 33

6 3 3

6 3 3

6 3 276 30

5

x

x

xxx

− =

− =

− ===


20.

( )

3

3

3 33

4 3 5 0

4 3 5

4 3 5

4 3 1254 128

32

x

x

x

xxx

− − =

− =

− =

− ===




426

22. 8 2x x− = −

( ) ( )2 28 2x x

x

− = −

8 x− =

( )22

4 4

8 4 4

12 4

12 44 43

3

9

x

x

x

x

x

x

x

− +

− = − +

− = −

− −=

− −=

=

=


24. 4 4 4x x− + + =

( ) ( )2 2

4 4 4

4 4 4

x x

x x

x

+ = − −

+ = − −

4 16 8 4x x+ = − − +

( )22

4

4 16 8 4 4

4 12 8 4

8 8 4

8 8 48 81 4

1 4

1 45

x

x

x

x

x

x

xx

−

= − − −

= − −

− = − −

− − −=

− −= −

= −

= −=


26. 2 3 4 1x x− + = +

( ) ( )

( )( )

( ) ( )

2 2

22 2

2

2

2

2 3 3

2 3 3

2 3 6 9

4 3 6 9

4 12 6 9

0 10 210 3 7

x x

x x

x x x

x x x

x x x

x xx x

− = −

− = −

− = − +

− = − +

− = − +

= − +

= − −

3 03

xx

− ==

or 7 07

xx

− ==

Check 3x = : 2 3 3 4 3 1

2 0 4 40 4 4

4 4

− + = +

+ =+ =

=

Check 7x = : 2 7 3 4 7 1

2 4 4 82(2) 4 8

4 4 88 8

− + = +

+ =+ =+ =

=

Both solutions check. The solution set is{ }3, 7 .

28. ( )( )

( )( )

1/ 3

1/ 3

31/ 3 3

3 6 5 8

3 6 3

3 6 3

3 6 273 33

11

x

x

x

xxx

− + =

− =

− =

− ===


30. ( )( )

( )( )

1/ 4

1/ 4

41/ 4 4

2 3 7 10

2 3 3

2 3 3

2 3 812 78

39

x

x

x

xxx

+ + =

+ =

+ =

+ ===


32. ( )( )

1/ 2

1/ 2

3 8 6

3 2

3 2

x

x

x

− + =

− = −

− = −

The square root of a number must be positive. The solution set is .∅



427

34. 2 3 7 1x x+ + + =

( ) ( )

( ) ( )

( ) ( )

2 2

22

2

2

2 1 3 7

2 1 3 7

2 1 2 3 7 3 7

2 8 2 3 7 3

2 2 8 2 3 7

2 6 2 3 72 2

3 3 7

3 3 7

6 9 3 7

3 2 02 1 0

x x

x x

x x x

x x x

x x

x x

x x

x x

x x x

x xx x

+ = − +

+ = − +

+ = − + + +

+ = − + +

− + = − +

− − − +=

− −+ = +

+ = +

+ + = +

+ + =

+ + =

1 02 02 1

xxx x

+ =+ == − = −

We disregard –1 because it does not check. The solution set is { }2 .−

36. ( ) ( )1/ 31/ 3 22 1 2x x x− = +

( )( ) ( )( )( )

( ) ( )

33 1/ 31/ 3 2

3 2

2

2

2

2 1 2

2 1 2

8 1 2

8 8 2

0 6 80 4 2

x x x

x x x

x x x

x x x

x xx x

⎛ ⎞− = +⎜ ⎟

⎝ ⎠

− = +

− = +

− = +

= − +

= − −

4 0 2 04 2

x xx x

− = − == =

Both solutions check. The solution set is { }2, 4 .

38. ( ) ( )

( )( ) ( )( )1/ 4 1/ 4

4 41/ 4 1/ 4

2 3 8

2 3 8

x x

x x

− = −

− = −

2 3 82 2 86 23

x xxx

x

− = −− = −

==


40. ( ) 2f x x x= − −

( ) ( )

( ) ( )

2 2

2

2

4 2

2 4

2 4

2 8 16

0 9 180 6 3

x x

x x

x x

x x x

x xx x

= − −

− = −

− = −

− = − +

= − +

= − −

6 06

xx

− ==

or 3 03

xx

− ==

Check 6x = : 6 6 2 6 44

− − = −=

Check 3x = : 3 3 2 3 12 4

− − = −= ≠

Discard 3. The solution is 6.

42. ( ) ( )1/ 49 2f x x= + ; ( ) ( )1/ 45 18g x x= +

( ) ( )

( ) ( )

1/ 4 1/ 4

4 41/ 4 1/ 4

9 2 5 18

9 2 5 18

9 2 5 184 16

4

x x

x x

x xxx

+ = +

⎡ ⎤ ⎡ ⎤+ = +⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦+ = +

==

Check: ( )( ) ( )( )( ) ( )

1/ 4 1/ 4

1/ 4 1/ 4

1/ 4 1/ 4

9 4 2 5 4 18

36 2 20 18

38 38

+ = +

+ = +

=

The solution is 4.

44.

22

2

2 2

4

4

44 or 4

Ar

Ar

Ar

r A A r

π

π

ππ π

=

⎛ ⎞= ⎜ ⎟⎜ ⎟⎝ ⎠

=

= =



428

46.

22

2

2

2

FRvm

FRvm

FRvm

mv FRFRmv

=

⎛ ⎞= ⎜ ⎟⎜ ⎟⎝ ⎠

=

=

=

48. Let x = the number. 3 5x x− = −

( ) ( )

( ) ( )

2 2

2

2

3 5

3 10 25

0 11 280 7 4

x x

x x x

x xx x

− = −

− = − +

= − +

= − −

7 07

xx

− ==

or 4 04

xx

− ==

Check 7x = : 7 3 7 5

4 22 2

− = −

==

Check 4x = : 4 3 4 5

1 11 1

− = −

= −≠ −

Discard 4. The number is 7.

50. ( ) 2 3 2 1f x x x= − − + To find the x-intercepts, set the function equal to 0 and solve for x.

( ) ( )

( )

2 2

2 2

0 2 3 2 1

2 1 2 3

2 1 2 3

2 2 2 1 2 3

2 2 4

2 2

2 2

2 42

x x

x x

x x

x x x

x

x

x

xx

= − − +

− = −

− = −

− + = −

− = −

=

=

==

Check 2x = :

( ) ( )2 2 3 2 2 1

4 3 4 1

1 4 11 2 10

− − +

= − − +

= − += − +=

The only x-intercept is 2.

52.

( )22

2

0.852

1.7

1.7

2.9

dt

d

d

d

d

=

=

=

=

≈

The vertical distance was about 2.9 feet.

54. It is represented by the point (2.9, 0.85).

56. a. ( ) 4.4 38f x x= − +

(60) 4.4 60 38 3.9f = − + ≈ About 3.9% of Americans earning $60 thousand annually report fair or poor health. This underestimates the percent displayed in the graph by 0.1%.

b. ( ) 4.4 38f x x= − +

( )2 2

24 4.4 38

14 4.4

14 4.44.4 4.4144.4

144.4

10

x

x

x

x

x

x

= − +

− = −

− −=

− −

=

⎛ ⎞ =⎜ ⎟⎝ ⎠

≈

$10 thousand is the annual income that corresponds to 24% reporting fair or poor health.



429

58. ( )

( )

1/ 3

1/ 3

1/ 3

1/ 3

33 1/ 3

29

58 29

58 2929 29

2

2

8

f x x

x

x

x

x

x

=

=

=

=

=

=

A Galapagos island with an area of 8 square miles will have 58 plant species.

60. ( ) 3/ 20.2f x x=

( )

3/ 2

3 / 2

3 / 2

22 3/ 2

3

3 33

88 0.2

88 0.20.2 0.2440

440

193, 600

193, 60058

x

x

x

x

x

xx

=

=

=

=

=

=

≈ The average distance of Mercury from the sun is approximately 58 million kilometers.


68. 2 2 3 5x x+ = −


70. 2 3 1x x+ = +


72. 4 2x + =

The graphs do not intersect. This means that the solution set is .∅

74. makes sense

76. makes sense

78. false; Changes to make the statement true will vary. A sample change is: The equation 4 5x + = − has no solution. By definition, absolute values are nonnegative.

80. false; Changes to make the statement true will vary. A sample change is: Note that if x is negative, then –x is positive. Thus an equation of this form can have a real solution. In this case the solution is

1 .3

x = −

82.

( )

( )

3

3 33

2 2

2

3

9

9

729

729

531441

53144181

x x

x x

x x

x x

x x

xx

=

=

=

=

=

==

Check: 3 81 81 9

9 9==




430

84. ( )

( )( )( )

2 / 3

3/ 22 / 3 3 / 2

3

3

4 25

4 25

4 25

4 54 125

129

x

x

x

xx

x

− =

− =

− =

− =− =

=

Check:

( )( )

( )

2 / 3

2 / 3

23

2

129 4 25

125 25

125 25

5 2525 25

− =

=

=

==


85. 3 4 3 2 1 112 45 141 426

4 15 47 142 425

− − − −− −− −

4 3 2

3 2

4 3 2 13

4254 15 47 1423

x x x xx

x x xx

− + − −+

= − + − ++

86.

( )( )

2

2

2

2

2

3 12 6 1842 8

3 12 46 182 8

3 4

4

x xxx x

x xxx x

x

x

− +÷

++ −

− += ⋅

++ −

−=

+ ( )4

2x

x+

⋅− ( )

( ) ( )

6 3

3 2 2

x

x x

+

+ −=

( )2x − ( )1

6 3x⋅

+

( )( )

3 2 11 6 3

xx

+= ⋅

+

( )( ) ( )

3 2 26 3 2 3

x xx x+ +

= =+ +

87.

( )2 2

2 2

6 9 25

6 9 25

y y x

y y x

− + −

= − + −

( )( ) ( )( )( ) ( )

3 5 3 5

3 5 3 5

y x y x

y x y x

= − + − −

= − + − −

88. ( 5 7 ) ( 11 6 ) 5 7 11 66 13

x x x xx

− + − − − = − + + += +

89.

2

2

(7 3 )( 2 5 )(7)( 2) (7)( 5 ) ( 3 )( 2) ( 3 )( 5 )

14 35 6 15

15 29 14

x xx x x x

x x x

x x

− − −= − + − + − − + − −

= − − + +

= − −

90. 7 4 2 7 4 2 2 5 22 5 2 2 5 2 2 5 2+ + +

= ⋅− − +

( )( )( ) ( )

( ) ( )

( ) ( )

( ) ( )

22

22

22

7 4 2 2 5 2

2 5 2 2 5 2

7 2 7 5 2 4 2 2 4 2 5 2

2 5 2

14 35 2 8 2 20 4

2 5 2

14 43 2 20 2

4 5 2

14 43 2 404 25 2

54 43 246

54 43 246

+ +=

− +

⋅ + ⋅ + ⋅ + ⋅=

−

+ + +=

−

+ + ⋅=

−

+ +=

− ⋅+

=−+

= −



431

7.7 Check Points

1. a. 64 64 1 64 1 8i− = ⋅− = ⋅ − =

b. 11 11 1 11 1 11i− = ⋅− = ⋅ − =

c. 48 48 1

48 1

16 3 1

4 3

4 3

i

i

− = ⋅−

= ⋅ −

= ⋅ ⋅ −

= ⋅ ⋅

=

2. a. (5 2 ) (3 3 ) 5 2 3 38

i i i ii

− + + = − + += +

b. (2 6 ) (12 4 ) 2 6 12 410 10

i i i ii

+ − − = + − += − +

3. a. 2

7 (2 9 ) 7 2 7 9

14 6314 63( 1)63 14

i i i i i

i ii

i

− = ⋅ − ⋅

= −= − −= +

b. 2(5 4 )(6 7 ) 30 35 24 2830 35 24 28( 1)30 28 35 2458 11

i i i i ii i

i ii

+ − = − + −= − + − −= + − += −

4.

2

5 7 5 1 7 1

5 7

35

( 1) 35

35

i i

i

− ⋅ − = − ⋅ −

= ⋅

=

= −

= −

5. 6 2 6 2 4 34 3 4 3 4 3

i i ii i i

+ + += ⋅

− − +

2

2 2

2

24 18 8 64 (3 )

24 18 8 6( 1)16 9

18 2616 9( 1)18 26

2518 2625 25

i i ii

i ii

i

i

i

+ + +=

−+ + + −

=−

+=

− −+

=

= +

6. 3 2 3 2 44 4 4

i i ii i i

− − −= ⋅

−

2

212 8

1612 8( 1)

16( 1)8 12168 12

16 161 32 4

i ii

i

i

i

i

− +=

−− + −

=− −

− −=

= − −

= − −

7. a. 16 2 8 8( ) ( 1) 1i i= = − =

b. 25 24 2 12 12( ) ( 1) 1i i i i i i i i= = = − = ⋅ =

c. 35 34 2 17 17( ) ( 1) ( 1)i i i i i i i i= = = − = − = −

7.7 Exercise Set

2. 49 49 1 7i− = ⋅− =

4. 21 21 1 21i− = ⋅− =

6. 125 25 5 1 5 5i− = ⋅ ⋅− =

8. 28 4 7 1 2 7i− = ⋅ ⋅ − =

10. 300 100 3 1 10 3i− − = − ⋅ ⋅ − = −

12. 7 4 7 4 1 7 2i+ − = + ⋅− = +

14. 20 5 20 5 1 20 5i+ − = + ⋅− = +

16. 3 27 3 9 3 1 3 3 3i− − − = − − ⋅ ⋅ − = − −

18. ( ) ( )6 5 4 3 6 5 4 3 10 8i i i i i+ + + = + + + = +

20. ( ) ( )2 6 4 2 6 4 2 5i i i i i− + + − = − + + − = +

22. ( ) ( )11 8 2 5 11 8 2 5 9 3i i i i i+ − + = + − − = +

24. ( ) ( )8 5 6 2 8 5 6 2 2 7i i i i i− − + = − − − = −



432

26. ( ) ( )7 5 9 11 7 5 9 1116 16

i i i ii

− + − − = − + − +

= − +

28. ( ) ( )9 2 17 6 9 2 17 68 8

i i i ii

− + − − − = − + + +

= +

30. ( )15 12 11 15 12 11 12 26i i i i i− − = − + = − +

32. ( ) ( )4 5 8 6 5

4 5 8 6 5 12 7 5

i i

i i i

+ + +

= + + + = +

34. ( ) ( )25 4 7 20 35 20 35 135 20

i i i i ii

+ = + = + −

= − +

36. ( ) ( )28 4 3 32 24 32 1 2432 24

i i i i ii

− = − = − −

= − −

38. ( ) ( )26 3 5 18 30 18 30 130 18

i i i i ii

− − = − + = − + −

= − −

40. ( ) ( )( )

24 5 6 20 24 5 6

20 29 6 1 14 29

i i i i i

i i

+ + = + + +

= + + − = +

42. ( ) ( )( )

28 4 3 2 24 16 12 8

24 28 8 1 16 28

i i i i i

i i

− − = − − +

= − + − = −

44. ( ) ( )( )

27 2 3 6 21 42 6 12

21 48 12 19 48

i i i i i

ii

− − + = − + + −

= − + − −

= − +

46. ( ) ( ) ( )2 7 2 7 4 49 1 4 4953 53 0

i ii

+ − = − − = +

= = +

48. ( ) ( ) ( )4 2 4 2 16 4 1 16 420 20 0

i ii

− + − − = − − = +

= = +

50. ( )( )( )

25 3 5 3 25 3

25 3 125 32828 0

i i i

i

− + = −

= − −

= +== +

52. ( )( )

2 23 2 9 2 3 2 4

9 12 4 1 5 12

i i i

i i

+ = + ⋅ ⋅ +

= + + − = +

54. ( )( )

2 25 3 25 30 9

25 30 9 1 16 30

i i i

i i

− = − +

= − + − = −

56.

( )

27 3 7 3 21

21 1 21 21 0

i i i

i

− ⋅ − = ⋅ =

= − = − = − +

58. ( )216 4 4 2 8 8 18 8 0i i i

i− ⋅ − = ⋅ = = −

= − = − +

60.

( )

23 36 3 6 6 3

6 3 1 6 3 6 3 0

i i i

i

− ⋅ − = ⋅ =

= − = − = − +

62.

( )

29 5 3 5 3 5

3 5 1 3 5 3 5 0

i i i

i

− ⋅ − = ⋅ =

= − = − = − +

64. ( )

3 3 4 12 3 12 34 4 4 16 116 1

12 3 12 317 17 17

i i ii i i

i i

− − −= ⋅ = =

+ + − +− −

−= = −

66. ( )

( )2

2 2

10 5 15 5 2 10 52 2 2 4 12

10 5 10 5 5 10 1 24 1 5 5 5

ii i i i ii i i i

i i i i

− −− −= ⋅ = =

+ + − − −−+ +

= = = + = ++

68. ( )2

9 9 1 2 9 18 9 181 2 1 2 1 2 1 4 11 4

9 18 9 18 9 181 4 5 5 5

i i ii i i i

i i i

+ + += ⋅ = =

− − + − −−+ +

= = = ++

70. ( )

( )( )

2

22

2

5 5 2 3 10 152 3 2 3 2 3 2 3

10 15 1 10 154 9 14 9

10 15 10 15 15 104 9 13 13 13

i i i i ii i i i

i ii

i i i

+ += ⋅ =

− − + −

+ − −= =

− −−− −

= = = − ++

72.

( )( )

2

2 21 1 1 1 2 11 1 1 1

1 2 1 2 or 021 1

i i i i ii i i i

i i i i

− − − − ⋅ ⋅ += ⋅ =

+ + − −− + − −

= = = − −− −



433

74.

( )( )

2

2 22 3 2 3 3 6 2 9 33 3 3 3

6 11 3 1 6 11 39 19 1

3 11 3 1110 10 10

i i i i i ii i i i

i i

i i

+ + + + + += ⋅ =

− − + −+ + − + −

= =+− −

+= = +

76. ( )

( )( )

2

22

2

6 3 6 3 4 2 24 12 12 64 2 4 2 4 2 4 2

24 24 6 1 24 24 616 4 116 4

18 24 18 2416 4 20

18 24 9 620 20 10 5

i i i i i ii i i i

i ii

i i

i i

− − − − − += ⋅ =

+ + − −

− + − − −= =

− −−− −

= =+

= − = −

78. ( )

( )( )

2

22

2

5 5 3 2 15 10 3 23 2 3 2 3 2 3 2

15 7 2 1 15 7 29 4 19 4

17 7 17 7 17 79 4 13 13 13

i i i i i ii i i i

i ii

i i i

− − + + − −= ⋅ =

− − + −

+ − − + += =

− −−+ +

= = = ++

80. ( )25 5 5 57 7 7 17

5 5 or 07 7

i i ii i i i

i i

= ⋅ = =−

= − −

82. ( )

( )2

2

3 4 13 4 3 4 3 45 5 5 15

3 4 4 3 4 35 5 5 5 5

ii i i i ii i i i

i i i

+ −+ + += ⋅ = =

−

− −= = + = −

− − −

84. ( )( )

2

2

5 15 5 54 4 4 14

5 1 1 54 4 4

ii i i i ii i i i

i i

+ −+ + += ⋅ = =

− − − −−−

= = − +

86. ( ) ( )7 714 2 1 1i i= = − = −

88. ( ) ( )7 715 2 1i i i i i= = − = −

90. ( ) ( )23 2346 2 1 1i i= = − = −

92. ( ) ( )200 200400 2 1 1i i= = − =

94. ( ) ( )10 1021 2 1i i i i i= = − =

96. ( ) ( ) ( )

( ) ( )

6 6 6 6 6

3 32

1 1

1 1

i i i i

i

− = − ⋅ = − =

= = − = −

98. ( ) ( ) ( ) ( )( ) ( ) ( )

613 13 13 2

6

1 1

1 1 1

i i i i

i i i

− = − = −

= − − = − = −

100. ( ) ( ) ( ) ( )( )

14 15 14 1528 30 2 2 1 1

1 1 0

i i i i+ = + = − + −

= + − =

102. ( ) ( ) ( ) ( )

( ) ( )

( )

2 2 2

2 2

2

8 9 2 1 1

16 8 18 9 1

16 10 9 1

15 10 815 10 8 123 10

i i i i

i i i i

i i i

i iii

+ − − − +

= − + − − −

= + − − +

= + −

= + − −

= +

104. ( ) ( )

( ) ( )

( )

2 2

2 2

2 2

2

4 1 2

16 8 1 4 4

16 8 1 4 4

15 12 315 12 3 118 12

i i

i i i i

i i i i

i iii

− − +

= − + − + +

= − + − − −

= − −

= − − −

= −

106. 5 8 3 18

5 4 2 1 3 9 2 1

5 2 2 3 3 2

10 2 9 2

19 2 or 0 19 2

i i

i i

i i

− + −

= − + −

= ⋅ + ⋅

= +

= +



434

108. ( ) ( )2 104 48 40

4 3 4 2

i ii ii i i i i

++=

+ + ⋅

( )

2 10

2 2

1 1 1 111 1

2 2 11 1 12 2 2 2

1 112 2 1

2

iii

i i ii i

ii i

+ += =

−+ −

+= = ⋅

− − ++ +

= =+−

+= = +

110. ( ) 2 2 5f x x x= − +

( ) ( ) ( )2

2

2

1 2 1 2 2 1 2 5

1 4 4 2 4 5

4 44 40

f i i i

i i i

i

− = − − − +

= − + − + +

= += −=

112. ( ) 12f x x i= − ; ( ) 6 3g x x i= +

1 112 43 31 16 3 2 33 3

f i i

g i i

⎛ ⎞ ⎛ ⎞− = − − = − −⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠⎛ ⎞ ⎛ ⎞− = − + = − +⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

( )( ) ( )( )

( )

13

2

4 2 3

8 12 2 38 10 3 111 10

fg i i

i i iii

− = − − − +

= − + −

= − − −

= −

114. ( )2 113

xf xx

+=

−

( ) ( )

( )

( )

( )

2 2

2 22

4 11 16 1143 4 3 4

16 1 11 53 4 3 45 3 4

3 4 3 415 20 15 20

9 163 415 20 15 20

259 16 115 20

25 253 45 5

i if ii i

i ii

i ii i

iii i

i

i

+ += =

− −− + −

= =− −

− += ⋅

− +− − − −

= =−−

− − − −= =

− −

−= −

= − −

116.

( ) ( )( )2

2 3 3 5

6 10 9 15 6 15 16 15 21

E IR i i

i i i ii i

= = − +

= + − − = + − −

= + + = +

The voltage of the circuit is ( )21 i+ volts.


132. ( )22

9 9 9

3 3 9 9i i i

− = − ⋅ −

= ⋅ = = −

134. does not make sense; Explanations will vary. Sample explanation: Imaginary numbers are defined.

136. makes sense

138. false; Changes to make the statement true will vary. A sample change is: ( )( )

( )( )

22 2

3 7 3 7

3 7 9 49

9 49 19 49 58

i i

i i

+ −

= − = −

= − −

= + =

140. true



435

142. 1 11 2 1 2

i ii i

+ −+

+ −

( ) ( )( ) ( )

( ) ( )( ) ( )

( ) ( ) ( ) ( )( ) ( )

( )( )( )

2 2

22

2

2

1 1 2 1 1 21 2 1 2 1 2 1 21 1 2 1 1 2

1 2 1 2

1 2 2 1 2 2

1 22 4 12 41 4 11 4

2 4 6 6 01 4 5 5

i i i ii i i i

i i i ii i

i i i i i i

i

i

i

i

+ − − += +

+ − + −+ − + − +

=+ −

− + − + + − −=

−− −−

= =− −−

+= = = +

+

144. 2 2 2 2

2 2

2 2

1 1

1 1

x xy yy y x y

y y x yx xx x

+ +

= ⋅+ +

2

x

y=

2 2x y⋅1y

+ 2 2x y⋅

2

y

x2x⋅ 2 1y

x+ 2x⋅

( )

2

23 2

3 2

y

x x yx x y

y xy

++= =

+ ( )2y y x+

2

2xy

=

145. 1 1 1

1x y z

x

+ =

x⋅ 1yzy

+ x y⋅1zz

= xy z⋅

( )

yz xz xyyz xy xzyz x y z

yzxy z

+ == −= −

=−

146. 3 1 528 2 2

x xx − +− = +

( )

3 1 58 2 88 2 2

16 3 4 4 515 3 4 4 2015 3 4 2411 3 24

11 212111

x xx

x x xx xx xx

x

x

− +⎛ ⎞ ⎛ ⎞− = +⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

− + = + +

+ = + ++ = ++ =

=

=


⎧ ⎫⎨ ⎬⎩ ⎭

147. 22 7 4 0( 4)(2 1) 0

x xx x

+ − =+ − =

4 0 or 2 1 04 2 1

12

x xx x

x

+ = − == − =

=

The solution set is 14, .2

⎧ ⎫−⎨ ⎬⎩ ⎭

148. 2

2

9

9 0( 3)( 3) 0

x

xx x

=

− =+ − =

3 0 or 3 03 3

x xx x

+ = − == − =


149. 2

2

2 2

3 18

3( 6) 18

3( 1) ( 6) 183 1 6 18

18 18, true

x =

− =

− =⋅ ⋅ =

=

6− is a solution.



436

Chapter 7 Review

1. 81 9= because 29 81=

2. 1 1100 10

− = − because 21 1

10 100⎛ ⎞− =⎜ ⎟⎝ ⎠

3. 3 27 3− = − because ( )33 27− = −

4. 4 16− not a real number The index is even and the radicand is negative.

5. 5 32 2− = − because ( )52 32− = −

6. ( ) ( )15 2 15 5 30 5

25 5

f = − = −

= =

( ) ( )4 2 4 5 8 5 3 1.73f = − = − = ≈

5 52 5 5 52 2

0 0

f ⎛ ⎞ ⎛ ⎞= − = −⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

= =

( ) ( )1 2 1 5 2 5 3f = − = − = − not a real number

7. ( ) ( ) 3 334 4 4 8 16 8 8 2g = − = − = =

( ) ( )0 4 0g = 33 8 8 2− = − = −

( ) ( ) 33

3

14 4 14 8 56 8

64 4

g − = − − = − −

= − = −

8. To find the domain, set the radicand greater than or equal to zero and solve the resulting inequality.

2 02

xx

− ≥≥

The domain of f is { }2x x ≥ or [ )2, .∞

9. To find the domain, set the radicand greater than or equal to zero and solve the resulting inequality. 100 4 0

4 1004 1004 4

25

xxx

x

− ≥− ≥ −− −

≤− −

≤

The domain of g is { }25x x ≤ or ( ], 25 .−∞

10. 225 5x x=

11. ( )214 14x x+ = +

12. ( )22 8 16 4 4x x x x− + = − = −

13. 3 364 4x x=

14. 4 416 2x x=

15. ( ) ( )55 32 7 2 7x x− + = − +

16. ( )1

335 5xy xy=

17. ( ) ( )3

3 3216 16 4 64= = =

18. ( ) ( )4

4 45532 32 2 16= = =

19. ( )127 7x x=

20. ( ) ( )553 319 19xy xy=

21. ( ) ( )

23

2 2 233

1 1 1 184288

−= = = =

22. ( )( )

45 4

5

33 xx abab

− =

( )

( )

45

45

4 45 5

3

3

3

x

ab

x

abx

a b

=

=

=

23. 1 1 11 4 3 73 3 44 12 12 12x x x x x

+ +⋅ = = =


Intermediate Algebra for College Students 5E Chapter 7 Review

437

24.

11 1 3 2 1

22 3 6 6 6

13

5 5 5 5

5

− −= = =

25. ( )1 1 11 6 36 3 23 3 338 8 2x y x y x y

⋅ ⋅= =

26.

12 2 11 1 123 3 24 4 2

11 1 83 8

13

x y x y

yx y

x

− − ⋅ ⋅

−

⎛ ⎞⎜ ⎟ =⎝ ⎠

= =

27. ( )1

9 12 9 123 3

1 19 12 3 43 3

x y x y

x y x y⋅ ⋅

=

= =

28. ( )1 11 3 93 9 3 99 9 99

133

x y x y x y

x y y x

⋅ ⋅= =

= =

29. 1 1 1 3 21

3 3 2 3 6 62

56 56

x x x x x x

x x

+ +⋅ = = =

= =

30.

22 13 2 33 2

24 24

4 3 166 6 6

x x xx

x

x x x

−

−

= =

= = =

31.

11 1 1 1555 3 3 3 3 5

11515

x x x x

x x

⋅⎛ ⎞⎜ ⎟= = =⎝ ⎠

= =

32. Since 2012 is 27 years after 1985, find ( )27 .f

( ) ( ) ( )( ) ( )

2 233

2

27 350 27 350 27

350 3 350 9 3150

f = =

= = =

Expenditures will be $3150 million or $3,150,000,000 in the year 2012.

33. 3 7 21x y xy⋅ =

34. 5 52 357 11 77x x x⋅ =

35. ( ) ( )4 56 6 65 5 5x x x− ⋅ − = −

36. ( )

( )( )

2

2

2

7 14 7

7 2 1

7 1 7 1

f x x x

x x

x x

= − +

= − +

= − = −

37. 3 2 220 4 5 4 5

2 5

x x x x x

x x

= ⋅ ⋅ ⋅ = ⋅

=

38. 8 6 6 2 63 3

6 6 23

32 2 2

54 27 2

27 2

3 2

x y x x y

x y x

x y x

= ⋅ ⋅ ⋅

= ⋅

=

39. 3 11 5 3 8 3 54 432 16 2x y z x y y z z= ⋅ ⋅ ⋅ ⋅ ⋅

8 4 3 34

2 3 34

16 2

2 2

y z x y z

y z x y z

= ⋅

=

40. 3 2 5 4

4 2

6 4 24 4 6

4 6 2 6

x x x x x

x x x x

⋅ = = ⋅ ⋅ ⋅

= ⋅ =

41. 2 4 3 53 3 3

3 3 23

3 3 23

23

4 4 16

8 2

8 2

2 2

x y xy x y

x y y

x y y

xy y

⋅ =

= ⋅ ⋅ ⋅ ⋅

= ⋅

=

42. 4 3 4 6 75 5

5 9 115

5 5 4 105

5 5 10 45

2 45

2 8

16

16

16

16

x y z xy z

x y z

x y y z z

x y z y z

xyz y z

⋅

=

= ⋅ ⋅ ⋅ ⋅ ⋅

= ⋅

=



438

43. ( ) ( )2

1 1 1 1

1

x x x x

x

+ ⋅ − = + −

= −

44. ( )3 3 3 36 3 2 3 6 2 3 8 3+ = + =

45. 5 18 3 8 5 9 2 3 4 2− = ⋅ − ⋅

( )

5 3 2 3 2 2

15 2 6 2

15 6 2 9 2

= ⋅ − ⋅

= −

= − =

46. 3 4 63

3 3 63

27

27

x xy

x x xy

+

= +

( )23 3

2 3

3

3

x x y x

x y x

= +

= +

47.

( )

3 3 3 3

3 3

3 3

3 3

2 6 5 48 2 6 5 8 6

2 6 5 2 6

2 6 10 6

2 10 6 8 6

− = − ⋅

= − ⋅

= −

= − = −

48. 33 316 8 2 2 2125 125 5

⋅= =

49. 3 2

4 4

2 2

100 100

or 10 10

x x xy y

x x xxy y

⋅=

=

50. 5 4

4 420 20

44

5 5

3 3

16 163

3 or 2 2

y y y

x xy yy

yx x

⋅=

=

51. 48 48 24 4 6 2 622

= = = ⋅ =

52. 3

33 33332 32 16 8 2 2 2

22= = = ⋅ =

53. 4 7 7

4 5424 2

4 4

4 4 4

64 64 3222

16 2

16 2 2 2

x x xxx

x x

x x x x

= =

= ⋅ ⋅ ⋅

= ⋅ =

54. 3 2 3 2

522

4 2

200 200 10022

100 10

x y x y x yx yx y

x xy x xy

−−= =

= =

55. ( )3 2 6 4 15 2 18 4 45

2 9 2 4 9 5

2 3 2 4 3 5

6 2 12 5

+ = +

= ⋅ + ⋅

= ⋅ + ⋅

= +

56. ( )33 3 3 3

3 3

3 3

5 50 2 250 10

125 2 10

5 2 10

− = −

= ⋅ −

= −

57. ( ) ( )7 3 5 7 6 5

7 6 35 3 35 18 5

7 3 35 90

3 35 83 or 83 3 35

− +

= + − − ⋅

= + −

= − − +

58. ( )( )11 11

11 11 11

x y

xy x y

− −

= − − +

59. ( )25 8 5 2 5 8 8+ = + ⋅ ⋅ +

13 2 40

13 2 4 10

13 2 2 10

13 4 10

= +

= + ⋅

= + ⋅

= +

60. ( )22 3 10

4 3 2 2 3 10 10

12 4 30 10 22 4 30

−

= ⋅ − ⋅ ⋅ +

= − + = −



439

61. ( ) ( )( ) ( )2 2

7 13 7 13

7 13 7 13 6

+ −

= − = − = −

62. ( ) ( ) ( )227 3 5 7 3 5 7 3 5

49 9 549 45 4

− + = −

= − ⋅= − =

63. 4 4 6 4 6 2 66 36 6 6

= ⋅ = =

64. 2 2 2 7 147 77 7 7= = ⋅ =

65. 3 3

3 33 32 3

33

12 12 3 12 39 33 3

12 3 4 33

= ⋅ =

= =

66. 2 2

5 10 102 25 55 5 5

y xy xyx xy yy y y= ⋅ = =

67. 3 32 2

3 3 3 32 2 2 3 3

3 3

14 14 2 14 2

2 2 2 214 4 7 4

2

x x

x x x xx x

x x

= ⋅ =

= =

68. 4 3 34 4

44 4 4 3 3

7 7 7 33 3 3 3

xx x x x= = ⋅

4 43 3 3

4 4 4

4 3

7 3 7 2733

1893

x xxx

xx

⋅ ⋅= =

=

69. 45

4 5 4 45 5 5

4 45 5

5 5 55

5 5

32 2

5 522

xy

x y x y xy

xy xyxyx y

= ⋅

= =

70.

( )( )

( )

( ) ( )

2 2

6 6 3 13 1 3 1 3 1

6 3 1 6 3 1

3 13 1

6 3 13 3 1

23 3 3

+= ⋅

− − +

+ += =

−−

+= = +

= +

71.

( ) ( )2 2

7 7 5 3

5 3 5 3 5 3

35 21

5 3

−= ⋅

+ + −

−=

−

35 21 35 215 3 2− −

= =−

72.

( )( ) ( )

( )2 2

10 2 5 3 2 10 2 5 3 2

4 5 9 22 5 3 2

102 5 3 2

10 2 5 3 22 5 3 2 2 5 3 2

+ += =

⋅ − ⋅−

−+

= ⋅− +

( ) ( )

( )

10 2 5 3 2 10 2 5 3 2

20 18 2

5 2 5 3 2 10 5 15 2

+ += =

−

= + = +

73.

( )2 2

5 5 33 3 33 5 15

3

8 159

x x xx x xx x x

x

x xx

+ + += ⋅

− − +

+ + +=

−

+ +=

−



440

74.

( ) ( )

( )

2 2

7 3 7 3 7 37 3 7 3 7 3

7 2 7 3 3

7 3

10 2 21 10 2 217 3 4

2 5 21 5 214 2

+ + += ⋅

− − +

+ ⋅ ⋅ +=

−

+ += =

−

+ += =

75. 2 3 6 2 3 6 2 6 32 6 3 2 6 3 2 6 3

+ + −= ⋅

+ + −

( ) ( )

( )

2 24 18 2 3 2 6 18

2 6 3

3 18 6 12 3 9 2 64 6 3 24 3

3 3 2 6 9 2 621 21

3 3 2 2 3 2 221 7

− ⋅ + ⋅ −=

−

− + ⋅ += =

⋅ − −⋅ + +

= =

+ += =

76. 2 2 2 2 27 7 7 2 14= = ⋅ =

77. 3 2 23 3

3 3 3 2 2

3 3 3

2 2 23 3

3 3 3

3

3 3

3 9

x x xy y x

x x

x y x y

= ⋅

= =

78. 7 7 75 3 5 3 7

735 21

= ⋅+ +

=+

79. 7 37 3+

−

( ) ( )

( )

2 2

7 3 7 37 3 7 3

7 3 7 37 2 7 3 3 10 2 21

4 410 2 21 2 5 21

25 21

+ −= ⋅

− −

− −= =

− + −

= =− −

=−

80.

( )2 2

2 4 6

2 4 6

2 4 362 32

16

x

x

xxx

+ =

+ =

+ ===


81. 5 9 4

5 5

x

x

− + =

− = −

The square root of a number is always nonnegative. The solution set is { } or .∅

82.

( ) ( )

( ) ( )

2 2

2

2

2

2 3 3

2 3 3

2 3 3

2 3 9 6

0 12 8

0 8 120 6 2

x x

x x

x x

x x x

x x

x xx x

− + =

− = −

− = −

− = − +

= − +

= − +

= − −

6 0 2 06 2

x xx x

− = − == =

6 is an extraneous solution. The solution set is { }2 .



441

83. 4 1 5x x− + + =

( ) ( )

( )

2 2

22

4 25 10 1 1

4 5 1

4 5 1

30 10 1

30 10 110 10

3 1

3 1

9 18

x x x

x x

x x

x

x

x

x

xx

− = − + + +

− = − +

− = − +

− = − +

− − +=

− −= +

= +

= +=


84. ( )1

2 36 2 0x x+ + =

( )

( )

( ) ( )

12 3

3 2

3 33 2

2

2

6 2

6 2

6 2

6 8

6 8 04 2 0

x x

x x

x x

x x

x xx x

+ = −

+ = −

⎛ ⎞+ = −⎜ ⎟⎝ ⎠

+ = −

+ + =

+ + =

4 0 2 04 2

x xx x

+ = + == − = −

Both solutions check. The solution set is { }4, 2 .− −

85. a. ( ) 0.23 0.43f x x= +

(6) 0.23 6 0.43 0.99f = + ≈ The average state cigarette tax per pack was approximately $0.99 6 years after 2001, or in 2007. This underestimates the tax displayed in the graph by $0.01.

b. ( ) 0.23 0.43f x x= +

( )22

1.12 0.23 0.43

0.69 0.23

0.69 0.230.23 0.23

3

3

9

x

x

x

x

x

x

= +

=

=

=

=

=

The average state cigarette tax will be $1.12 per pack 9 years after 2001, or 2010.

86.

( )22

20, 000 5000 100

20, 000 5000 1005000 5000

4 100

4 100

16 1008484

x

x

x

x

xx

x

= −

−=

= −

= −

= −− = −

=

20,000 people in the group will survive to 84 years old.

87. 81 81 1 81 1 9i− = ⋅− = − =

88. 63 9 7 1

9 7 1 3 7i

− = ⋅ ⋅−

= − =

89. 8 4 2 1

4 2 1 2 2i

− − = − ⋅ ⋅ −

= − − = −

90. ( ) ( )7 12 5 107 12 5 10 12 2

i ii i i

+ + −

= + + − = +

91. ( ) ( )8 3 17 7 8 3 17 79 4

i i i ii

− − − = − − +

= − +



442

92. ( )

( )

2

4 3 2 4 3 4 2

12 812 1 8

12 8

i i i i i

i ii

i

− = ⋅ − ⋅

= −

= − −

= − −

93. ( ) ( )( )

27 5 2 3 14 21 10 15

14 11 15 114 11 1529 11

i i i i i

iii

− + = + − −

= + − −

= + += +

94. ( ) ( )

( )

2 22

2

3 4 3 2 3 4 4

9 24 169 24 16 19 24 16

7 24

i i i

i iii

i

− = − ⋅ ⋅ +

= − +

= − + −

= − −= − −

95. ( ) ( ) ( )

( )

22

2

7 8 7 8 7 8

49 6449 64 149 64113 or 113 0

i i i

i

i

+ − = −

= −

= − −

= += +

96.

( )

2

8 3 4 2 1 3 1

2 2 3 2 6

2 6 1 2 6

2 6 or 2 6 0

i i i

i

− ⋅ − = ⋅ ⋅ − ⋅ ⋅ −

= ⋅ =

= − = −

= − − +

97.

( )

26 6 5 30 6

5 5 5 2530 6 30 6

25 125 130 6 30 6

26 26 2615 313 13

i ii i i i

i i

i i

i

− −= ⋅ =

+ + − −− −

= =+− −

−= = −

= −

98. 3 4 3 4 4 24 2 4 2 4 2

i i ii i i

+ + += ⋅

− − +

( )( )

2

212 6 16 8

16 412 22 8 1

16 4 112 22 8 4 22

16 4 204 22 1 1120 20 5 10

i i ii

i

i i

i i

+ + +=

−+ + −

=− −

+ − += =

+

= + = +

99.

( )( )

2

25 5 53 3 35 1 5 1

33 11 5 1 53 3 3 3

i i i i ii i i ii i

i i

+ + += ⋅ =

+ − −= =

−−

−= + = −− −

100. ( ) ( )8 816 2 1 1i i= = − =

101. ( ) ( ) ( )11 1123 22 2 1 1i i i i i i i i= ⋅ = = − = − = −

Chapter 7 Test

1. a. ( ) ( )14 8 2 14

8 28 36 6

f − = − −

= + = =

b. To find the domain, set the radicand greater than or equal to zero and solve the resulting inequality. 8 2 0

2 84

xxx

− ≥− ≥ −

≤

The domain of f is { }4x x ≤ or ( ], 4 .−∞

2. ( ) ( )

43

4 4 433

1 1 1 1278132727

−= = = =


Intermediate Algebra for College Students 5E Chapter 7 Test

443

3.

111 1 1 1282 4 2 4

1184

18

14

25 25

5

5

x y x y

x y

y

x

− −

−

⎛ ⎞⎜ ⎟ =⎜ ⎟⎝ ⎠

=

=

4. ( )1 11 48 4 4 8 28x x x x x⋅

= = = =

5. 1 1 1 5 41

54 5 4 5 20 204

920 920

x x x x x x

x x

+ +⋅ = ⋅ = =

= =

6. 2 275 25 3 5 3x x x= ⋅ =

7. ( )22 10 25 5

5

x x x

x

− + = −

= −

8. 4 8 3 6 23 3

3 6 23

2 23

16 8 2

8 2

2 2

x y x x y y

x y xy

xy xy

= ⋅ ⋅ ⋅ ⋅ ⋅

= ⋅

=

9.

( )5

5 510 5 22

32 2 2x xx

− = − = −

10. 3 2 2335 10 50x y x y⋅ =

11. 3 2 4 34 4 4

4 34

4 34

34

8 4 32

16 2

16 2

2 2

x y xy x y

x y

x y

x y

⋅ =

= ⋅ ⋅ ⋅

= ⋅

=

12. 3 18 4 32 3 9 2 4 16 2

3 3 2 4 4 2

9 2 16 2 7 2

− = ⋅ − ⋅

= ⋅ − ⋅

= − = −

13.

( )

3 34 6 3 63 3

23 3

2 3

8 8

2

2

x xy x x xy

x x y x

x y x

+ = ⋅ +

= +

= +

14. 3 8 8

3 4343 4

3 3 3

16 16 822

8 2

x x xxx

x x x x

= =

= ⋅ =

15. ( )3 4 6 5 3 4 6 3 5

4 18 15

4 9 2 15

4 3 2 15

12 2 15

− = ⋅ − ⋅

= −

= ⋅ −

= ⋅ −

= −

16. ( ) ( )5 6 2 2 6 2

5 6 5 12 2 12 2 2

30 3 12 4 26 3 4 3

26 3 2 3 26 6 3

− +

= ⋅ + − − ⋅

= + − = + ⋅

= + ⋅ = +

17. ( )27 3 49 2 7 3 3

52 14 3

− = − ⋅ ⋅ +

= −

18. 5 5 5x xx xx x= ⋅ =

19. 3 32 2

3 3 3 32 2 2 3 3

5 5 5 5 5

5 5 5 5

x x

x x x x= ⋅ =

3 35 25 255

x xx x

= =

20. 2 3 2 3 2 32 3 2 3 2 3

2 2 2 3 32 3

5 2 6 5 2 61

− − −= ⋅

+ + −

− +=

−−

= = − +−



444

21.

( ) ( )

( ) ( )

2 2

2

2

3 2 3

2 3 3

2 3 3

2 3 6 9

0 8 120 6 2

x x

x x

x x

x x x

x xx x

+ − =

− = −

− = −

− = − +

= − +

= − −

6 0 2 06 2

x xx x

− = − == =

2 is an extraneous solution. The solution set is { }6 .

22. 9 7 2x x+ − − =

( ) ( )

( )

2 2

22

9 4 2 2 7 7

9 2 7

9 2 7

9 4 7 312 4 7

3 7

3 79 7

16

x x x

x x

x x

x x xx

x

xxx

+ = + ⋅ ⋅ − + −

+ = + −

+ = + −

+ = − + −= −= −

= −

= −=


23. ( )

( )

( ) ( )

13

13

3

3 33

11 6 3 0

11 6 3

11 6 3

11 6 3

11 6 2711 33

3

x

x

x

x

xxx

+ + =

+ = −

+ = −

+ = −

+ = −= −= −

The solution set is { }3 .−

24.

( )22

40.4 2.9 20.1

20.3 2.9

7

7

49

x

x

x

x

x

= +

=

=

=

=

Boys who are 49 months of age have an average height of 40.4 inches.

25. 75 25 3 1

25 3 1 5 3i

− = ⋅ ⋅−

= ⋅ ⋅ − =

26. ( ) ( )5 3 6 9 5 3 6 95 6 3 9

1 6

i i i ii i

i

− − − = − − +

= − − += − +

27. ( ) ( )( )

23 4 2 5 6 15 8 20

6 7 20 16 7 2026 7

i i i i i

ii

i

− + = + − −

= + − −

= + += +

28.

( )2

9 4 9 1 4 1

9 1 4 1

3 2 6 6 16 or 6 0i i i

i

− ⋅ − = ⋅− ⋅ ⋅ −

= ⋅ − ⋅ ⋅ −

= ⋅ ⋅ ⋅ = = −

= − − +

29. 3 3 1 21 2 1 2 1 2

i i ii i i

+ + += ⋅

− − +

( )( )

2

23 6 2

1 43 7 2 1

1 4 13 7 2

1 41 7 1 7

5 5 5

i i ii

i

i

i i

+ + +=

−+ + −

=− −

+ −=

++

= = +

30. ( )( )( )

1735 34 2

171

1

i i i i i

i

i i

= ⋅ = ⋅

= − ⋅

= − = −


Intermediate Algebra for College Students 5E Cumulative Review

445

Cumulative Review (Chapters 1 – 7)

1. 2 5 2 3 6 2 1

x y zx y zx y z

− + = −− − =+ − =

Add the first and third equations to eliminate y.

2 5 2 1

3 4

x y zx y z

x z

− + = −+ − =

− = −

Multiply the third equation by 2 and add to the second equation.

2 3 62 2 4 2

3 7 8

x y zx y z

x z

− − =+ − =

− =

We now have a system of two equations in two variables.

3 43 7 8

x zx z− = −− =

Multiply the first equation by –1 and add to the second equation.

3 4 3 7 8 6 12 2

x zx z

zz

− + =− =− =

= −

Back-substitute –2 for z to find x.

3 43 2 4

3 62

x zx

xx

− + =− − =

− == −

Back-substitute –2 for x and z in one of the original equations to find y.

( )2 5

2 2 2 54 2 5

6 51

1

x y zyyy

yy

− + = −

− − − = −

− − − = −− − = −

− == −

The solution is ( )2, 1, 2− − − or the solution set is

( ){ }2, 1, 2 .− − −

2.

( ) ( )

2

2

3 11 4

3 11 4 03 1 4 0

x x

x xx x

− =

− − =

+ − =

Apply the zero product principle.

3 1 0 4 0

3 1 413

x xx x

x

+ = − == − =

= −

The solution set is 1 , 4 .3

⎧ ⎫−⎨ ⎬⎩ ⎭

3. ( ) ( )2 4 5 3 2

2 8 5 3 62 8 8 66 8 6

6 26 26 6

13

x x xx x xx xx

xx

x

+ < + +

+ < + ++ < +

− + <− < −− −

>− −

>

The solution set is 13

x x⎧ ⎫

>⎨ ⎬⎩ ⎭

or 1 , .3

⎛ ⎞∞⎜ ⎟⎝ ⎠



446

4.

( ) ( )

21 15 5

2 241 15 5

2 22 2

x xx

x xx x

+ =+ −−

+ =+ −+ −

So that denominators will not equal zero, x cannot equal 2 or –2. To eliminate fractions, multiply by the LCD, ( ) ( )2 2x x+ − .

( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

1 15 52 2 2 22 22 2

1 152 2 2 2 2 52 2 2

2 15 5 1013 5 10

4 13 104 3

34

x x x xx xx x

x x x x xx x x

x xx xx

x

x

⎛ ⎞ ⎛ ⎞+ − + = + −⎜ ⎟ ⎜ ⎟⎜ ⎟+ −+ − ⎝ ⎠⎝ ⎠⎛ ⎞⎛ ⎞+ − + + − = +⎜ ⎟⎜ ⎟ ⎜ ⎟+ + −⎝ ⎠ ⎝ ⎠

− + = ++ = +

− + =− = −

=


⎧ ⎫⎨ ⎬⎩ ⎭

5.

( ) ( )2 2

2 1 1

2 1 1

2 1 1

2 1 2 1 1

2 2 2 1

x x

x x

x x

x x x

x x x

+ − + =

+ = + +

+ = + +

+ = + + + +

+ = + + +

( )( )

220 2 1

0 4 10 4 44 41

x

xxx

x

= +

= +

= +− =− =

The solution checks. The solution set is { }1 .−



447

6. 2 22 4x y

y x+ <− >

First consider 2 2x y+ < . Replace the inequality symbol with an equal sign and we have 2 2x y+ = . Solve for y to put the equation in slope-intercept form.

2 22 2

1 12

x yy x

y x

+ == − +

= − +

slope = 12

− y–intercept = 1

Now, use the origin as a test point.

( )0 2 0 20 2

+ <

<

This is a true statement. This means that the point ( )0, 0 will fall in the shaded half-plane.

Next consider 2 4y x− > . Replace the inequality symbol with an equal sign and we have 2 4y x− = . Solve for y to put the equation in slope-intercept form.

2 4

2 41 22

y xy x

y x

− == +

= +

slope = 12

y–intercept = 2

Now, use the origin as a test point.

( )2 0 0 40 4

− >

>

This is a false statement. This means that the point ( )0, 0 will not fall in the shaded half-plane.

Next, graph each of the inequalities. The solution to the system is the intersection of the shaded half-planes.



448

7.

( )

2

2

2

2

2

2

8 40 lim23 12

8 2403 12

8 2403 4

x

xxx

x xx

x x

x

→∞÷

−−

−= ⋅

−

−= ⋅

−

8=

( )

12

3 2 2x

x x+ −( )2x −

⋅40

( ) ( )

52 2

3 5 2 15 2x xx x

= =⋅ + +

8.

( )( )

2

2

1 1 1

1 1 1

11

x x xy x xyy xy y y

xyy y xy y xyx x x

x xyx y x xyy xyxy y

+ + ⋅ + ⋅= ⋅ =

+ + ⋅ + ⋅

++= = =

++

9. ( ) ( )2

2 2

3 2 2

3 2

2 3 4 5 2

2 4 2 5 2 2 3 4 3 5 3 2

8 10 4 12 15 6

8 22 11 6

x x x

x x x x x xx

x x x x x

x x x

− − −

= ⋅ − ⋅ − ⋅ − ⋅+ ⋅ + ⋅

= − − − + +

= − + +

10. 2

7 252 15

xxx x

−−− −

( ) ( )

( ) ( )( )

( ) ( )( )

( ) ( ) ( ) ( )

( ) ( )

7 255 3

2 375 3 5 3

7 2 3 7 2 65 3 5 3

5 65 3

xxx x

xxx x x x

x x x xx x x x

xx x

= −−− +

+= −

− + − +

− + − −= =

− + − +

−=

− +

11. ( ) ( )( ) ( )

3

3

7 8 10 7 3 3

7 2 7 3 3

− − + ÷ −

= − − + ÷ −

( ) ( )( )

7 8 7 3 3

56 7 1 64

= − − + ÷ −

= − − + − = −



449

12. 80 5 20 2 45

16 5 5 4 5 2 9 5

x x x

x x x

− +

= ⋅ − ⋅ + ⋅

4 5 5 2 5 2 3 5

4 5 10 5 6 5 0

x x x

x x x

= − ⋅ + ⋅

= − + =

13. 3 2 3 2 2 3 52 3 5 2 3 5 2 3 5

− − −= ⋅

+ + −

2 3 5 3 4 3 104 3 25

6 9 3 1012 25

16 9 313

16 9 313

⋅ − − +=

⋅ −− +

=−

−=

−−

= −

14.

2

3 2

3 2

2

2

2 5 2 2 3 3 4

2 4

3

2 5 4 5 10 6

x xx x x x

x x

x x

x xxx

+ +− − + −

−

+

−−−

3 2

22 3 3 4 62 52 2

x x x x xx x

− + −= + + +

− −

15. ( ) ( )2 3 5 2 3 4 2

2 3 8 6 5 6 20 2

6 3 6 40 34 3 6

+ −

= ⋅ − + − ⋅

= − − = − −

16. ( )( ) ( )

2 224 10 4 2 12 5 2

2 3 2 4 1

x x x x

x x

+ − = + −

= + −

17. ( ) ( )( ) ( ) ( )

4 2 2

2

16 1 4 1 4 1

4 1 2 1 2 1

x x x

x x x

− = + −

= + + −



450

18. Since light varies inversely as the square of the distance, we have 2

.kld

=

Use the given values to find k.

2

2120

10

120100

12, 000

kldk

k

k

=

=

=

=

The equation becomes 2

12, 000 .ld

= When d = 15, 2

12, 000 53.3.15

l = ≈

At a distance of 15 feet, approximately 53 lumens are provided.

19. Let x = the amount invested at 7%. Let y = the amount invested at 9%.

60000.07 0.09 510

x yx y+ =+ =

Solve the first equation for y. 60006000

x yy x

+ == −

Substitute and solve. ( )0.07 0.09 6000 510

0.07 540 0.09 510540 0.02 510

0.02 301500

x xx x

xxx

+ − =

+ − =− =− = −

=

Back-substitute 1500 for x to find y. 60006000 1500 4500

y xy= −= − =

$1500 was invested at 7% and $4500 was invested at 9%.

20. Let x = the number of students enrolled last year. 0.12 23320.88 2332

2650

x xxx

− ===

2650 students were enrolled last year.


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