CHAPTER 1 Equations and Inequalitiescollege.cengage.com/mathematics/larson/college... · variable. For example, in Exercise 47, we multiplied by x(x − 3). If x or x − 3 are actually

C H A P T E R 1 Equations and Inequalities

Section 1.1 Linear Equations...................................................................................22

Section 1.2 Mathematical Modeling .......................................................................26

Section 1.3 Quadratic Equations .............................................................................35

Section 1.4 The Quadratic Formula ........................................................................39

Mid-Chapter Quiz Solutions ......................................................................................45

Section 1.5 Other Types of Equations.....................................................................47

Section 1.6 Linear Inequalities ................................................................................52

Section 1.7 Other Types of Inequalities..................................................................56

Review Exercises ..........................................................................................................62

Chapter Test Solutions ................................................................................................67

Cumulative Test Solutions ..........................................................................................69

Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.

22

C H A P T E R 1 Equations and Inequalities

Section 1.1 Linear Equations

1. The equation ( )2 1 2 2x x− = − is an identity because

(by the Distributive Property) it is true for every real value of x.

3. The equation ( )2 1 3 4x x− = + is conditional since

there are real number values of x for which the equation is not true.

5. The equation ( )2 1 2 1x x+ = + is conditional since

there are no real number values of x for which the equation is true.

7. 5 3 3 5x x− = +

(a) ( ) ( )?5 0 3 3 0 5

3 5

− = +

− ≠

x = 0 is not a solution.

(b) ( ) ( )?

5 5 3 3 5 528 10

− − = − +

− ≠ −

x = −5 is not a solution.

(c) ( ) ( )?5 4 3 3 4 5

17 17− = +

=

x = 4 is a solution.

(d) ( ) ( )?5 10 3 3 10 5

47 35− = +

≠


9. 2 23 2 5 2 2x x x+ − = −

(a) ( ) ( ) ( )?2 23 3 2 3 5 2 3 2

16 16− + − − = − −

=

x = −3 is a solution.

(b) ( ) ( ) ( )?2 23 1 2 1 5 2 1 2

0 0+ − = −

=


(c) ( ) ( ) ( )?2 23 4 2 4 5 2 4 2

51 30+ − = −

≠


(d) ( ) ( ) ( )?2 23 5 2 5 5 2 5 2

60 48

− + − − = − −

≠


Skills Review

1. ( ) ( )2 4 5 6 2 4 5 6

3 10

x x x x

x

− − + = − − −

= − −

2. ( ) ( )3 5 2 7 5 12x x x− + − = −

3. ( ) ( )2 1 2 2 2 2x x x x x+ − + = + − − =

4. ( ) ( )3 2 4 7 2 6 12 7 14

26

x x x x

x

− − + + = − + + +

= +

5. 5 3 83 5 15 15x x x x x++ = =

6. 4 34 4 4x x x xx −

− = =

7. ( )

( ) ( ) ( )11 1 1 1

1 1 1 1x x x x

x x x x x x x x− + − −

− = = = −+ + + +

8. 2 3 2 3 5x x x x

++ = =

9. ( )( ) ( )

( )

4 2 34 3 4 8 32 2 2

7 82

x x x xx x x x x x

xx x

− + − ++ = =

− − −

−=

−

10. ( )( )( ) ( )( )

2

1 11 1 1 11 1 1 1 1 1

21

x x x xx x x x x x

x

− − + − − −− = =

+ − + − + −

= −−


Section 1.1 Linear Equations 23

11. 5 4 32x x

− =

(a) ( ) ( )

?5 4 32 1 2 1 2

3 3

− =− −

=

12

x = − is a solution.

(b) ( )

?5 4 32 4 4

3 38

− =

− ≠


(c) ( )5 4

2 0 0− is undefined.


(d) ( )

?5 4 32 1 4 1 4

6 3

− =

− ≠

14

x = is not a solution.

13. ( )( )5 3 20x x+ − =

(a) ( )( ) ?3 5 3 3 20

0 20+ − =

≠


(b) ( )( ) ?2 5 2 3 20

15 20

− + − − =

− ≠


(c) ( )( ) ?0 5 0 3 20

15 20

+ − =

− ≠


(d) ( )( ) ?7 5 7 3 20

20 20

− + − − =

=

x = −7 is a solution.

15. 2 3 3x − =

(a) ( ) ?

?

2 6 3 3

9 33 3

− =

=

=


(b) ( )?

?

2 3 3 3

9 39 is undefined

− − =

− =

−


(c) ( )132 3− − is undefined.

13x = − is not a solution.

(d) ( )2 2 3− − is undefined.


17. 10 1510 10 15 10

5

xx

x

+ =

+ − = −

=

19. 7 2 157 2 7 15 7

2 84

xx

xx

− =

− − = −

− =

= −

21. 8 5 3 108 3 5 5 3 3 10 5

5 153

x xx x x x

xx

− = +

− − + = − + +

=

=

23. ( ) ( )2 5 7 3 2

2 10 7 3 62 3 3 6

99

x x

x xx x

xx

+ − = −

+ − = −

+ = −

− = −

=

25. ( )[ ]

6 2 3 8 56 3 8 5

6 18 8 526

26

x x xx xx x

xx

⎡ − + ⎤ = −⎣ ⎦− − = −

− − = −

− =

= −


24 Chapter 1 Equations and Inequalities

27.

( )

5 1 14 2 2

5 1 14 4 4 44 2 2

5 2 4 24

x x

x x

x xx

+ = −

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

+ = −

= −

29. ( ) ( )( )( ) ( )( ) ( )

( ) ( )

3 12 4

3 12 4

65

5 24 0

4 5 4 24 4 0

6 5 24 0

6 30 24 05 6

z z

z z

z z

z zz

z

+ − + =

+ − + =

+ − + =

+ − − =

= −

= −

31. ( )( ) ( )( ) ( )

( )

0.25 0.75 10 34 0.25 4 0.75 10 4 3

3 10 1230 3 12

2 189

x xx x

x xx x

xx

+ − =

+ − =

+ − =

+ − =

− = −

=

33. ( )8 2 28 2 48 48 4

x x xx x xx x

+ = − −

+ = − −

+ = −

≠ −

No solution

35.

( )

( ) ( )

100 4 5 6 63 4

100 4 5 612 12 12 63 4

4 100 4 3 5 6 72400 16 15 18 72

31 31010

u u

u u

u uu uuu

− += +

− +⎛ ⎞ ⎛ ⎞= +⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

− = + +

− = + +

− = −

=

37.

( ) ( )

5 4 25 4 3

3 5 4 2 5 415 12 10 8

5 204

xxx xx x

xx

−=

+− = +

− = +

=

=

39.

( ) ( )

13 510 4

13 510 4

10 13 4 56 18

3

x x

x x x xx x

x xxx

− = +

⎛ ⎞ ⎛ ⎞− = +⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠− = +

=

=

41.

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

2

2

1 1 103 3 9

1 1 103 3 3 3 3 33 3 9

3 3 102 10

5

x x x

x x x x x xx x x

x xxx

+ =− + −

− + + − + = − +− + −

+ + − =

=

=

43. ( )( )

( ) ( )

6 3 43 1 3 1

6 3 1 4 36 3 3 4 126 7 15

7 213

x x x x

x xx xx

xx

= +− − − −

= − + −

= − + −

= −

− = −=

A check reveals that x = 3 is an extraneous solution, so there is no solution.

45.

( ) ( ) ( )( )2 2

7 8 42 1 2 1

7 2 1 8 2 1 4 2 1 2 114 7 16 8 16 4

6 11116

xx x

x x x x xx x x x

x

x

− = −+ −

− − + = − + −

− − − = − +

=

=


Section 1.1 Linear Equations 25

47. ( )

( )

3 4 13 3

3 4 33 4 12

3 93

x x x x

x xx x

xx

+ =− −

+ − =

+ − =

==

A check reveals that x = 3 is an extraneous solution, so there is no solution.

49. ( ) ( )2 2

2 2

2 5 34 4 5 6 9

4 9 6 92 0

0

x xx x x x

x xxx

+ + = +

+ + + = + +

+ = +

− =

=

51. ( ) ( )2 2

2 2

2 4 14 4 4 4

4 4

x x xx x x x

+ − = +

+ + − = +

=

The equation is an identity; every real number is a solution.

53. ( ) ( )2 2

2 2

2 1 4 1

4 4 1 4 4 41 4

x x x

x x x x

+ = + +

+ + = + +

≠

No solution

55. When you check x = 2 in the original equation, you get division by zero, which is undefined. So, x = 2 is an extraneous solution and the equation has no solution.

57. Extraneous solutions may arise when a fractional expression is multiplied by factors involving the variable. For example, in Exercise 47, we multiplied by x(x − 3). If x or x − 3 are actually zero, an extraneous solution is introduced. In Exercise 47, an extraneous solution was x = 3 since multiplying the equation by (x − 3) actually multiplied the equation by zero.

59. Equivalent equations are equations that have the same solutions as the original equation. Usually, as part of the process of solving an equation, we rewrite the equation (using the properties and rules of algebra) into an equivalent form. For example, to solve Exercise 49, we would rewrite ( ) ( )2 22 5 3x x+ + = + as

2 4 4 5x x+ + + 2 6 9.x x= + + Next we might combine terms on the left so we would have equivalent equation 2 24 9 6 9.x x x x+ + = + + We would continue the solution process to the equation 2 0x = and finally 0.x =

61. ( )0.275 0.725 500 3000.275 362.5 0.725 300

0.45 62.562.50.45138.889

x xx x

x

x

+ − =

+ − =

− = −

=

≈

63.

( )( )

10000.6321 0.0692

0.0692 0.6321 1000 0.6321 0.06920.7013 43.74132

43.741320.7013

62.372

x x

x xx

x

+ =

+ =

=

=

≈

65.

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

2 4.405 17.398

2 4.405 7.398 7.3982 4.405 7.398 7.3982 5.405 7.398

5.405 7.3982

19.993

x xx

xx

x

x

− =

− =

= +

=

=

≈

67. Use the table feature in ASK mode or evaluate using the scientific calculator part of a graphing utility. (Answers will vary.)

69. 1 0.732051 0.73205+−

(a) 6.46

(b) 1.73 6.410.27

≈

The second method introduced an additional round-off error.

71.

1.983330.74

6.2543.15

+

+

(a) 56.09

(b) 335.68 56.135.98

≈

The second method introduced an additional round-off error.


26 Chapter 1 Equations and Inequalit ies

73. 944.7 19,898, 8 1532,000 944.7 19,89812,102 944.712,102944.7

13

y t ttt

t

t

= + ≤ ≤

= +

=

=

≈

The per capita personal income was $32,000 in 2003.

75. 15 0.432 10.4425.44 0.43225.440.432

58.89 inches

xx

x

x

= −

=

=

≈

77. 129.51 320.5, 8 152000 129.51 320.5

1679.5 129.51679.5129.5

13

y t tt

t

t

t

= + ≤ ≤

= +

=

=

≈

Credit extended to consumers was $2 trillion in 2003.

79. 0.112 5.83, 7 164.60 0.112 5.831.23 0.112

1.230.112

11

y t ttt

t

t

= − + ≤ ≤

= − +

− = −

−=

−≈

In 2001 the value of the federal minimum wage was $4.60 in 1996 dollars.

Section 1.2 Mathematical Modeling

Skills Review

1. 3 42 03 42

14

xxx

− =

=

=

2. 64 16 016 64

4

xxx

− =

− = −

=

3. 2 3 142 4 14

4 123

x xxxx

− = +

− =

− =

= −

4. 7 5 7 17 2 1

2 84

x xxxx

+ = −

− = −

− = −

=

5. ( ) ( )[ ]

( )

5 1 2 3 6 3 15 1 2 6 6 3 3

5 2 7 3 910 35 3 913 35 9

13 262

x xx xx xx xx

xx

⎡ + + ⎤ = − −⎣ ⎦+ + = − +

+ = − +

+ = − +

+ =

= −

= −

6. ( ) ( )[ ]( )

2 5 1 2 10 12 5 5 2 10 10

5 7 2 11 105 7 22 20

27 7 2027 27

1

x x xx x xx xx xx

xx

− − = ⎡ + − ⎤⎣ ⎦− + = + −

− + = −

− + = −

− + = −

− = −

=

7. 13 2 3

16 6 63 2 3

2 3 25 2

25

x x

y y

x xx

x

+ =

⎛ ⎞ ⎛ ⎞ ⎛ ⎞+ =⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠

+ =

=

=

8. 2 2 152 3

53 10

103

x

xx

x

+ =

=

=

=


Section 1.2 Mathematical Modeling 27

1. Model: (first number) + (second number) Labels: first number = n, second number = n + 1 Expression: ( )1 2 1n n n+ + = +

3. Model: (rate) × (time) Labels: rate = 50 mph, time = t Expression: 50t

5. Model: 20% × (amount of solution) Labels: amount of solution (in gallons) = x Expression: 0.2x

7. Model: 2(width) + 2(length) Labels: width = x, length = 2(width) = 2x Expression: ( )2 2 2 6x x x+ =

9. Model: (shipping fee) + (unit cost)(number of units) Labels: (shipping fee) = $1200, unit cost = $25,

number of units = x Expression: 1200 25x+

11. 5 + x = 8

13. 92r=

15. n + 2n = 15

17. Model: sum = (first number) + (second number) Labels: sum = 525, first number = n,

second number = n + 1

Equation: ( )525 1524 2

262

n nn

n

= + +

=

=

Answer: first number = n = 262, second number = n + 1 = 263

19. Model: difference = (one number) − (another number) Labels: difference = 148, one number = 5x,

another number = x Equation: 148 5

148 437

x xx

x

= −

=

=

Answer: one number = 5x = 185, another number = x = 37

21. Model: product = (first number) × (second number) Labels: product = (first number)2 − 5 = n2 − 5, first number = n, second number = n + 1

Equation: ( )2

2 2

5 15

5

n n nn n n

n

− = +

− = +

= −

Answer: first number = n = −5, second number = n + 1 = −4

23. Model: (Total of paychecks) = (coworker's paycheck) + (your paycheck) Labels: Total of paychecks = $848, coworker's paycheck = x,

your paycheck = (coworker's paycheck) + 12% × (coworker's paycheck) = x + 0.12x = 1.12x Equation: 848 1.12

848 2.12400

x xx

x

= +

=

=

Answer: coworker's paycheck = x = $400, your paycheck = 1.12x = $448

Skills Review —continued—

9.

( )( ) 2

2 2

213

22 3

2 366 0

6

zz z

z zz

z z zz z z

zz

− =+

−=

+− + =

+ − =

− =

=

10.

( )

1 41 2 3

111 6

6 11 16 11 115 1

115

xx

xx

x xx xx

x

− =+

=+

= +

= +

− =

= −



25. Model: (Total profit) = (January profit) + (February profit) Labels: Total profit = $129,000, January profit = x,

February profit = (January profit) + 5% × (January profit) = x + 0.05x = 1.05x Equation: 129,000 1.05

129,000 2.0562,926.83

x xx

x

= +

=

≈

Answer: January profit = x = $62,926.83, February profit = 1.05x = $66,073.17

27. Model: (1980 Star Wars) = (percent change)(1977 Star Wars) + (1977 Star Wars) Labels: 1980 Star Wars = $290,271,960; percent change = p, 1977 Star Wars = $460,998,007 Equation: 290,271,960 (460,998,007) 460,998,007

170,726,047460,998,007

0.370 37%

p

p

p

= +

−=

≈ − = −

Answer: The percent decrease in revenues was about 37%.

29. Models: (1999 Star Wars) = (percent change)(1983 Star Wars) + (1983 Star Wars) Labels: 1999 Star Wars = $431,088,295; percent change ,p= 1983 Star Wars = $309,209,079

Equation: 431,088,295 (309,209,079) 309,209,079121,879,216309,209,079

0.394 39.4%

p

p

p

= +

=

≈ =

Answer: The percent increase in revenues was about 39.4%.

31. Model: (2005 Star Wars) = (percent change)(2002 Star Wars) + (2002 Star Wars) Labels: 2005 Star Wars = $380,262,555; percent change ,p= 2002 Star Wars = $310,675,583

Equation: 380,262,555 (310,675,583) 310,675,58369,586,972310,675,583

0.224 22.4%

p

p

p

= +

=

≈ =

Answer: The percent increase in revenues was about 22.4%.

33. Model: (2006 size) = (percent change)(Past size) + (Past size) Labels: 2006 size = 16, percent change = p, Past size = 7 Equation: 16 (7) 7

97

1.286 128.6%

p

p

p

= +

≈ =

Answer: The percent increase in size was about 128.6%.

35. Model: (2006 size) = (percent change)(Past size) + (Past size) Labels: 2006 size = 6, percent change ,p= Past size = 3.5

Equation: 6 (3.5) 3.52.53.5

0.714 71.4%

p

p

p

= +

=

≈ =

Answer: The percent increase in size was about 71.4%.


29

37. Model: (Smaller lunch) = (percent change)(Larger lunch) + Labels: Smaller lunch = 660, percent change ,p= Larger lunch = 1440

Equation: 660 (1440) 1440780

14400.542 54.2%

p

p

p

= +

−=

≈ − = −

Answer: The percent decrease in calories is about 54.2%

39. Model: (Salary) = (percent increase)(Salary previous year) + (Salary previous year) Labels: 2

3

4

Salary second yearSalary third yearSalary fourth year

SSS

=

=

=

(a) S2 = 0.08(35,000) + 35,000 = 37,800 Your salary for the second year is $37,800. (b) S3 = 0.078(37,800) + 37,800 = 40,748.40 Your salary for the third year is $40,748.40. (c) S4 = 0.094(40,748.40) + 40,748.40 ≈ 44,578.75 Your salary for the fourth year is $44,578.75.

41. Model: (Number of world Internet users) = (percent change)(Number of users in previous year) + (Number of users in previous year)

Labels: 1

2

3

number of world Internet users in 2003number of world Internet users in 2004number of world Internet users in 2006

NNN

=

=

=

(a) 1

1

(0.438)(500) 500719

NN

= +

=

The number of world internet users in 2003 was about 719 million. (b) 2

2

(0.136)(719) 719816.784

NN

= +

=

The number of world Internet users in 2004 was about 816.8 million. (c) 3

3

(0.338)(816.8) 816.81092.8784

NN

= +

=

The number of world Internet users in 2006 was about 1092.9 million. (d) 1092.9 (500) 500

592.9500

1.1858 118.6%

p

p

p

= +

=

≈ ≈

The percent increase in the number of users from 2001 to 2006 was about 118.6%.


Section 1.2 Mathematical Modeling


43. Model: (Media type usages) = (percent)(3555) Labels: 1

2

3

4

5

6

Hours spent watching TVHours spent listening to radio or recorded musicHours spent using the InternetHours spent playing non-Internet video gamesHours spent reading print mediaHours

HHHHHH

=

=

=

=

=

= spent using other media

H1 = (0.44)(3555) = 1564.2 In 2009, the average person will spend 1564.2 hours watching TV. H2 = (0.32)(3555) = 1137.6 In 2009, the average person will spend 1137.6 hours listening to the radio or recorded music. H3 = (0.06)(3555) = 213.3 In 2009, the average person will spend 213.3 hours using the Internet H4 = (0.03)(3555) = 106.65 In 2009, the average person will spend 106.65 hours playing non-Internet video games. H5 = (0.11)(3555) = 391.05 In 2009, the average person will spend 391.03 hours reading print media. H6 = (0.04)(3555) = 142.2 In 2009, the average person will spend 142.2 hours using other media.

45. Model: perimeter = 2(width) + 2(length) Labels: perimeter = x= , length (in feet) = 1.5(width) 1.5x=

Equation: 75 2 2(1.5 )75 5

15

x xx

x

= +

=

=

Answer: width 15x= = feet, length 1.5 22.5x= = feet

47. Model: Interest = (principal)(rate)(time) Labels: Interest = $1000, principal = $2500, rate = 0.07, time x=

Equation:

57

1000 (2500)(0.07)1000 175

5 5.714

xx

=

=

= ≈

Answer: About 5.7 years

49. Model: (test #1) (test #2) (test #3) (test #4)average4

+ + +=

Labels: average = 90, test #1 = 87, test #2 = 92, test #3 = 84, test #4 = x

Equation: 87 92 84904

360 26397

x

xx

+ + +=

= +

=

Answer: You must score 97 or more on test 4 to have an average of at least 90%.



51. Model: (Sale Price) = (list price) (discount percent)(list price) Labels: x = list price, 1200 = sale price, 15% = discount Equation:1200 (0.15)

1200 0.851200 $1411.760.85

x xx

x

= −

=

= ≈

Answer: The list price was $ 1411.76.

53. Model: percent discount discount amount100 original price

=

Labels: discount amount $30original price 119 + 30 $149

percent discountp

=

= =

=

Equation: 30100 149

3000 20.1342149

p

p

=

= ≈

Answer: The satellite radio system was discounted 20.13%.

55. Model: (Sale price) = (List price) − (discount percent)(List price) Labels: Sale price = 21.60, List price = (Whole price)(0.60) + Wholesale price, discount percent = 0.25,

wholesale price = w

Equation: ( )( )

21.60 (0.60) 0.25 0.6021.60 1.60 0.25 1.6021.60 1.221.601.2

18

w w w ww w

w

w

w

= + − ⎡ + ⎤⎣ ⎦= −

=

=

=

Answer: The whole sale price of the power drill is $18.

57. (Reduced salary) = (Original weekly salary) − (discount percent)(original weekly salary)

425 0.15(425)$361.25

xx= −

=

The reduced salary is $361.25.

59. ( )

( )( )

(distance)(time)rate

30 miles 30 miles(rate)28 minutes 7 15 hour

150 12 hours330 7 15

t

=

= =

= =⎡ ⎤⎣ ⎦

The entire trip will take about 123

hours.



61.

1

2

distance rate time40 mph55 mph

d td t

= ×

= ×

= ×

(distance between cars) = (second distance) − (first distance)

2 1

13

55 55 40 15

hour

d dt t t

t

= −

= − =

=

The cars will be 5 miles apart after 13 hour, or 20 minutes.

63.

8

8

distancetimerate

3.84 10 meters3.0 10 meters per second

= 1.28 seconds

t

t

=

×=

×

It will take 1.28 seconds for a radio wave to travel from Houston to the surface of the moon.

65. height of tree height of lamp posttree's shadow lamp post's shadow

=

525 2

525 62.5 feet tall2

x

x

=

⎛ ⎞= =⎜ ⎟⎝ ⎠

The tree is 62.5 feet tall.

67. ( )Total expenses 12 Monthly expenses

325,45012 $781,0805

T

=

⎛ ⎞= =⎜ ⎟⎝ ⎠

If the monthly expense rate continues, the total expenses for the year will be $781,080.

69. Model: (Interest from 6.5%) + (Interest from 7.5%) = (Total interest) Labels: Amount invested at 6.5% = x, Amount invested at 7.5% = 15,000 − x,

Interest from 6.5% = 0.065x, Interest from 7.5% = 0.075(15,000 x− ), Total interest = 1020

Equation: ( )0.065 0.075 15,000 10200.065 1125 0.075 1020

0.01 10510,500

x xxxx

+ − =

+ − =

− = −

=

Answer: The amount invested at 6.5% is $10,500 and the amount invested at 7.5% is 15,000 15,000 10,500 $4500x− = − =

71. Model: (Amount earned by stock A) + (Amount earned by stock B) = (Total amount earned) Labels: Amount invested in stock A = ,x Amount invested in stock B = 5000 ,x− Amount earned by stock A = 0.098 ,x

Amount earned by stock B = 0.062 ( )5000 ,x− Total amount earned = 389.20

Equation: ( )0.098 0.062 5000 389.200.098 310 0.062 389.20

0.036 79.202200

x xxxx

+ − =

+ − =

=

=

Answer: The amount invested in stock A is $2200 and the amount invested in stock B is 5000 5000 2200 $2800.x− = − =



73. ( )1

2

interest = interest rate principal9.5% $12,000

$8000ii r

×

= ×

= ×

( ) ( )

( )1 2

total interest = interest in first account interest in second account$2054.402054.40 0.095 12,000 8000

914.40 0.1143 11.43%8000

i ir

r

+

= +

= +

= = =

An interest rate of 11.43% yields the same interest amount as the variable rate fund.

75. ( ) ( )( )cost fixed costs variable cost number of units

$90,000 $15,000 $8.7575,000 85718.75

x

x

= +

= +

= ≈

The company can manufacture about 8571 units.

77.

( )

2

2

Volumediameter 2 4 2

603.2 2

603.2 48 feet long4

r hr r

h

h

π

π

π

=

= = ⇒ =

=

= ≈

79. ( )( ) ( )( ) ( )( )( )( ) ( )( ) ( )

final concentration amount sol. 1 concentration amount sol. 2 concentration amount

75% 55gal 40% 55 100%

41.25 0.60 22 32.1 gal

x x

xx

= +

= − +

= +

≈

81. distancerate = time

38526 miles17607.752 hours60

12.31 miles per hour

⎛ ⎞+⎜ ⎟⎝ ⎠=⎛ ⎞+⎜ ⎟⎝ ⎠

≈

83. 12

22

A bh

A bhA h

b

=

=

=

85. V lwhV lwh

=

=

87. 2

2

V r hV hr

π

π

=

=

89.

( )1

1

S C RCS C R

S CR

= +

= +

=+

91. A P PrtA P PrtA P r

Pt

= +

− =

−=

93. ( )12

2

2

A a b h

A a bh

A ah bh

= +

= +

−=

95. ( )1L a n dL a nd d

nd L d aL d an

d

= + −

= + −

= + −

+ −=



97. 2

2

A rhA h

r

π

π

=

=

99.

( )

( )

( )( )

( )( )

( )

1 2

1 2

2 1

2

2 1

21

2

1 1 11

1 1 11

1 1 11

1 11

11

nf R R

n f R R

n f R R

R n fn f R R

n f RR

R n f

⎛ ⎞= − −⎜ ⎟

⎝ ⎠

= −−

+ =−

+ −=

−

−=

+ −

11

1 is the reciprocal of .RR

⎛ ⎞⎜ ⎟⎝ ⎠

101. 15,000 18,800 22,300 56,100Williams' average $18,7003 3

20,900 17,500 25,600 64,0003 3

18,600 25,000 16,400 60,000Walters' average $20,0003 3

18,100 18,700 23,Gilbert's average

+ += = =

+ += = ≈

+ += = =

+ +=

000 59,800 $19,9333 3

13,000 20,500 20,000 53,500Hart's average $17,8333 3

= ≈

+ += = ≈

15,000 20,900 18,600 18,100 13,000 85,600January's average $17,1205 5

18,800 17,500 25,000 18,700 20,500 100,500February's average $20,1005 5

22,300 25,600 16,400 23,000 20,000 107,300March's average5 5

+ + + += = =

+ + + += = =

+ + + += = $21,460=

103.

Williams' average $25,033Gonzalez's average $22,867Walters' average = $25,400Gilbert's average $27,467Hart's average = $28,100Reges' average $24,967Sanders' average $13,633July's average $24,514August's a

≈

≈

≈

≈

≈

≈

verage $25,157September's average = $22,100

≈

105. The time the diver takes to descend from 25 feet to 150 feet and the depth of 150 feet are red herrings.


Section 1.3 Quadratic Equations 35

Section 1.3 Quadratic Equations

1. 2

2

2 3 52 5 3 0

x xx x

= −

+ − =

3. 2

2

2525 0

x xx x

=

− =

5. ( )2

2

2

3 26 9 26 7 0

xx xx x

− =

− + =

− + =

7. ( ) 2

2 2

2

2

2 3 12 3 1

2 2 1 02 2 1 0

x x xx x x

x xx x

+ = +

+ = +

− + − =

− + =

9. 2

2

2

3 10 125

3 10 603 60 10 0

x x

x xx x

−=

− =

− − =

11.

( )( )

2 2 8 0

4 2 0

4 0 or 2 04 or 2

x x

x x

x xx x

− − =

− + =

− = + =

= = −

13.

( )

2

12

6 3 03 2 1 0

3 0 or 2 1 00 or

x xx x

x xx x

+ =

+ =

= + =

= = −

15.

( )

2

2

10 25 0

5 05 0

5

x x

xx

x

+ + =

+ =

+ =

= −

17.

( )( )

2

12

3 5 2 03 1 2 0

3 0 or 1 2 03 or

x xx x

x xx x

+ − =

− + =

− = + =

= = −

19.

( )( )

2

2

4 124 12 0

6 2 0

6 0 or 2 06 or 2

x xx xx x

x xx x

+ =

+ − =

+ − =

+ = − =

= − =

21.

( )( )

2

2

7 100 7 100 5 2

5 0 or 2 05 or 2

x xx xx x

x xx x

− − =

= + +

= + +

+ = + =

= − = −

23. 2 164

xx=

= ±

25. 2 77

2.65

xx=

= ±

≈ ±

Skills Review

1. 7 7 50 7 5050 5050 50

7 25 2 5 7 2 1450 50 10

⋅= ⋅ =

⋅ ⋅ ⋅= = =

2. 32 16 2 4 2= ⋅ =

3. 2 27 3 7 49 147 196 14+ ⋅ = + = =

4. ( )1 3 5 5 8 5 84 8 8 88 8

5 4 2 2 5 2 108 8 4

⋅+ = = ⋅ =

⋅ ⋅ ⋅= = =

5. ( )23 7 3 7x x x x+ = +

6. ( )( )24 25 2 5 2 5x x x− = + −

7. ( ) ( ) ( )( )( )( )( )

216 11 4 11 4 11

7 15

7 15

x x x

x x

x x

− − = ⎡ + − ⎤⎡ − − ⎤⎣ ⎦⎣ ⎦= − − +

= − − −

8. ( )( )2 7 18 9 2x x x x+ − = + −

9. ( )( )210 13 3 5 1 2 3x x x x+ − = − +

10. ( )( )26 73 12 6 1 12x x x x− + = − −



27. 2

2

3 3612

2 33.46

xxx

=

=

= ±

≈ ±

29. ( )212 1812 3 2

12 3 216.24 or7.76

xx

xxx

− =

− = ±

= ±

≈

≈

31. ( )22 122 2 3

2 2 31.46 or

5.46

xx

xxx

+ =

+ = ±

= − ±

≈

≈ −

33. 2

2

12 30025

5

xxx

=

=

= ±

35. 2

2

5 19038

386.16

xxxx

=

=

= ±

≈ ±

37.

( )2 2

2 2

2

2

3 2 4 15

3 2 8 155 23

235

235

1155

2.14

x x

x xx

x

x

x

x

+ − =

+ − =

=

=

= ±

= ±

≈ ±

39.

( )2 2

2 2

2

2

6 3 1 23

6 3 3 233 26

263

263

783

2.94

x x

x xx

x

x

x

x

− + =

− − =

=

=

= ±

= ±

≈ ±

41. 2 648

xx=

= ±

43.

( )

2

2

2 1 0

1 01 0

1

x x

xx

x

− + =

− =

− =

=

45.

( )( )

2

3 34 4

16 9 0

4 3 4 3 0

4 3 0 or 4 3 0

or

x

x x

x x

x x

− =

+ − =

+ = − =

= − =

47.

( )

2

2

32

4 12 9 0

2 3 02 3 0

x x

xx

x

− + =

− =

− =

=

49. ( )24 49

4 74 7 or 4 7

3 or 11

x

xx x

x x

+ =

+ = ±

+ = + = −

= = −

51.

( )( )

2

2

312 2

4 4 34 4 3 0

2 1 2 3 0

2 1 0 or 2 3 0

or

x xx x

x x

x x

x x

= −

− − =

+ − =

+ = − =

= − =

53.

( )( )

2

2

103

50 5 33 5 50 0

3 10 5 0

3 10 0 or 5 0

or 5

x xx x

x x

x x

x x

+ =

− − =

+ − =

+ = − =

= − =

55.

( )( )

2

2

12 270 12 27

0 9 3

9 0 or 3 09 or 3

x xx x

x x

x xx x

= +

= − +

= − −

− = − =

= =

57.

( )( )( )

2

2

15

50 60 10 0

10 5 6 1 0

10 5 1 1 0

5 1 0 or 1 0

or 1

x x

x x

x x

x x

x x

− + =

− + =

− − =

− = − =

= =


37

59. ( )( )

2

2

3 4 0

3 43 2

3 23 2 or 3 21 or 5

x

xx

xx xx x

+ − =

+ =

+ = ±

= − ±

= − + = − −

= − = −

61. ( )( )

2 2

2 2

2 2

12

1

1 02 1 0

2 1 0

x x

x xx x x

xx

+ =

+ − =

+ + − =

+ =

= −

63. Answers will vary. Sample answer: Algebra Argument:

( ) ( )( )2

2

2

2 2 2 Definition of an exponent2 2 4 F.O.I.L.4 4 Simplify

x x xx x xx x

+ = + +

= + + +

= + +

So, ( )2 22 4 for any real .x x x+ ≠ +

Graphing utility argument: Let ( )2 2

1 22 and 4.y x y x= + = + Use the table feature with a value of x other than zero.

The table will show y1 is not the same as y2. OR Use a scientific calculator to show that if x = 5 then ( )2 25 2 49 and 5 4 29+ = + = so ( )22x +

is not the same as x2 + 4.

65. ( )( )( )( )

( )( )

2

Area length width

1632 14

0 14 1632

0 48 34

48 0 or 34 048 or 34

w w

w w

w w

w ww w

=

= +

= + −

= + −

+ = − =

= − =

Extraneous Solution: length 14 48 feet,

width 34 feetww

= + =

= =

The building has a length of 48 feet and a width of 34 feet.

67.

( )( )

( )

12

12

2

Area (base)(height)4 = 88

2 2 2 2 is extraneous.

b bbb

b

=

=

± =

= −

Solution: base height

2 2 2.83 feet

=

= ≈

The sign has a base length of about 2.83 feet and a height of about 2.83 feet.

69. ( )( )( )( )

( )( )( )

2

2

2

Area = length width1200 = 2 30 2 201200 4 100 600

0 4 100 600

0 4 25 150

0 4 30 5

30 0 or 5 030 or 5

x xx xx x

x x

x x

x xx x

+ +

= + +

= + −

= + −

= + −

+ = − =

= − =

Extraneous Solution: width = x = 5 feet The width of the path is 5 feet.


Section 1.3 Quadratic Equations


71. 20 0

0 0

160 and 200

0 when the rock hits the ocean.

s t v t sv ss

= − + +

= =

=

2

2

2

0 16 20016 200

252

52

5 53.54 is extraneous2 2

tt

t

t

t

= − +=

=

= ±

⎛ ⎞= ≈ −⎜ ⎟⎝ ⎠

The rock hits the ocean in about 3.54 seconds.

73. 20 0

0 0

160 and 10 meters 32.808 feet

0 when the diver hits the water.

s t v t sv ss

= − + +

= = ≈

=

( )

2

2

2

0 16 32.80816 32.808

2.0505

2.0505 2.0505 is extraneous

1.43

ttt

t

t

= − +==

= ± −

≈

The diver will be in the air about 1.43 seconds.

75.

( )

0 0

0 02

2

2

163000, 0, and 13,000

3000 16 13,00016 10,000

625625 25 is extraneous

25

s t v t ss v s

tttt tt

= − + += = =

= − +

=

=

= ± = −

=

The ball reaches a height of 3000 feet 67 − 25 = 42 seconds faster than the skydiver.

77. 2 2 2

2 2 2

2

2

62 36

184.24

a b cx x

xxx

+ =

+ =

=

=

≈

The sides of the isosceles right triangle are about 4.24 centimeters in length.

79. 2 2 2

2 2 2

2

2

583 703339,889 494,209

154,320393

a b ca

aaa

+ =

+ =

+ =

=

≈

The flying distance from Atlanta to Buffalo by way of Chicago = 976a b+ = miles.

81. Since the angle is 45°, the triangle is isosceles.

2 2 2

2

2

7002 490,000

245,000494.97

x xxxx

+ =

=

=

≈

The whale shark is about 494.97 meters deep.

83. ( )

( )

2

2

2

2

36 0.0003

1,080,000 36 0.0003

0.0003 36 1,080,000 0

120,000 3,600,000,000 0

60,000 060,000

R xp x x

x x

x x

x x

xx

= = −

= −

− + =

− + =

− =

=

To produce a revenue of $1,080,000, a total of 60,000 units must be sold.

85. 235.65 7205, 0C t t= + ≥

(t = 0 corresponds to 2000.)

2

2

2

12,000 35.65 72054795 35.65

134.5011.60

tt

tt

= +

=

≈

≈

The average monthly cost will reach $12,000 in 2012.

87. 2694.59 6179, 0 9P t t= + ≤ ≤ (t = 0 corresponds to 1800, t = 1 corresponds to 1810, etc.)

(a) 2

2

2

250,000 694.59 6179243,821 694.59351.0318.74

tt

tt

= +

=

≈

≈

The resident population would have reached 250,000,000 in 1987.

( )( )18.74 represents 1800 18.74 10 1987.4.t = + =

(b) Because the lengths of the bars that represent the actual data are close to the lengths of the bars that represent the data values given by the model, the model is a good representation of the resident population through 1890.

(c) When 20.6:t =

( )2694.59 20.6 6179P = +

300,935 300,935,000≈ ≈ people. Because the population given by the model is close to the actual population in 2006, the model is a good representation of the resident population through 2006.

x

x

6 cm

ac = 703 mi.

b = 583 mi.

B

A C

x

45°

45°

x

700 m


Section 1.4 The Quadratic Formula 39

89. For this model t = 15 corresponds to 2050.

( )21951.00 15 97,551 536,526+ =

The model is not a good predictor for the population in 2050, since 536,526,000 is larger than 419,854,000.

91. 20.31 32.9, 7 15T t t= + ≤ ≤

(t = 7 corresponds to 7 A.M.)

2

2

2

85 0.31 32.952.1 0.31

168.0612.96

tt

tt

= +

=

≈

≈

The temperature was 85 F° at about 1 P.M.

When t = 19 (7 P.M.):

( )20.31 19 32.9 144.81 FT = + = °

Because the temperature is extremely high, the answer is unreasonable.

So, the model should not be used to predict the temperature at 7 P.M.

93. 20.74 25, 0 5H t t= + ≤ ≤

(t = 0 corresponds to 2000.)

2

2

2

32 0.74 257 0.74

9.463.08

tt

tt

= +

=

≈

≈

The height of the tree was about 32 inches in 2003.

Section 1.4 The Quadratic Formula

Skills Review

1. ( )( ) ( )9 4 3 12 9 144 3 17− − = − − =

2. ( )( )36 4 2 3 36 24 12 2 3− = − = =

3. ( )( )212 4 3 4 144 48 96 4 6− = − = =

4. ( )( )215 4 9 12 225 432 657 3 73− = − = =

5.

( )( )

2 2 02 1 0

2 0 or 1 02 or 1

x xx x

x xx x

− − =

− + =

− = + =

= = −

6.

( )( )

2

32

2 3 9 02 3 3 0

2 3 0 or 3 0or 3

x xx x

x xx x

+ − =

− + =

− = + =

= = −

7.

( )( )

2

2

4 54 5 0

5 1 0

5 0 or 1 0 5 or 1

x xx x

x x

x xx x

− =

− − =

− + =

− = + =

= = −

8.

( )( )

2

2

12

2 13 72 13 7 02 1 7 0

2 1 0 or 7 0

or 7

x xx xx x

x x

x x

+ =

+ − =

− + =

− = + =

= = −

9.

( )( )

2

2

5 65 6 0

2 3 0

2 0 or 3 0 2 or 3

x xx x

x x

x xx x

= −

− + =

− − =

− = − =

= =

10. ( )

( )( )

2

2

3 43 4

3 4 01 4 0

1 0 or 4 01 or 4

x xx x

x xx x

x xx x

− =

− =

− − =

+ − =

+ = − =

= − =



1. 24 4 1 0x x− + =

( ) ( )( )22 4 4 4 4 1 0b ac− = − − =

One real solution

3. 23 4 1 0x x+ + =

( ) ( )( )22 4 4 4 3 1 4 0b ac− = − = >

Two real solutions

5. 2

2

2 5 5

2 5 5 0

x x

x x

− = −

− + =

( ) ( )( )22 4 5 4 2 5 15 0b ac− = − − = − <

No real solutions

7. 2 615 5 8 0x x+ − =

( ) ( )( )22 6 19615 5 254 4 8 0b ac− = − − = >

Two real solutions

9. 22 1 0x x+ − =

( )( )

( )

2

2

42

1 1 4 2 12 2

1 3 1, 14 2

b b acxa

− ± −=

− ± − −=

− ±= = −

11. 216 8 3 0x x+ − =

( )( )( )

2

2

42

8 8 4 16 32 16

8 16 1 3, 32 4 4

b b acxa

− ± −=

− ± − −=

− ±= = −

13. 2

2

2 2 0

2 2 0

x x

x x

+ − =

− + + =

( )( )

( )

2

2

42

2 2 4 1 22 1

2 2 3 1 32

b b acxa

− ± −=

− ± − −=

−

− ±= = ±

−

15. 2 14 44 0x x+ + =

( )( )

( )

2

2

42

14 14 4 1 442 1

14 2 5 7 52

b b acxa

− ± −=

− ± −=

− ±= = − ±

17. 2 8 4 0x x+ − =

( )( )

( )

2

2

42

8 8 4 1 42 1

8 4 5 4 2 52

b b acxa

− ± −=

− ± − −=

− ±= = − ±

19. 2

2

12 9 39 12 3 0

x xx x

− = −

− + + =

( )( )

( )

2

2

42

12 12 4 9 32 9

12 6 7 2 718 3 3

b b acxa

− ± −=

− ± − −=

−

− ±= = ±

−

21. 2

2

36 24 7

36 24 7 0

x x

x x

+ =

+ − =

( )( )

( )

2

2

42

24 24 4 36 72 36

24 12 11 1 1172 3 6

b b acxa

− ± −=

− ± − −=

− ±= = − ±

23. 2

2

4 4 74 4 7 0

x xx x

+ =

+ − =

( )( )

( )

2

2

42

4 4 4 4 72 4

4 8 2 1 28 2

b b acxa

− ± −=

− ± − −=

− ±= = − ±



25. 2

2

28 49 449 28 4 0

x xx x

− =

− + − =

( )( )

( )

2

2

42

28 28 4 49 42 49

28 0 298 7

b b acxa

− ± −=

− ± − − −=

−

− ±= =

−

27. 2

2

8 5 22 8 5 0

t tt t

= +

− + − =

( )( )( )

2

2

42

8 8 4 2 52 2

8 2 6 624 2

b b acta

− ± −=

− ± − − −=

−

− ±= = ±

−

29. ( )2

2

5 212 25 0

y yy y

− =

− + =

( ) ( )( )( )

2

2

42

12 12 4 1 252 1

12 2 11 6 112

b b acya

− ± −=

± − −=

±= = ±

31. 25.1 1.7 3.2 0x x− − =

( ) ( )( )

( )

21.7 1.7 4 5.1 3.22 5.1

0.976, 0.643

x

x

± − − −=

≈ −

33. 27.06 4.85 0.50 0x x− + =

( ) ( )( )

( )

24.85 4.85 4 7.06 0.502 7.06

0.561, 0.126

x

x

± − −=

≈

35. 20.003 0.025 0.98 0x x− + − =

( )( )

( )

20.025 0.025 4 0.003 0.982 0.003

x− ± − − −

=−

No real solution

37. 2 22 7 2 47 4

11

x x xx

x

+ = − −

= − −

= −

39. 2

2

2

4 15 254 40

1010

xxxx

− =

=

=

= ±

41. 2 3 1 0x x+ + =

( )( )

( )

23 3 4 1 1 3 52 1 2

x− ± − − ±

= =

43. ( )22 1 9

1 31 31 3 4

OR1 3 2

x

xxx

x

− =

− = ±

= ±

= + =

= − = −

45. ( )2

2

2

3 5 11 4 23 5 11 4 8

3 3 0

x x xx x x

x x

+ − = −

+ − = −

+ − =

( )( )( )

21 1 4 3 32 3

1 37 1 376 6 6

x− ± − −

=

− ±= = − ±

47.

( )

( )

2

2

Let one integer100 the other integer

100 25000 100 2500

0 5050

100 50

xx

x xx x

xxx

=− =− =

= − +

= −=

− =

Verbal models will vary.

49.

( )

( )( )

22

2

2

Let an integer1 next integer

1 1132 2 112 0

56 07 8 0

7 OR 81 8 7

xx

x xx x

x xx x

x xx x

=

+ =

+ + =

+ − =

+ − =

− + =

= = −

+ = = −

Verbal models will vary.



51. 2

2

14,000 0.125 20 5000

0.125 20 9000 0

C x x

x x

= = + +

+ − =

( )( )( )

220 20 4 0.125 90002 0.125

20 700.25

x− ± − −

=

− ±=

Choosing the positive value of x, we have 200x = units.

53. 2

2

1680 800 0.04 0.002

0.0002 0.04 880 0

C x x

x x

= = + +

+ − =

( ) ( )( )( )

20.04 0.04 4 0.002 8802 0.002

0.04 7.04160.004

x− ± − −

=

− ±=

Choosing the positive value for x, we have 0.04 7.0416 653

0.004x − += ≈ units.

55. Number of rows72Number of seats/row

x

x

=

=

( )

( )( )2

3 2 72

72 3 2 723 6 144 0

xx

x x xx x

⎛ ⎞+ − =⎜ ⎟⎝ ⎠

+ − =

− − =

( ) ( )( )

( )

26 6 4 3 1442 3

6 42 8, 66

72 72 9 seats/row8

x

x

± − − −=

±= = −

= =

The original number of seats in each row was nine.

57. 2

2

Volume 2 200 100 10

xx

x

= ===

The original piece of material was x + 4 = 14 inches by 14 inches.

59. (a) 20 016s t v t s= − + +

The initial velocity is 20 miles per hour, or 883 feet per second. So, 88

0 3 ,v = and the initial height

is 2 880 3984. 16 984.s s t t= = − + +

(b) When 4:t = ( ) ( )2 88316 4 4 984 845.3 feets = − + + ≈

(c)

( )( )

( )

2

2

880 16 9843

88 88 884 16 984 63,836.43 3 3 8.81 or 6.982 16 32

t t

t

= − + +

⎛ ⎞− ± − −⎜ ⎟ − ±⎝ ⎠= ≈ ≈ −

− −

It will take the coin about 8.81 seconds to strike the ground.

61.

( )( )( )

20 0

0 0

2

2

Moon: 2.7Let 40, 5 and set 0.

0 2.7 40 5

40 40 4 2.7 52 2.7

40 16545.4

14.9 seconds

s t v t sv s s

t t

t

t

t

= − + +

= = =

= − + +

− ± − −=

−

− −=

−

≈

( )( )( )

20 0

0 0

2

2

Earth: 16Let 40, 5 and set 0

0 16 40 5

40 40 4 16 52 16

40 192032

2.6 seconds

s t v t sv s s

t t

t

t

t

= − + +

= = =

= − + +

− ± − −=

−

− −=

−

≈

On the moon, it would take about 14.9 seconds to hit the surface while on Earth it would take only about 2.6 seconds.



63.

( )( )( )

20 0

0 0

2

2

Moon: 2.7Let 27, 6 and set 0.

0 2.7 27 6

27 27 4 2.7 62 2.7

27 793.85.4

10.2 seconds

s t v t sv s s

t t

t

t

t

= − + +

= = =

= − + +

− ± − −=

−

− −=

−≈

( )( )( )

20 0

0 0

2

2

Earth: 16Let 27, 6 and set 0.

0 16 27 6

27 27 4 16 62 16

27 111332

1.9 seconds

s t v t sv s s

t t

t

t

t

= − + +

= = =

= − + +

− ± − −=

−

− −=

−≈

The rock will take longer to reach the ground on the moon.

65. Distance between and Distance between and

A C xC B y

=

=

Total distance 600 1400x y= + + =

( )

( ) ( )( )( )

22 2

2

2

800

800 6002 1600 280,000 0

1600 1600 4 2 280,000 1600 400 2 400 100 22 2 4

y x

x xx x

x

= −

+ − =

− + =

± − − ±= = = ±

The other two distances are 400 100 2 259 miles and 400 100 2 541 miles.− ≈ + ≈

67. 258.155 612.3 2387.1, 6 15S t t t= − + ≤ ≤ (t = 6 corresponds to 1996.)

(a)

( ) ( )( )( )

2

2

2

4000 58.155 612.3 2387.10 58.155 612.3 1612.9

612.3 612.3 4 58.155 1612.92 58.155

t tt t

t

= − +

= − −

± − − −=

Choosing the positive solution, t ≈ 12.71, you can estimate that total sales were about $4 billion in 2003.

(b)

( ) ( )( )( )

2

2

2

6200 58.155 612.3 2387.10 58.155 612.3 3812.9

612.3 612.3 4 58.155 3812.92 58.155

t tt t

t

= − +

= − −

± − − −=

Choosing the positive solution t ≈ 14.92, you can predict that total sales were about $6.2 billion in 2005.

(c)

( ) ( )( )( )

2

2

2

9450 58.155 612.3 2387.10 58.155 612.3 7062.9

612.3 612.3 4 58.155 7062.92 58.155

t tt t

t

= − +

= − −

± − − −=

Choosing the positive solution, t ≈ 17.48, you can predict that total sales will reach $9.45 billion in 2007. So, the model agrees with the original prediction.



69. 20.270 3.59 83.1, 2 7L t t t= − + + ≤ ≤

(t = 2 corresponds to 2:00 P.M.)

( )( )( )

2

2

2

93 0.270 3.59 83.10 0.270 3.59 9.9

3.59 3.59 4 0.270 9.92 0.270

t tt t

t

= − + +

= − + −

− ± − − −=

−

t ≈ 4 or t ≈ 9 Because t = 9 is not in the domain, choose t ≈ 4. The patient′s blood oxygen level was 93% at about 4:00 P.M.

71. ( )( )( )( )3 hours 50 mph3 hours mph

E

S

d rd r

= +

=

( )

2 2 2

2 2 2

2

2440

9 50 9 244018 900 5,931,100 0

E Sd d

r rr r

+ =

+ + =

+ − =

( )( )

( )

2900 900 4 18 5,931,100 900 60 118,8472 18 36

r− ± − − − ±

= =

Thus, the eastbound plane is moving at r + 50 ≈ 600 mph and the southbound at r ≈ 550 mph.

73. 20.45 1.65 50.75, 10 25C x x x= − + ≤ ≤

(a)

( ) ( )( )( )

2

2

2

150 0.45 1.65 50.750 0.45 1.65 99.25

1.65 1.65 4 0.45 99.2516.80

2 0.45

x xx x

x

= − +

= − −

± − − −= ≈

The air temperature is about 16.80°C.

(b) When 10:x = ( ) ( )20.45 10 1.65 10 50.75 79.25C = − + =

When 20:x = ( ) ( )20.45 20 1.65 20 50.75

197.75197.75 2.579.25

C = − +

= = ≈

Oxygen consumption increased by a factor of 2.5.

75. ( )

( ) ( ) ( )( )( )

2

2

2

Revenue 50 0.0005

250,000 50 0.00050.0005 50 250,000 0

50 50 4 0.0005 250,0002 0.0005

50 20000.001

94,721 units or

50 20000.001

5279 units

xp x x

x xx x

x

x

x

x

x

= = −

= −

− + =

− − ± − −=

+=

≈

−=

≈

To produce a revenue of $250,000, approximately 5279 or approximately 94,721 units must be sold.

77. Answers will vary.


Mid-Chapter Quiz Solutions for Chapter 1 45

Mid-Chapter Quiz Solutions

1. ( ) ( )3 2 4 2 5 43 6 8 20 4

5 306

x xx x

xx

− − + =− − − =

− == −

2.

( ) ( )

3 3 35 2 4

4 3 3 3 5 212 12 15 6

3 186

xxx xx x

xx

+=

−+ = −

+ = −

− = −

=

3. ( )

2 1 11 4x x x x

+ =− −

( )( ) ( ) ( )( )

( ) ( )( ) ( )2 2

2 2

2 1 11 4 1 41 4

2 4 1 4 12 8 5 4

3 42 4

2

x x x x x xx x x x

x x x x xx x x x x

x x x xxx

⎡ ⎤ ⎛ ⎞− − + = − −⎢ ⎥ ⎜ ⎟− −⎝ ⎠⎢ ⎥⎣ ⎦− + − − = −

− + − + = −

− − = −

− =

= −

4. ( ) ( )2 2

2 2

3 6 26 9 6 12

6 9 6 129 12

x x xx x x x

x x

+ − = +

+ + − = +

+ = +

≠

Contradiction: No solution

5. One method would be to use the scientific calculator portion of a graphing utility to check for a true statement. Another method would be to use a table feature.

6.

( )( ) ( )( )( )

1002.004 5.128

2.004 5.128 2.004 5.128 1002.004 5.128

5.128 2.004 1027.65123.124 1027.6512

1027.65123.124

328.954

x x

x x

x xx

x

x

− =

⎡ ⎤− =⎢ ⎥⎣ ⎦− =

=

=

≈

7. ( )0.378 0.757 500 2150.378 378.5 0.757 215

0.379 163.5163.50.379431.398

x xx x

x

x

x

+ − =

+ − =

− = −

=

≈

8. ( ) ( ) ( )( )Total cost Fixed costs Unit cost Number of units200,000 30,000 8.50170,000 8.5020,000

xx

x

= +

= +

=

=

The company can manufacture 20,000 units.



9.

( )

( ) ( )( )( )

2

2

2

300,000 75 0.0002

300,000 75 0.0002

0.0002 75 300,000 0

75 75 4 0.0002 300,000 75 53852 0.0002 0.0004

4044 or 370,956

R xp

x x

x x

x x

x

x x

=

= −

= −

− + =

± − − ±= =

≈ ≈

To produce a revenue of $300,000 about 4044 units or about 370,956 units must be sold.

10.

( )( )

2

2

23

3 13 103 13 10 03 2 5 0

3 2 0 or 5 0

or 5

x xx xx x

x x

x x

+ =

+ − =

− + =

− = + =

= = −

11. 2

2

3 155

5 2.24

xxx

=

=

= ± ≈ ±

12. ( )23 173 17

3 171.12 or 7.12

xx

xx x

+ =

+ = ±

= − ±

≈ ≈ −

13. 2

2

2 52 5 0x x

x x+ =

+ − =

( ) ( )( )( )

22 2 4 1 5 2 242 1 2

2 2 62

1 6 1.45, 3.45

x− ± − − − ±

= =

− ±=

= − ± ≈ −

14. 23 7 2 0x x+ − =

( ) ( )( )( )

27 7 4 3 22 3

7 73 0.26, 2.596

x− ± − −

=

− ±= ≈ −

15. 23 4.50 0.32 0x x− − =

( ) ( ) ( )( )( )

24.50 4.50 4 3 0.322 3

4.50 24.096

1.568 or 0.068

x

x x

− − ± − − −=

±=

≈ ≈ −

16.

( ) ( )( )

2

22

2 4 9 0

4 4 4 2 9 56 0

x x

b ac

− + =

− = − − = − <

No real solutions

17.

( ) ( )( )

2

22

4 12 9 0

4 12 4 4 9 0

x x

b ac

− + =

− = − − =

One repeated real solution

18. Answers will vary. One method would be to use the definition of an exponent, multiply and simplify:

( ) ( )( )2 2 23 3 3 3 3 9 6 9x x x x x x x x+ = + + = + + + = + +

Another method would be to use the table feature of a graphing utility and let

( )2 21 23 and 6 9.y x y x x= + = + +

As real values of x are selected y1 will always equal y2.

A third method would be to use the scientific calculator portion of a graphing utility and demonstrate that ( )2 23 6 9x x x+ = + + with real values of x.


Section 1.5 Other Types of Equations 47

19. 0 016s t v t s= − + +

Since 0 0v = and 0 300,s = we have

216 300s t= − +

When it hits the ground, the height, s, is zero.

2

2

2 30016

0 16 30016 300

ttt

= − +

=

=

Since t represents time, it must be nonnegative.

300 5 12 4.33 seconds16 4

t = = ≈

The rock will hit the ground in about 4.33 seconds.

20. ( )( )( )( )( )( )

2

2

Volume length width height384 6384 664

x xx

x

=

=

=

=

Since x must be nonnegative, we have 8.x = Dimensions of box: 8 inches × 8 inches × 6inches

Section 1.5 Other Types of Equations

Skills Review

1.

( )( )( )

2

2

22 121 011 11 0

11 011 0

11

x xx x

xx

x

− + =

− − =

− =

− =

=

2. ( ) ( )( )( )20 3 20 0

20 3 020 0 or 3 0

20 or 3

x x xx x

x xx x

− + − =

− + =

− = + =

= = −

3. ( )220 62520 25

20 25 or 20 255 or 45

xx

x xx x

+ =

+ = ±

= − + = − −

= = −

4.

( )

2

15

5 05 1 0

0 or 5 1 0

x xx x

x xx

+ =

+ =

= + =

= −

5.

( )( )

2

23

3 4 4 03 2 2 0

3 2 0 or 2 0or 2

x xx x

x xx x

+ − =

− + =

− = + =

= = −

6.

( )( )

2

5116 2

12 8 55 06 11 2 5 0

6 11 0 or 2 5 0or

x xx x

x xx x

+ − =

− + =

− = + =

= = −

7.

( )( )

2 4 5 05 1 0

5 0 or 1 05 or 1

x xx x

x xx x

+ − =

+ − =

+ = − =

= − =

8.

( )( )

2

5 32 2

4 4 15 02 5 2 3 0

2 5 0 or 2 3 0or

x xx x

x xx x

+ − =

+ − =

+ = − =

= − =

9.

( ) ( )( )( )

2

2

2

3 1 0

42

3 3 4 1 1 3 5 3 52 1 2 2 2

x x

b b acxa

x

− + =

− ± −=

± − − ±= = = ±

10.

( ) ( )( )( )

2

2

2

4 2 0

42

4 4 4 1 22 1

4 8 4 2 2 2 22 2

x x

b b acxa

x

− + =

− ± −=

± − −=

± ±= = = ±

x

x

6 in.



1.

( )( )( )

3 2

2

2 3 0

2 3 0

3 1 0

x x x

x x x

x x x

− − =

− − =

− + =

03 0 31 0 1

xx xx x

=

− = ⇒ =

+ = ⇒ = −

3.

( )4 2

2 2

2

2 32

4 18 0

2 2 9 0

2 0 02 9 0 2

x x

x x

x xx x

− =

− =

= ⇒ =

− = ⇒ = ±

5.

( )( )( )

4

2

81 0

9 3 3 0

x

x x x

− =

+ + − =

2 9 0 No real solution3 0 33 0 3

xx xx x

+ = ⇒

+ = ⇒ = −

− = ⇒ =

7.

( )( )

3 2

2

2

5 30 45 0

5 6 9 0

5 3 0

5 0 03 0 3

x x x

x x x

x x

x xx x

+ + =

+ =

= ⇒ =

+ = ⇒ = −

9.

( ) ( )( )( )

( )( )( )

3 2

2

2

7 4 28 07 4 7 0

4 7 0

2 2 7 0

2 0 22 0 27 0 7

x x xx x x

x x

x x x

x xx xx x

− − + =

− − − =

− − =

− + − =

− = ⇒ =

+ = ⇒ = −

− = ⇒ =

11.

( ) ( )( )( )

( )( )( )

4 3

3

3

2

2

1 0

1 1 0

1 1 0

1 1 1 0

1 0 1 1 0 1

1 0 No real solution

x x x

x x x

x x

x x x x

x xx x

x x

− + − =

− + − =

− + =

− + − + =

− = ⇒ =

+ = ⇒ = −

− + = ⇒

13.

( )( )4 2

2 2

2

2

12 11 0

11 1 0

11 0 11 3.317

1 0 1

x x

x x

x x

x x

− + =

− − =

− = ⇒ = ± ≈ ±

− = ⇒ = ±

15. 4 2

2 2

2

2

5 36 0

( 9)( 4) 0

( 9)( 2)( 2) 02 0 22 0 29 0 No real solution

x x

x x

x x xx xx x

x

+ − =

+ − =

+ + − =

+ = ⇒ = −

− = ⇒ =

+ = ⇒

17.

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

4 2

2 2

12

12

4 65 16 0

4 1 16 0

2 1 2 1 4 4 0

2 1 0

2 1 0

4 0 44 0 4

x x

x x

x x x x

x x

x x

x xx x

− + =

− − =

+ − + − =

+ = ⇒ = −

− = ⇒ =

+ = ⇒ = −

− = ⇒ =

19.

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

6 3

3 3

2 2

7 8 0

8 1 0

2 2 4 1 1 0

x x

x x

x x x x x x

+ − =

+ − =

+ − + − + + =

2

2

2 0 2

2 4 0 No real solution1 0 1


x x

x xx x

x x

+ = ⇒ = −

− + = ⇒

− = ⇒ =

+ + = ⇒

21. 2 10 02 102 100

50

xxxx

− =

=

=

=

23. 10 4 010 410 16

26

xxx

x

− − =

− =

− =

=

25. 3

3

2 5 3 02 5 32 5 27

2 3216

xxx

xx

+ + =

+ = −

+ = −

= −

= −



27.

( )( )

2

2

14

2 9 5 09 5 2

81 25 20 40 25 101 40 25 1 4

1 4 025 is not a solution

x xx xx x x

x xx x

x xx

+ − =

= −

= − +

= − +

= − −

− = ⇒ =

=

29.

( )( )

2

2

11 3011 30

11 30 05 6 0

5 0 56 0 6

x xx x

x xx x

x xx x

= −

= −

− + =

− − =

− = ⇒ =

− = ⇒ =

31.

( )( )

2

2

26 11 44 26 11

16 8 26 113 10 0

5 2 0

5 0 52 0 2

x xx x

x x xx xx x

x xx x

− − + =

− = −

− + = −

+ − =

+ − =

+ = ⇒ = −

− = ⇒ =

33.

( )

2

2

59

1 3 11 3 11 9 6 10 9 50 9 5

0

is not a solution

x xx xx x x

x xx x

x

x

+ − =

+ = +

+ = + +

= +

= +

=

= −

35. ( )2 3

3 2

5 165 165 64

69, 59

xxx

x

− =

− = ±

− = ±

= −

37. ( )3 2

2/3

3 83 83 4

1

xxx

x

+ =

+ =

+ =

=

39. ( )2 32

2 3 2

2

2

2

5 16

5 165 64


x

xx

x xx

− =

− = ±

− = ±

= ⇒ = ±

= − ⇒

41.

( )2

1 1 31

1 3 1 , 0, 1

3 3 1 0

x xx x x x x

x x

− =+

+ − = + ≠ −

− − + =

( ) ( ) ( )( )( )

2

2

42

3 3 4 3 12 3

3 21 3 216 6

b b acxa

− ± −=

− − ± − − −=

−

± − ±= =

−

43.

2

2

20

20 , 0

20 0

x xx

x x x

x x

−=

− = ≠

− − + =

( ) ( ) ( )( )( )

2

2

42

1 1 4 1 202 1

1 9 5, 42

b b acxa

− ± −=

− − ± − − −=

−

±= = −

−

45.

( )

( )

2

2

2

1 4 11

1 4 1 , 0, 11 3

2 1 0

1 01 0 1

x xx x x x xx x x

x x

xx x

= +−

− = + − ≠

− = +

+ + =

+ =

+ = ⇒ = −

47.

( ) ( ) ( )( )

( )( )

2

2

4 3 11 2

4 2 3 1 1 2 , 2, 1

4 8 3 3 3 2

2 3 0

1 3 0

1 0 13 0 3

x xx x x x x

x x x x

x x

x x

x xx x

− =+ +

+ − + = + + ≠ − −

+ − − = + +

+ − =

− + =

− = ⇒ =

+ = ⇒ = −

49.

( )

1 21 2 1

1 2 3

xx x

x x

+ =

+ = ⇒ =

− + = ⇒ = −

51.

( )

2 1 52 1 5 3

2 1 5 2

xx x

x x

− =

− = ⇒ =

− − = ⇒ = −



53.

( )( )

2

2

2

2

2

33

3 03

OR3

2 3 01 3 0

1 0 13 0 3

x x xx x x

xx

x x xx x

x x

x xx x

= + −

= + −

− =

= ±

− = + −

+ − =

− + =

− = ⇒ =

+ = ⇒ = −

Only 3 and 3 are solutions.x x= = −

55.

( )( )

2

2

2

10 1010 10

11 10 01 10 0

1 0 110 0 10

x x xx x x

x xx x

x xx x

− = −

− = −

− + =

− − =

− = ⇒ =

− = ⇒ =

OR

( )

( )( )

2

2

2

10 1010 10

9 10 01 10 0

1 0 110 0 10

x x xx x x

x xx x

x xx x

− − = −

− + = −

− − =

+ − =

− = ⇒ =

Only 1 and 10 are solutions.x x= − =

57. The quadratic equation was not written in standard form. As a result, the substitutions in the quadratic formula are incorrect. The solution should be:

( ) ( ) ( )( )( )

2

2

3 7 4 0

7 7 4 3 42 3

7 97 is the only solution.6

7 97 is not a solution.6

x x

x

x

x

− − =

− − ± − − −=

+=

−=

59.

( ) ( )( )( )

4 2

22

3.2 1.5 2.1 0

1.5 1.5 4 3.2 2.12 3.2

1.5 29.13 1.0386.4

x x

x

x

− − =

± − − −=

+= ± ≈ ±

61.

( ) ( )( )( )

2

2

1.8 6 5.6 0

6 6 4 1.8 5.62 1.8

6 76.32 16.7563.6

x x

x

x

− − =

± − − −=

⎡ ⎤+= ≈⎢ ⎥⎢ ⎥⎣ ⎦

63. Number of studentsCost per student

xf

=

=

( )( )

( )

( )( )

( ) ( )( )( )

2

2

17001700

7.50 6 1700

1700 7.50 6 1700

3400 15 6 3400

15 90 20,400 0

90 90 4 15 20,400 90 11102 15 30

f x fx

f x

xx

x x x

x x

x

= ⇒ =

− + =

⎛ ⎞− + =⎜ ⎟⎝ ⎠

− + =

− − + =

± − − − ±= =

− −

The original number was 34.x =



65.

( )

( ) ( )

12(6)

72

1 72

1 72

1

4296.16 3000 112

1.43205 112

1.43205 112

1.43205 1 12

ntrA Pn

r

r

r

r

⎛ ⎞= +⎜ ⎟⎝ ⎠

⎛ ⎞= +⎜ ⎟⎝ ⎠

⎛ ⎞≈ +⎜ ⎟⎝ ⎠

= +

⎡ ⎤− =⎣ ⎦

0.06 6%r ≈ =

The annual interest rate is about 6%.

67.

( )12 1 2

6

1 6

1

320 300 112

16 115 12

16 115 12

ntrA Pn

r

r

r

⎛ ⎞= +⎜ ⎟⎝ ⎠

⎛ ⎞= +⎜ ⎟⎝ ⎠

⎛ ⎞= +⎜ ⎟⎝ ⎠

⎛ ⎞ = +⎜ ⎟⎝ ⎠

( )1 616 1 12

15r

⎡ ⎤⎛ ⎞ − =⎢ ⎥⎜ ⎟⎝ ⎠⎣ ⎦

0.1298 12.98%r ≈ =

You are paying an annual interest rate of about 12.98%.

69.

( )

0.25 13.5 0.25 1

12.25 0.25 111.25 0.25

4545 1000 45,000

C xx

xx

x

= +

= +

= +

=

=

=

There were 45,000 passengers that flew in the month.

71.

( ) ( )( )( )

2

2

2

2

2

0.874 140.07 5752.5, 48 65

20 0.874 140.07 5752.5400 0.874 140.07 5752.5

0 0.874 140.07 5352.5

140.07 140.07 4 0.874 5352.52 0.874

62.9 or 97.4

y x x x

x xx xx x

x

x x

= − + ≤ ≤

= − +

= − +

= − +

± − −=

≈ ≈

The solution 97.4x ≈ does not make sense in this situation. So, the person is about 62.9 years old.

73.

13.95 40

40 0.01 1, 0 159,900

0.01 1

0.01 1 26.050.01 1 678.6025

0.01 677.602567,760.25 67,760 units

p x x

x

xx

xx

= −

= − + ≤ ≤

+

+ =

+ =

=

= ≈

When the price is set at $13.95, the demand for the product is 67,760 units.

This model is only valid for 0 159,900x≤ ≤ because it does not make sense in the context of the problem to have a demand less than 0 or a price less than 0.

75. By the Pythagorean Theorem, we have:

( )

( )

22 2

2 2 2

2 2 2 2

2 2

135070

35 10 and

15

Thus 35 10 15

1225 70 100 22570 1350 0

.

h x

h x

x x

x x xx

x

= − −

= −

− − = −

− + − = −

− + =

=

( )2 21357Thus 15 12.12 feet.h = − ≈

The stays are attached about 12.12 feet up the mast.

77.

( )

80 1.2

80 1.2 81.2

80 1.2 78.8

x

x x

x x

− =

− = ⇒ =

− − = ⇒ =

The least acceptable weight is 78.8 ounces and the greatest acceptable weight is 81.2 ounces.

79. Let t = time to paint the house then 15t = portion painted by you, 18t = portion painted by your friend.

115 18

18 15 27033 270

90 8.18 hours11

t t

t tt

t

+ =

+ =

=

= ≈

Working together, it will take about 8.18 hours to paint the house.

xh

15 10

35 − x



Section 1.6 Linear Inequalities

1. [ ]1, 5 corresponds to 1 5.

The interval is bounded.

x− − ≤ ≤

3. ( )11, corresponds to 11.x∞ >

The interval is unbounded.

5. ( ), 2 corresponds to 2.x−∞ − < −

The interval is unbounded.

7. 4x < indicates all points to the left of 4;x = graph (c).

9. 2 5x− < ≤ indicates all points from, but not including 2x = − up to and including 5;x = graph (f ).

11. 4x < indicates all points less than 4 units from

0x = in both directions; graph (g).

13. 5 2x − > indicates all points more than 2 units from

5;x = graph (b).

15. 5 12 0x − >

(a) ( )5 3 12 3 0 3 is a solution of the inequality.x− = > ⇒ =

(b) ( )5 3 12 27 0 3 is not a solution of the inequality.x− − = − < ⇒ = −

(c) ( )5 512 2 25 12 0 is a solution of the inequality.x− = > ⇒ =

(d) ( )3 9 32 2 25 12 0 is not a solution of the inequality.x− = − < ⇒ =

Skills Review

1.

Because 7− is to the left of 1 12 2,− − is larger than −7.

2.

Because 13− is to the left of 1 1

6 6,− − is larger than 13.−

3.

Because ( )3.14π π− ≈ is to the left of 3, 3− − is larger than .π−

4.

Because 132− is to the left of – 6, – 6 is larger than

132.−

5. The statement “x is nonnegative” can be represented by 0.x ≥

6. The statement “z is strictly between − 3 and 10” can be represented by 3 10.z− < <

7. The statement “P is no more than 2” can be represented by 2.P ≤

8. The statement “W is at least 200” can be represented by 200.W ≥

9. 10

When 12: 12 10 2 2

When 3: 3 10 7 7

x

x

x

−

= − = =

= − = − =

10.

( )( )

3 32 2

2 3

When : 2 3 3 3 0 0

When 1: 2 1 3 2 3 1 1

x

x

x

−

= − = − = =

= − = − = − =

−1−2−3−4−5−6−7−8 0

12

−

x

−1 012

−

13

− 16

−x

−4 −3

− πx

x

−7 −6

132

−


Section 1.6 Linear Inequalit ies 53

17. 20 24

x −< <

(a) 4 2 10 2 0 2 4 is a solution of the inequality.4 2

x−< < ⇒ < < ⇒ =

(b) 10 20 2 0 2 2 10 is not a solution of the inequality.4

x−< < ⇒ < ⇒ =�

(c) 0 2 10 2 0 2 0 is not a solution of the inequality.4 2

x−< < ⇒ < − < ⇒ =

(d) 7 2 2 3 70 2 0 2 is a solution of the inequality.4 8 2

x−< < ⇒ < < ⇒ =

19. 10 3x − ≥

(a) 13 10 3 3 3 13 is a solution of the inequality.x− ≥ ⇒ ≥ ⇒ =

(b) 1 10 3 11 3 1 is a solution of the inequality.x− − ≥ ⇒ ≥ ⇒ = −

(c) 14 10 3 4 3 14 is a solution of the inequality.x− ≥ ⇒ ≥ ⇒ =

(d) 9 10 3 1 − ≥ ⇒ ≥ 3 9 is not a solution of the inequality.x⇒ =

21. 2 10x + ≤

(a) 15 2 13 13 13− + = − = ⇒ ≤ 10 15 is not a solution of the inequality.x⇒ = −

(b) 4 2 2 2 2 10 4 is a solution of the inequality.x− + = − = ⇒ ≤ ⇒ = −

(c) 1 2 3 3 3 10 1 is a solution of the inequality.x+ = = ⇒ ≤ ⇒ =

(d) 8 2 10 10 10 10 10 is a solution of the inequality.x+ = = ⇒ ≤ ⇒ =

23. If 2 6,x > then 3.x >

25. If 2 8,x ≤ − then 4.x ≤ −

27. 2 4 104 12

3

xxx

− > −

− > −

<

If 2 4 10,x− > − then 3.x <

29. If 23 6,x− ≥ − then 9.x ≤

31.

( ) ( )

32

32 23 2 3

9

9

6

x

x

x

≥

≥

≥

33.

( ) ( )1 110 10

10 40

10 40

4

x

x

x

− <

− − > −

> −

35. 35

35

7 8

15

25

x

x

x

− <

<

<

37. 2 7 3 42 7 3

2 42

x xx

xx

+ < +

− + <

− < −

<

39.

13

2 1 53 1

x xxx

− ≥

− ≥

≤ −

4 5 6 7 8

x

x

−2−3−4−5−6

x

23 24 25 26 27

0 1 2 3 4

x

x

10−1−2−3−4



41. ( )3 2 7 2 53 6 7 2 5

3 13 2 518

x xx x

x xx

+ + < −

+ + < −

+ < −

< −

43. ( )

25

3 1 7 2 83 3 7 2 8

3 10 2 85 2

x xx x

x xxx

− − + < +

− + + < +

− + < +

− < −

>

45. 3 2 1 74 2 82 4

xx

x

≤ − <

≤ <

≤ <

47. 1 2 3 92 2 61 3

xx

x

< + <

− < <

− < <

49. 2 34 43

12 2 3 129 2 159 152 2

x

xx

x

−− < <

− < − <

− < <

− < <

51. 3 14 4

314 4

1xx

> + >

− > > −

53. 66 6x

x<

− < <

55. 32

3 or 32 2

6 6

x

x x

x x

>

< − >

< − >

57. 3 55 3 58 2

xxx

+ <

− < + <

− < <

59. 20 44 20 4

16 24

xxx

− ≤

− ≤ − ≤

≤ ≤

61.

1 112 2

2 5 62 5 6 or 2 5 6

2 1 2 11

xx x

x xx x

− >

− < − − >

< − >

< − >

63. 3 52

3 35 or 52 2

3 10 3 107 13

x

x x

x xx x

−≥

− −≤ − ≥

− ≤ − − ≥

≤ − ≥

65. 9 2 2 19 2 1

1 9 2 110 2 8

5 4

xx

xx

x

− − < −

− <

− < − <

− < − < −

> >

67. 92

9 92 229 112 2

2 10 91010 or 10

xxx x

x x

+ ≥

+ ≥

+ ≤ − + ≥

≤ − ≥ −

−20 −19 −18 −17 −16

x

1 20−1

x

1 2 3 4 5

x

320 1

x

−1

5 100

x

−5

10x

−1

0 2 4 6

x

−2−4−6

8640 2

x

−2−4−6−8

20 4

x

−2−4−6−8−10

x

16 18 20 22 24

5 643210

x

−1

0 5 10 15

x

−5−10

4 5 63

x

x

−4−8−12−16


Section 1.6 Linear Inequalit ies 55

69. 5 0x − <

No solution

71. All real numbers no more than 2 units from 0 yields 0 2 or 2.x x− ≤ ≤

73. All real numbers no less than 3 units from 9 yields 9 3x − ≥ .

75. 12 10x − ≤

77. ( 3) 53 5

xx− − >

+ >

79. Rental fee for Company B Rental fee for Company A199 0.32 279

0.32 80250

xxx

>

+ >

>

>

You must drive more than 250 miles in a week for the rental fee for Company B to be greater than that for Company A.

81. I = Prt; P = 1500, t = 3, I = 1890 − 1500 = 390

( )( )

3901500 3 390

4500 3900.0878.7%

Ir

rrr

>

>

>

>

>

The simple interest rate must be greater than 8.7%.

83.

13

180 1.5 13050 1.5

33

xx

x

− ≥

≥

≥

The person must be in the program 1333 or less weeks.

85. Length Girth 132 in68 4 132

4 6416

xxx

+ ≤

+ ≤

≤

≤

The sides of the package’s cross sections cannot be more than 16 inches.

87. (a)

(b) 139.95 97 85042.95 850

20

R Cx xxx

>

> +

>

≥

The product will return a profit for 20 units.x ≥

89. C = 0.32 2500 12,0000.32 9500

29,687.5

mmm

+ <

<

<

Less than 29,687.5 miles traveled will yield an operating cost that is less than $12,000.

91. 0.068 4.753 3.00.068 7.753

114.015

xxx

− ≥

≥

≥

According to the model, an IQ score greater than 114 would produce a grade point average of 3.0 or higher.

93. 0.1527 0.295, 5 16S t t= + ≤ ≤ (t = 5 corresponds to 1995.)

30.1527 0.295 3

0.1527 2.70517.7

St

tt

>

+ >

>

>

According to the model, the average professional baseball player’s salary will exceed $3,000,000 sometime during 2007.

95. 10.4 0.06250.0625 10.4 0.0625

10.3375 10.4625

xxx

− ≤

− ≤ − ≤

≤ ≤

2Area x=

Interval: 2 2106.86 in. , 109.46 in.⎡ ⎤⎣ ⎦

97. 132

1 132 32

5.725.72

5.68875 5.75125

xxx

− ≤

− ≤ − ≤

≤ ≤

You could have been undercharged ( ) ( )5.75125 7.99 5.72 7.99 $0.25− ≈

or you could have been overcharged ( ) ( )5.72 7.99 5.68875 7.99 $0.25.− ≈

99. 68.5 2.72.7 68.5 2.7

65.8 71.2

hhh

− ≤

− ≤ − ≤

≤ ≤

Interval: [ ]65.8, 71.2

x 10 20 30

R $1399.50 $2799 $4198.50

C $1820 $2790 $3760

x 40 50 60

R $5598 $6997.50 $8397

C $4730 $5700 $6670



101. 50 3030 50 30

20 80

hh

h

− ≤

− ≤ − ≤

≤ ≤

The minimum relative humidity is 20% and the maximum relative humidity is 80%.

103. 6.928 3.45, 8 15B t t= − ≤ ≤ (t = 8 corresponds to 1998.) 75

6.928 3.45 756.928 78.45

11.3

Bt

tt

>

− >

>

>

The average price of a brand name prescription drug exceeded $75 sometime during 2001.

105. 276.4 16,656, 5 15D t t= + ≤ ≤ (t = 5 corresponds to 1995.)

(a) 18,000276.4 16,656 18,000

276.4 13444.9

Dt

tt

>

+ >

>

>

The demand for U.S. oil exceeded 18 million barrels a day in 1995. (b) 22,000

276.4 16,656 22,000276.4 5344

19.3

Dt

tt

>

+ >

>

>

The demand for U.S. oil will exceed 22 million barrels a day sometime during 2009.

Section 1.7 Other Types of Inequalities

Skills Review

1. 23

6

y

y

− >

< −

2. 92

6 27zz

− <

> −

3. 3 2 3 56 2 23 1

xx

x

− ≤ + <

− ≤ <

− ≤ <

4. 3 5 203 15

5

xxx

− + ≥

− ≥

≤ −

5. ( )( )

10 4 3 16 3 12 13 or 3

xx

xx x

> − +

> − +

− < +

− < > −

6. ( )( )

3 1 2 4 72 2 4 61 4 35 7

xx

xx

< + − <

< − <

< − <

< <

7. 72

7 72 2

2 7xx

x

≤

≤

− ≤ ≤

8. 3 13 1 or 3 1

2 or 4

xx x

x x

− >

− < − − >

< >

9. 4 24 2 or 4 2

6 or 2

xx x

x x

+ >

+ < − + >

< − > −

10. 2 44 2 4

6 2 6 2 2 6

xxxxx

− ≤

− ≤ − ≤

− ≤ − ≤

≥ ≥ −

− ≤ ≤


Section 1.7 Other Types of Inequalities 57

1. 2

2

2

25 025 0

255

xx

xx

− <

− =

=

= ±

Critical numbers: 5x = ±

Test intervals: ( ) ( ) ( ), 5 , 5, 5 , 5, −∞ − − ∞

3.

( )( )

2

2

2

12

2 7 16 202 7 16 202 7 4 0

2 1 4 02 1 0

4 0 4

x xx xx x

x xx xx x

+ + ≥

+ + =

+ − =

− + =

− = ⇒ =

+ = ⇒ = −

Critical numbers: 124, x x= − =

Test intervals: ( ) ( ) ( )1 12 2, 4 , 4, , , −∞ − − ∞

5.

( )

3 213 2, 113 2 13 2 21

xxx xxx xx x

x

−<

−−

= ≠−− = −

− = −

− =

Critical numbers: 1, 1x x= − =

Test intervals: ( ) ( ) ( ), 1 , 1, 1 , 1, −∞ − − ∞

7.

( )( )

2

2

99 0

3 3 0

xx

x x

≤

− ≤

+ − ≤


Test intervals: ( ) ( ) ( ), 3 , 3, 3 , 3, −∞ − − ∞

Test: ( )( )Is 3 3 0?x x+ − ≤

Solution set: [ ]3, 3−

9.

( )( )

2

2

44 0

2 2 0

xx

x x

>

− >

+ − >

Critical numbers: 2x = ± Test intervals: ( ) ( ) ( ), 2 , 2, 2 , 2, −∞ − − ∞

Test: 2Is 4 0?x − > Solution set: ( ) ( ), 2 2, −∞ − ∪ ∞

11. ( )

( )( )

2

2

2

2 254 4 25

4 21 07 3 0

xx x

x xx x

+ <

+ + <

+ − <

+ − <


Test: intervals ( ) ( ) ( ), 7 , 7, 3 , 3, −∞ − − ∞

Test: Is ( )( )7 3 0?x x+ − <

Solution set: ( )7, 3−

13.

( )( )

2

2

4 4 94 5 0

5 1 0

x xx x

x x

+ + ≥

+ − ≥

+ − ≥

Critical numbers: 5, 1x x= − = Test intervals: ( ) ( ) ( ), 5 , 5, 1 , 1, −∞ − − ∞

Test: Is ( )( )5 1 0?x x+ − ≥

Solution set: ( ] [ ), 5 1, −∞ − ∪ ∞

15.

( )( )

2

2

66 0

3 2 0

x xx x

x x

+ <

+ − <

+ − <


Test intervals: ( ) ( ) ( ), 3 , 3, 2 , 2, −∞ − − ∞

Test: Is ( )( )3 2 0?x x+ − <


17. ( )( )3 1 1 0x x− + >


Test intervals: ( ) ( ) ( ), 1 , 1, 1 , 1, −∞ − − ∞

Test: ( )( )Is 3 1 1 0?x x− + >

Solution set: ( ) ( ), 1 1, −∞ − ∪ ∞

x

0 1 2 3−1−2−3

420

x

−2−4−6−8

x

20−2−4−6

x

210−1−2−3

x

0 1 2−1−2

x

0 1 2 3−1−2−3



19.

( )( )

2 2 3 03 1 0

x xx x

+ − <

+ − <


Test intervals: ( ) ( ) ( ), 3 , 3, 1 , 1, −∞ − − ∞

Test: 2Is 2 3 0?x x+ − < Solution set: ( )3, 1−

[Note: Compare this problem to #16.]

21.

( )

3 2

2

4 6 02 2 3 0

x xx x

− <

− <

Critical numbers: 320, x x= =

Test intervals: ( ) ( ) ( )3 32 2, 0 , 0, , , −∞ ∞

Test: ( )2Is 2 2 3 0?x x − <

Solution set: ( ) ( )32, 0 0, −∞ ∪

23.

( )( )

3 4 0x xx x x

− ≥

+ − ≥

Critical numbers: 0, 2x x= = ±

Test intervals: ( ) ( ) ( ) ( ), 2 , 2, 0 , 0, 2 , 2, −∞ − − ∞

Test: ( )( )Is 2 2 0?x x x+ − ≥

Solution set: [ ] [ ]2, 0 2, − ∪ ∞

25.

( )( )( )

3 22 2 01 1 2 0

x x xx x x

− − + ≥

− + − ≥

Critical numbers: 2, 1x = ±

Test intervals: ( ) ( ) ( ) ( ), 1 , 1, 1 , 1, 2 , 2, −∞ − − ∞

Test: ( )( )( )Is 1 1 2 0?x x x− + − ≥

Solution: [ ] [ ]1, 1 , 2, − ∞

27.

2

1

1 0

1 0

xx

xx

xx

>

− >

−>


Test intervals: ( ) ( ) ( ) ( ), 1 , 1, 0 , 0, 1 , 1, −∞ − − ∞

Test: 21Is 0?x

x−

>

Solution set: ( ) ( ), 1 0, 1−∞ − ∪

29.

( )

6 21

6 2 01

6 2 10

14 0

1

xx

xx

x xx

xx

+<

++

− <+

+ − +<

+−

<+


Test intervals: ( ) ( ) ( ), 1 , 1, 4 , 4, −∞ − − ∞

Test: 4Is 0?1x

x−

<+

Solution set: ( ) ( ), 1 4, −∞ − ∪ ∞

31.

( )

3 5 45

3 5 4 05

3 5 4 50

515 0

5

xx

xx

x xx

xx

−>

−−

− >−

− − −>

−−

>−

Critical numbers: 5, 15x x= =

Test intervals: ( ) ( ) ( ), 5 , 5, 15 , 15, −∞ ∞

Test: 15Is 0?5x

x−

>−

Solution set: ( )5, 15

x

20 1−1−2−3−4

x

210−1−2

x

3210−1−2

x

3210−1−2

x

10−1−2

x

43210−1

x

0 5 10 15 20


Section 1.7 Other Types of Inequalities 59

33.

( ) ( )( )( )

( )( )

4 15 2 3

4 1 05 2 3

4 2 3 50

5 2 3

05 2 3

x x

x xx xx x

xx x

>+ +

− >+ ++ − +

>+ +

+>

+ +

Critical numbers: 321, 5, x x x= − = − = −

Test intervals: ( ) ( )( ) ( )

32

32

, 5 , 5,

, 1 , 1,

−∞ − − −

− − − ∞

Test: Is ( )

( )( )7 1

0?5 2 3x

x x+

>+ +

Solution set: ( ) ( )325, 1, − − ∪ − ∞

35.

( )( )( )

( )( )

1 93 4 3

1 9 03 4 3

4 3 9 30

3 4 3

30 5 03 4 3

x x

x xx xx x

xx x

≤− +

− ≤− ++ − −

≤− +

−≤

− +

Critical numbers: 343, , 6x x x= = − =

Test intervals: ( ) ( ) ( ) ( )3 34 4, , , 3 , 3, 6 , 6,−∞ − − ∞

Test: ( )

( )( )5 6

Is 0?3 4 3

xx x

−≤

− +

Solution set: ( ) [ )34, 3 6, − ∪ ∞

37.

( )( )

2 9 03 3 0

xx x

− ≥

+ − ≥


Test intervals: ( ) ( ), 3 , ( 3, 3), 3, −∞ − − ∞

Test: 2Is 9 0?x − ≥ Domain: ( ] [ ), 3 3, −∞ − ∪ ∞

39. 26 0x+ ≥

No critical numbers because ( )( )2 24 0 4 1 6 0.b ac− = − <

Test interval: ( , )−∞ ∞

Test: 2Is 6 0?x+ ≥

Domain: ( ), −∞ ∞ or all real numbers

41.

( )( )

281 4 09 2 9 2 0

xx x

− ≥

+ − ≥


Test intervals: ( ) ( ) ( )9 9 9 92 2 2 2, , , , , −∞ − − ∞

Test: ( )( )Is 9 2 9 2 0?x x+ − ≥

Domain: 9 92 2, ⎡ ⎤−⎣ ⎦

43.

( )( )

2 7 10 05 2 0

x xx x

− + ≥

− − ≥

Critical numbers: 5, 2x x= = Test intervals: ( ) ( ) ( ), 2 , 2, 5 , 5, −∞ ∞

Test: ( )( )Is 5 2 0?x x− − ≥

Domain: ( ] [ ), 2 5, −∞ ∪ ∞

45. 2 3 3 0x x− + ≥ No critical numbers since

( ) ( )( )22 4 3 4 1 3 0b ac− = − − <

Test interval: ( ), −∞ ∞

Test: 2Is 3 3 0?x x− + ≥ Domain: ( ), −∞ ∞ or all real numbers

47. You can take the cube root of a negative number and you will get a real number. For example, if 2 7 12x x− + was equal to 8− , then

( )( )( )3 38 2 2 2 2.− = − − − = −

49.

( )

3 2

2

6 10 02 3 5 0

x xx x

− >

− >

Critical numbers: 530,x x= =

Test intervals: ( ) ( ) ( )5 53 3, 0 , 0, , , −∞ ∞

Test: Is ( )22 3 5 0?x x − >

Solution set: ( )53, ∞

x

0−1−2−3−4−5

x

86420−2



51.

( )( )( )

3

2

9 0

9 0

3 3 0

x x

x x

x x x

− ≤

− ≤

+ − ≤


Test intervals: ( ) ( ) ( ) ( ), 3 , 3, 0 , 0, 3 , 3, −∞ − − ∞

Test: Is ( )( )3 3 0?x x x+ − ≤

Solution set: ( ] [ ], 3 0, 3−∞ − ∪

53. ( ) ( )2 31 2 0x x− + ≥

Critical numbers: 1, 2x x= = −

Test intervals: ( ) ( ) ( ), 2 , 2, 1 , 1, −∞ − − ∞

Test: Is ( ) ( )2 31 3 0?x x− + ≥

Solution set: [ )2, − ∞

55.

( )

2

2

2

0.4 5.26 10.20.4 4.94 0

0.4 12.35 0

xx

x

+ <

− <

− <

Critical numbers: 3.51x ≈ ±

Test intervals: ( ) ( ) ( ), 3.51 , 3.51, 3.51 , 3.51, −∞ − − ∞

Solution set: ( )3.51, 3.51−

57. 0.5 12.5 1.6 0x x− + + >

The zeros are ( ) ( )( )

( )

212.5 12.5 4 0.5 1.62 0.5

x− ± − −

=−

Critical numbers: 0.1325.13

xx≈ −

≈

Test intervals: ( ) ( ) ( ), 0.13 , 0.13, 25.13 , 25.13, −∞ − − ∞

Solution set: ( )0.13, 25.13−

59. 1 3.42.3 5.2

1 3.4 02.3 5.2

7.82 18.68 02.3 5.2

x

xxx

>−

− >−

− +>

−

Critical numbers: 2.39, 2.26x x≈ ≈

Test intervals: ( ) ( ) ( ), 2.26 , 2.26, 2.39 , 2.39, −∞ − ∞

Solution set: ( )2.26, 2.39

61. 20 0

2

1616 200 , 0 12.5

s t v t st t t

= − + +

= − + ≤ ≤

( )( )( )

2

2

2

16 200 40016 200 400 0

8 2 25 50 0

8 2 5 10 0

t tt t

t t

t t

− + >

− + <

− + <

− − <

2.5 seconds 10 secondst< <

63.

( )

( ) ( )2

2 2 100 50500

50 50050 500 0

25 5 5 25 5 5 0

l w w llw

l ll l

l l

+ = ⇒ = −

≥

− ≥

− + − ≥

⎡ ⎤⎡ ⎤− − − + ≤⎣ ⎦⎣ ⎦

13.8 meters 25 5 5 25 5 5 36.2l≈ − ≤ ≤ + ≈ meters. (Use the Quadratic Formula to find the critical numbers.)

65. (a)

( ) ( )2

2

Profit Revenue Cost 1,650,000

50 0.0002 12 150,000 1,650,0000.0002 38 1,800,000 00.0002 38 1,800,000 0

x x xx xx x

= − ≥

− − + ≥

− + − ≥

− + ≤

Critical numbers: ( )( )

( )

238 38 4 0.0002 1,800,0002 0.0002

x± −

=

100,000 or 90,000x x= =

90,000 units 100,000 unitsx≤ ≤


Section 1.7 Other Types of Inequalit ies 61

(b) 50 0.0002p x= −

When 90,000, $32.x p= =

When 100,000, $30.x p= =

$30 $32.p≤ ≤

(c)

( )2

2

Revenue Cost

50 0.0002 12 150,000

50 0.0002 12 150,000

0 0.0002 38 150,000

x x x

x x x

x x

=

− = +

− = +

= − +

( ) ( )( )

( )

238 38 4 0.0002 150,0002 0.0002

4033 or 185,967

x

x x

± − −=

≈ ≈

Choose the larger value of x. After revenue starts to decrease, the revenue is approximately equal to the cost when 185,967 units are sold. Producing more than 185,967 units will result in the cost being greater than revenue. So, the company would

incur a loss instead of making a profit.

67. ( )31A P r= +

( )

( )

3

3

3

3

1500

1000 1 1500

1 1.5

1 1.5

1.5 1

0.145 14.5%

A

r

r

r

r

r

>

+ >

+ >

+ >

> −

> ≈

The interest rate must be greater than 14.5%.

69.

( )

20.18 80.30 5288, 5 16

5 corresponds to 1995.

P t t t

t

= − + + ≤ ≤

=

2

2

70000.18 80.30 5288 70000.18 80.30 1712 0

Pt tt t

>

− + + >

− + − >

( )( )

( )

280.30 80.30 4 0.18 17122 0.18

22.45 or 423.66

t

t t

− ± − − −=

−

≈ ≈

Choose 22.45t ≈ (because 423.66t ≈ does not make sense in this situation). So, the world population will exceed 7,000,000,000 sometime during 2012.

71.

( )

242.93 68.0 15,309, 6 15

6 corresponds to the 1995/1996 academic year.

C t t t

t

= + + ≤ ≤

=

2

2

32,00042.93 68.0 15,309 32,00042.93 68.0 16,691 0

Ct tt t

>

+ + >

+ − >

( )( )

( )

268.0 68.0 4 42.93 16,6912 42.93

20.53 or 18.94

t

t t

− ± − −=

≈ − ≈

Choose the positive solution 18.94.t ≈ So, the average yearly cost of higher education at private institutions will exceed $32,000 during the 2008/2009 school year.

73.

( )

1

1 1

1 1

1

1

1 1 12

2 2

2 22

2

R RR R RR

R R RR R

R

= +

= +

= +

=+

Since 1,R ≥ we have

1

1

1

1

1

1

2 12

2 1 02

2 0.2

RR

RRR

R

≥+

− ≥+

−≥

+

Since 1 0R > , the only critical number is 1 2.R = The inequality is satisfied when 1 2R ≥ ohms.



Review Exercises for Chapter 1

1. ( )5 3 2 95 15 2 9

3 248

x xx x

xx

− = +

− = +

=

=

Conditional

3. (a) ( ) ( ) ( )?2 23 0 7 0 5 0 9 5 9+ + = +

≠

0 a solution.x is not=

(b) ( ) ( ) ( )2 2?1 1 12 2 23 7 5 9

9.25 9.25+ + = +

=

12 a solution.x is=

(c) ( ) ( ) ( )?2 23 4 7 4 5 4 9 25 25

− + − + = − +

=

4 a solution.x is= −

(d) ( ) ( ) ( )?2 23 1 7 1 5 1 9 1 10

− + − + = − +

≠

1 a solution.x is not= −

5. 7 2013

xx

+ =

=

7. ( ) ( )

12

4 3 3 2 4 3 44 12 3 8 6 4

4 9 6 410 5

x xx x

x xxx

+ − = − −

+ − = − −

+ = − +

= −

= −

9.

( ) ( )

3 2 35 1 4

4 3 2 3 5 112 8 15 3

3 553

xx

x xx x

x

x

−=

−− = −

− = −

− =

= −

11.

( )

4 2 03 3

4 2 3 0; 33 2 0

23

xx x

x x xx

x

− + =+ +− + + = ≠ −

+ =

= −

13. ( )0.375 0.75 300 2000.375 225 0.75 200

1.125 425377.778

x xx x

xx

− − =

− + =

=

≈

15. 10.055 0.085

0.085 0.055 0.0046750.14 0.004675

0.033

x x

x xxx

+ =

+ =

=

≈

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

Sum first integer second integer third integer

2 2 2 2 442 6 636 6

62 12 is the smallest of these integers.

s n n nnn

nn

= + +

= + + + +

= +===

19. Let x = amount of weight set 150 120

3030

xxx

− =

− = −

=

A person weighing 150 pounds would set the weight at 30 pounds in order to pull 120 pounds.

21. ( ) ( )( )

Perimeter = 2 length 2 width177 2 2 2177 629.5

width 29.5 feetlength 2 59 feet

w ww

www

+= +=== == =

23.

( )( )500 0.04 1 20I PrtI=

= =

After 1 year, you will earn $20 in interest.

25. Sale Price List Price 15% of List Price139 0.15139 0.85

163.53

L LL

L

= −= −=≈

The outdoor grill was originally priced $163.53.


Review Exercises for Chapter 1 63

27. ( ) ( ) ( ) ( )Distance Difference Rate of Faster Car Time Rate of Slower Car Time10 50 4510 5

2 hours

t tt

t

= × − ×= −==

29. Total Revenue 12 Monthly Revenue375,83212

6$751,664

= ×

⎛ ⎞= ⎜ ⎟⎝ ⎠

=

If the monthly revenue rate continues, the total revenue for the year will be $751,644.

31. ( )( ) ( )

10% of 10 100% of 30% of 100.10 10 1.00 0.30 10

1 0.10 1.00 30.90 2

2.2 quarts

x xx xx x

xx

− + =− + =

− + ==≈

About 2.2 ( )29or 2 quarts will have to be replaced

with pure anti-freeze.

33.

( )( )

2

2

4 13 2

6 5 46 5 4 0

3 4 2 1 0

3 4 0 or 2 1 0or

x xx x

x x

x xx x

= +

− − =

− + =

− = + =

= = −

35.

( )( )

2 11 24 03 8 0

3 0 or 8 03 or 8

x xx x

x xx x

− + =

− − =

− = − =

= =

37. 2 1111 3.32

xx=

= ± ≈

39. ( )24 184 3 2

4 3 28.24 or 0.24

xx

xx x

+ =

+ = ±

= − ±

≈ − ≈

41. (1) Use the table feature in ask mode with the variable equal to a solution.

(2) Use the scientific calculator portion to evaluate the quadratic equation at a particular value. (Answers will vary.)

43. ( )( )( )

( )( )

2

Area length height405 12

0 12 4050 27 15

27 027

15 015

Length 12 27 feetHeight 15 feet

h hh hh h

hh

hh

hh

=

= +

= + −

= + −

+ =

= −

− =

=

= + =

= =

45. ( )60 0.0001 8,000,00020 0.0001 60 8,000,000

x x

x x

− =

= − +

( ) ( )( )

( )

260 60 4 0.0001 8,000,0002 0.0001

200,000 units or 400,000 units

x

x x

± − −=

= =

To produce a revenue of $8,000,000, the company should produce 200,000 units or 400,000 units.

47.

( ) ( )( )

2

22

11 24 0

4 11 4 1 24 25 0

x x

b ac

+ + =

− = − = >

Two real solutions

49. 2 12 30 0x x− + =

( ) ( ) ( )( )

( )

212 12 4 1 302 1

12 242

12 2 62

6 6

x− − ± − −

=

±=

±=

= ±

h + 12

h 405 ft2



51. ( )2

2

2

7 514 49 519 49 0

y yy y yy y

+ = −

+ + = −

+ + =

( )( )

( )

219 19 4 1 492 1

19 1652

y− ± −

=

− ±=

53. 2 6 3 0x x+ − =

( )( )

( )

26 6 4 1 3 6 482 1 2

6 4 3 3 2 32

x− ± − − − ±

= =

− ±= = − ±

55. 23.6 5.7 1.9 0x x− − =

( ) ( ) ( )( )

( )

25.7 5.7 4 3.6 1.92 3.6

1.866 or 0.283

x

x x

− − ± − − −=

≈ ≈ −

57. 234 296 47 0x x− + =

( ) ( ) ( )( )

( )

2296 296 4 34 472 34

296 81,22468

8.544 or 0.162

x

x x

− − ± − −=

±=

≈ ≈

59. On the moon

2

2 2002.7

200 8.61 seconds2.7

t

t

t

− + =

=

= ≈

On the Earth

2

2

16 200 0

25 3.54 seconds2

t

t

− + =

= ≈

When the rock is dropped off a 200 foot cliff on the moon it takes about 8.61 seconds for it to hit the lunar surface. On Earth it would only take about 3.54 seconds.

61.

( )( )( )

3 2

2

3 9 12 0

3 3 4 0

3 4 1 0

0, 4 or 1

x x x

x x x

x x x

x x x

− − =

− − =

− + =

= = = −

63.

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

4 2

2 2

5 4 0

1 4 0

1 1 2 2 0

1, 2

x x

x x

x x x x

x x

− + =

− − =

+ − + − =

= ± = ±

65. 52

254

2 5 0x

x

x

− =

=

=

67.

( ) ( )2 2

2

2

2 3 4 3

2 3 3 4

4 12 9 24 169 20 28 0

x x

x x

x x xx x

− − =

− = +

− = + +

+ + =

There are no real zeros since

( )( )2 24 20 4 9 28 0.b ac− = − <

No solution

69. ( )2 32

2 3 2

2

2 2

5 9

5 9

5 27

32 or 22 no real solution

32 4 2

x

x

x

x x

x

− =

− =

− = ±

= = −

= ± = ±

71.

75

5 4 11

5 4 11 or 5 4 115 7 or 5 15

or 3

x

x xx x

x x

+ =

+ = + = −

= = −

= = −

73.

( ) ( ) ( )( )

( ) ( ) ( )( )( )

2

2

2

5 3 11 3

5 3 3 1 1 3 , 1, 3

5 15 3 3 4 3

0 4 15

4 4 4 1 152 1

4 76 4 2 192 2

2 19

x xx x x x x x

x x x x

x x

x

+ =+ +

+ + + = + + ≠ − ≠ −

+ + + = + +

= − −

− − ± − − −=

± ±= =

= ±


Review Exercises for Chapter 1 65

75. Let r = rent per student per week if three students rent the condominium.

Weekly rent: ( )3 4 753 4 300

300

r rr r

r

= −

= −

=

The weekly rent is )3 300 $900.=

77.

( )

( )

12

112 4

3

3

3

112

535.76 500 112

1.07152 112

1.07152 112

1.07152 1 12

0.280 28.0%

trA P

r

r

r

r

r

⎛ ⎞= +⎜ ⎟⎝ ⎠

⎛ ⎞= +⎜ ⎟⎝ ⎠

⎛ ⎞= +⎜ ⎟⎝ ⎠

= +

⎡ ⎤− =⎣ ⎦≈ =

The annual percentage rate for the cash advance is about 28.0%.

79. ( )3 1 2 83 3 2 8

11

x xx x

x

− < +

− < +

<

81. 2 13 34

12 2 1 1213 2 1113 112 2

x

xx

x

+− < <

− < + <

− < <

− < <

83. 10 3 5 10 2

2 10 2 12 8

xxx

x

+ + <

+ <

− < + <

− < < −

85. 89.95 35 250054.95 2500

45

R Cx xxx

>

> +

>

≥

To maximize the profit, the company should produce 45 or more units.

87. ( )( )5 1 3 0x x+ − <

Critical number: 1, 3x x= − =

Test intervals: ( ) ( ) ( ), 1 , 1, 3 , 3, −∞ − − ∞


89.

( )( )

3 9 03 3 0

x xx x x

− <

+ + <


Test intervals: ( ) ( ) ( ) ( ), 3 , 3, 0 , 0, 3 , 3, −∞ − − ∞

Solution set: ( ) ( ), 3 0, 3−∞ − ∪

91.

( )

2 3 24

2 3 2 04

2 3 2 40

45 6 04

xx

xx

x xx

xx

+<

−+

− <−

+ − −<

−−

<−


Test intervals: ( ) ( ) ( )6 65 5, , , 4 , 4, −∞ ∞

Solution set: ( )65, −∞ ∪ (4, ∞)

93. 2

2

2

1.2 4.76 1.321.2 3.44

2.86671.69 1.69

xxx

x

− + >

− > −

<

− < <

95.

( )

1 2.93.7 6.1

1 2.9 03.7 6.1

1 2.9 3.7 6.10

3.7 6.110.73 18.69 0

3.7 6.1

x

xx

xx

x

>−

− >−

− −>

−− +

>−

Critical numbers: 1.74, 1.65x x≈ ≈ 1.65 1.74x< <

97.

)

10 010

Domain: 10,

xx

− ≥

≥

⎡ ∞⎣

99. Since you can take the cube root of any real number, the domain is ( ), −∞ ∞ or all real numbers.

x

9 10 11 12 13

0 2 4 6

x

−2−4−6

x

−8−9−10−11−12

x

0 1 2 3−1

x

0 1 2 3 4 5

x

0 1 2 3−1−2−3



101.

( )( )

2 15 54 06 9 0

x xx x− + ≥

− − ≥


Test intervals: ( ) ( ) ( ), 6 , 6, 9 , 9, −∞ − ∞

Domain: ( ] [ ), 6 9, −∞ − ∪ ∞

103.

( )

20 0

0 0

2

2

2

16Let 134 and 0.

16 134 27616 134 276 0

2 8 67 138 0

S t v t sv s

t tt t

t t

= − + +

= =

− + >

− + − >

− − + >

Critical numbers:

( ) ( ) ( )( )( )

267 67 4 8 138 67 732 8 16

4.72 or 3.65

t

t t

− − ± − − ±= =

≈ ≈

3.65 seconds 4.72 secondst< <

105. (a) 2

8

0.054 1.43 8

y

x x

<

− + <

(b)

( )( )( )

2

2

2

0.054 1.43 80.054 1.43 8 0

1.43 1.43 4 0.054 82 0.054

8.0 or 18.5

x xx x

x x

− + <

− + − <

− ± − − −=

−

≈ ≈

Solution set: ( ) ( )0, 8.0 18.5, ∪ ∞

The value of x must be greater than 0 and less than 8, or greater than 18.5.

(c) Because x = 15 is not part of the solution set, the player will not score a goal.

107.

( )

( ) ( )( )( )

2

2

2

2 260 2 2

60 2 230

150150

30 15030 150

30 150 0

30 30 4 1 1502 1

6.3 or 23.7

P l wl w

l wl wA

lwl l

l ll l

l

l l

= +

= +

− =

− =

≥

≥

− ≥

− ≥

− + ≤

± − −=

≈ ≈

Solution set: [ ]6.3, 23.7

The length must be between 6.3 feet and 23.7 feet.

109. ( )2 8

16

2000 1 42002

1 2.12

1 1.0474632

0.0474632

0.0964926 9.5%

r

r

r

r

r

⎛ ⎞+ >⎜ ⎟⎝ ⎠

⎛ ⎞+ >⎜ ⎟⎝ ⎠

+ >

>

> ≈

The interest must be greater than 9.5%.

111.

( ) ( )

( )( )( )

2

2

2

1,000,00075 0.0005 25 100,000 1,000,00075 0.0005 25 100,000 1,000,000

0.0005 50 1,100,000 0

50 50 4 0.0005 1,100,0002 0.0005

32,679 or 67,321

R Cx x x

x x xx x

x

x x

− ≥

− − + ≥

− − − ≥

− + − ≥

− ± − − −=

−

≈ ≈

Solution set: [ ]32,679, 67,321

( )( )

75 0.0005 32,679 $58.6675 0.0005 67,321 $41.34

pp= − ≈

= − ≈

The company should set the price between $41.34 and $58.66 to obtain a profit of at least $1,000,000.

113.

( )

20.0399 0.244 1.61, 6 15


R t t t

t

= − + ≤ ≤

=

(a) t 6 10 13 15

R 1.58 3.16 5.18 6.93


Chapter Test Solutions for Chapter 1 67

(b)

( ) ( )( )( )

2

2

2

8.800.0399 0.244 1.61 8.800.0399 0.244 7.19 0

0.244 0.244 4 0.0399 7.192 0.0399

10.7 or 16.8

Rt tt t

t

t t

≥

− + ≥

− − ≥

± − − −≈

≈ − ≈

Choose the positive value for t. So, according to the model, Sonic′s revenue per share will be at least $8.80 in 2007. The model supports the prediction.

(c)

( ) ( )( )( )

2

2

2

11.100.0399 0.244 1.61 11.100.0399 0.244 9.49 0

0.244 0.244 4 0.0399 9.492 0.0399

12.7 or 18.8

Rt tt t

t

t t

≥

− + ≥

− − ≥

± − − −≈

≈ − ≈

Choose the positive value for t. So, according to the model, Sonic′s revenue per share will be at least $11.10 in 2009. The model supports the prediction.

Chapter Test Solutions

1. ( ) ( )

1723

3 2 8 4 2 5 73 6 8 8 20 7

3 2 20 1523 17

x xx x

x xxx

+ − = − +

+ − = − +

− = − +

=

=

2. (a) Since you can take the cube root of any real number the domain is ( ), .−∞ ∞

(b)

( )( )

29 03 3 0

xx x

− ≥

− + ≥


Test intervals: ( ) ( ) ( ), 3 , 3, 3 , 3, −∞ − − ∞

Test: Is ( )( )3 3 0?x x− + ≥

Domain: [ ]3, 3−

3. Total profit April profit May profit625,509.12 0.92625,509.12 1.92

325,786299,723.12 0.92

x xx

xx

= +

= +

=

=

=

The profit in April was $325,786 and the profit in May was $299,723.12.

4.

( )( )

2

2

6 7 5

6 7 5 0

3 5 2 1 0

x x

x x

x x

+ =

+ − =

+ − =

5 13 2

3 5 0 or 2 1 03 5 or 2 1

or

x xx x

x x

+ = − =

= − =

= − =

5.

( )( )

212 5 2 03 2 4 0

x xx x

+ − =

+ − =

32

3 2 0 or 4 02 3 or 4

or 4

x xx xx x

+ = − =

= − − = −

= − =

6. 2

2

5 1015

15

xxx

− =

=

= ±

7. ( )

( )( )( )

2

2

2

2

5 310 25 313 25 0

13 13 4 1 252 1

13 169 1002

13 692

x xx x xx x

x

+ = −

+ + = −

+ + =

− ± −=

− ± −=

− ±=



8.

( ) ( ) ( )( )( )

2

2

2

3 11 23 11 2 0

11 11 4 3 22 3

11 121 246

11 1456

x xx x

x

− =

− − =

− − ± − − −=

± +=

±=

9. 25.4 3.2 2.5 0x x− − =

( ) ( ) ( )( )( )

23.2 3.2 4 5.4 2.52 5.4

3.2 10.24 5410.8

3.2 64.2410.8

3.2 64.24 1.03810.8

3.2 64.24 0.44610.8

x

x

x

− − ± − − −=

± ±=

±=

+= ≈

−= ≈ −

10.

7 132 2

2 3 102 3 10 or 2 3 10

2 7 or 2 13 or

xx x

x xx x

− =

− = − − =

= − =

= − =

11.

( )( )

2

2

3 53 53 25 100 11 280 4 7

x xx xx x x

x xx x

− + =

− = −

− = − +

= − +

= + −

4 0 or 7 0

4 or 7x x

x x− = − =

= =

x = 4 is the only solution. (Note: x = 7 is extraneous.)

12.

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

4 2

2 2

10 9 0

1 9 0

1 1 3 3 0

1, 1, 3, 3

x x

x x

x x x x

x

− + =

− − =

+ − + − =

= − −

13. ( )( )

2 32

22 3

2

2

2

2 2

9 9

9 9

9 7299 27

9 2736 or 18

6 No real solution

x

x

xx

xx xx

− =

− =

− = ±

− = ±

= ±

= = −

= ±

14. Revenue = xp = ( )40 0.0001x x−

( ) ( ) ( )( )( )

2

2

2

2,000,000 40 0.00010.0001 40 2,000,000 0

40 40 4 0.0001 2,000,0002 0.0001

40 8000.0002

40 20 2 341,421 units0.0002

or

40 20 2 58,579 units0.0002

x xx x

x

x

x

= −

− + =

− − ± − −=

±=

±= ≈

−= ≈

Since the equation is quadratic, it is possible to have two distinct solutions. In this case they would both have to be positive since x represents the number of units sold.

15. 3 1 25

3 1 103 9

3

x

xxx

+<

+ <

<

<

16.

285

4 5 24

4 5 24 or 4 5 245 28 or 5 20

or 4

x

x xx xx x

− ≥

− ≤ − − ≥

− ≤ − − ≥

≥ ≤ −

x

−1 0 1 2 3 4

6420 8

x

−2−4−6


Cumulative Test Solutions for Chapters 0–1 69

17. 3 27

3 2 07

xx

xx

+>

++

− >+

( )3 2 70

711 07

x xx

xx

+ − +>

+− −

>+

Critical numbers: 11, 7x = − −

Test intervals: ( ) ( ) ( ), 11 , 11, 7 , 7, −∞ − − − − ∞

Solution set: ( )11, 7− −

11 7x− < < −

18.

( )( )( )

3

2

3 12 0

3 4 0

3 2 2 0

x x

x x

x x x

− ≤

− ≤

+ − ≤

Critical numbers: 0, 2x = ±

Test intervals: ( ) ( ) ( ) ( ), 2 , 2, 0 , 0, 2 , 2, −∞ − − ∞

Solution set: ( ] [ ], 2 0, 2−∞ − ∪

2, 0 2x x≤ − ≤ ≤

19.

( ) ( )2

2

800,00090 0.0004 25 300,00 800,000

90 0.0004 25 300,000 800,0000.0004 65 1,100,000 0

R Cx x x

x x xx x

− ≥

− − + ≥

− − − ≥

− + − ≥

( )( )

( )

265 65 4 0.0004 1,100,0002 0.0004

19,189 or 143,311

x

x x

− ± − − −=

−

≈ ≈

Solution set: [ ]19,189, 143,311

The company must sell between 19,189 units and 143,311 units to obtain a profit of at least $800,000.

20.

( )

27.71 136.9 2433, 0 50 corresponds to 2000.

C t t tt= + + ≤ ≤

=

2

2

40007.71 136.9 2433 40007.71 136.9 1567 0

Ct tt t

>

+ + >

+ − >

( )( )

( )

2136.9 136.9 4 7.71 15672 7.71

25.7 or 7.9

t

t t

− ± − −=

≈ − ≈

Choose the positive value for t. So, average dormitory costs exceeded $4000 in 2008.

Cumulative Test Solutions

1. ( ) ( )32 6

6

4 2 4 8

32

x x

x

− = −

= −

2. 5 4 1

2

18 9 23 2

x x xx x

= ⋅ ⋅ ⋅

=

3. 2 2 3 53 5 3 5 3 5

6 2 59 5

6 2 54

3 52

+= ⋅

− − +

+=

−

+=

+=

4. ( ) ( )( )( )

( )( )( )

3 2 2

2

6 3 18 6 3 6

3 6

3 3 6

x x x x x x

x x

x x x

− − + = − − −

= − −

= − + −

5. ( )( )( )

2 4 4165 20 5 4

4, 45

x xxx x

x x

− +−=

− −

+= ≠

6.

1 11 1

1 1 1 1

, 0, 0

x y xyx yxy

y x y x

y x x yx y

⎛ ⎞−− ⎜ ⎟

⎝ ⎠= ⋅⎛ ⎞+ +⎜ ⎟⎝ ⎠−

= ≠ ≠+

x

−7−8−9−10−11

x

0 1 2 3−1−2−3



7.

( )11.9 243, 0 50 corrosponds to 2000.

C t tt= + ≤ ≤

=

(a) When ( )5: 11.9 5 243 302.5t C= = + =

The average monthly retail sales in 2005 was $302.5 billion.

(b) 36011.9 243 360

11.9 1179.8

Ct

tt

>

+ >

>

>

The average monthly retail sales will exceed $360,000,000,000 in 2010.

8.

( )( )

2

2

12

2 11 52 11 5 02 1 5 0

2 1 0 is a solution.5 0 5 is a solution.

x xx xx x

x xx x

− = −

− + =

− − =

− = ⇒ =

− = ⇒ =

9.

( ) ( )( )( )

2

2

5.2 1.5 3.9 0

1.5 1.5 4 5.2 3.92 5.2

0.734, 1.022

x x

x

x x

+ − =

− ± − −=

≈ ≈ −

10.

8 103 3

3 1 93 1 9 or 3 1 9

3 8 or 3 10 or

xx x

x xx x

+ =

+ = + = −

= = −

= = −

11.

( ) ( ) ( )( )( )

2

2

2

2 1 42 1 42 1 16 8

0 10 17

10 10 4 1 17 10 4 2

5 2 22.172, 7.828

x xx xx x x

x x

x

xx x

− + =

− = −

− = − +

= − +

− − ± − − ±= =

= ±

≈ ≈

The only solution is 5 2 2 2.172x = − ≈

12.

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

4 2

4 2

2 2

17 1617 16 0

1 16 0

1 1 4 4 0

1, 4

x xx x

x x

x x x x

x x

− = −

− + =

− − =

− + − + =

= ± = ±

13. ( )3 22

2 2 3

2

2

14 8

14 814 4

183 2

x

xx

xx

− =

− =

− =

=

= ±

14. 1 32 25

10 1 3 1011 3 911 33

1133

x

xx

x

x

−− < <

− < − <

− < − <

> > −

− < <

15.

( )( )( )

3

2

2 16 0

2 8 0

2 2 2 2 2 0

x x

x x

x x x

− ≥

− ≥

− + ≥

Critical numbers: 0, 2 2x x= = ±

Test intervals: ( ) ( )( ) ( )

, 2 2 , 2 2, 0 ,

0, 2 2 , 2 2,

−∞ − −

∞

Test: Is ( )( )2 2 2 0?x x x− + ≥

Solution set: )2 2, 0 , 2 2, ⎡ ⎤ ⎡− ∞⎣ ⎦ ⎣

16.

26 163 316 263 3

5 3 2121 5 3 2126 3 16

xx

xxx

− ≤

− ≤ − ≤

− ≤ − ≤

≥ ≥ −

− ≤ ≤

−3 −2 −1 0 1 2 3 4

x

x

−1−2−3−4 1 20 3 4

x

0−2−4−6 4 62 8 10


Cumulative Test Solutions for Chapters 0–1 71

17. 600,000

P R CP

= −

≥

( ) ( )2

2

2

120 0.0002 40 200,000 600,000120 0.0002 40 200,000 600,000

0.0002 80 200,000 600,0000.0002 80 800,000 600,000

x x xx x x

x xx x

− − + ≥

− − − ≥

− + − ≥

− + − ≥

( )( )

( )

280 80 4 0.0002 800,0002 0.0002

x− ± − − −

=−

10,263 or 389,737x x≈ ≈

Solution set: [ ]10,263, 389,737

To obtain a profit of at least $600,000, between 10,263 units and 389,737 units must be sold.

18.

( )

2228.57 323.3 34,808, 0 5


D t t t

t

= + + ≤ ≤

=

2

2

50,000228.57 323.3 34,808 50,000228.57 323.3 15,192 0

Dt tt t

>

+ + >

+ − >

( )( )

( )

2323.3 323.3 4 228.57 15,1922 228.57

8.89 or 7.48

t

t t

− ± − −=

≈ − ≈

Choose the positive value for t. Per capita gross domestic product will f000 during 2007.


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CHAPTER 1 Equations and Inequalitiescollege.cengage.com/mathematics/larson/college... · variable. For example, in Exercise 47, we multiplied by x(x − 3). If x or x − 3 are actually