Answer Key 61

Chapter 9

Lesson 9.1.1

9-2. a. Inequalities have multiple solutions, but equalities only have one solution. b. infinite c. The result of !1 " x " 4 does not extend infinitely. It has two endpoints.

The result of x2 ! 4 has two endpoints (2 and –2) and extends infinitely in both directions.

9-3. a. x ! 1 b. x < 1 c. x ! 3 d. ! 4 < x " 2

9-4. a. infinite b. 4 c. It is the solution to the equality 2x ! 5 = 3 ; it is the “starting point” for the solution. d. x ! 4 ,

4 x

9-5. a. Solve the corresponding equality for the variable. 3! 2x = 1, x = 1 . b. There should be an unfilled circle at x = 1 .

c. If the point is not a solution, then the solutions lie on the other side of the

boundary; x > 1; 1 2 3 40-1-2-3-4 x

9-6. x < 2 , 1 2 3 40-1-2-3-4 x

9-7. Let n = # of North American countries. Then n + (2n) + (2n + 7) = 122 and n = 23.

9-8. a. p > !1 b. k < 2 c. 1 ! k or k ! 1

9-9. a. k = 1.5 or –2 b. m = 3 or –3 c. w = 2 or –6 d. n " 2.12 or –0.79 9-10. a. always true b. sometimes true c. never true d. sometimes true

e. always true f. never true

9-11. 1; Any non-zero number divided by itself is 1.

9-12. a. (5, 0) and (8, 0); Robbie backed up 5m and the rocket landed 8m away from him. b. 3 m

9-13. a. 5± 13

2 " 0.7 or 4.3 b. !1± 7 " –3.6 or 1.6

62 Algebra Connections

Lesson 9.1.2

9-14. a. x < 5 b. m ! 2 c. p > !3 d. x ! 3

e. all numbers f. no possible solution

9-15. Let t = number of Turks and k = number of Kurds. Then t + k = 66,000,000 and

t = 4k . There are 13,200,000 Kurds and 52,800,000 Turks.

9-16. a. Since trees bloom when they are taller than 150 cm, an inequality is fitting. b. 6 + 9x > 150 , x > 16 , so the trees should be more than 16 years old.

c. 6 + 9x ! 240 , x ! 26 , so the trees should be less than 26 years old. Thus, the trees should be between 16 and 26 years of age or 26 years old, which can be written. 16 < x $ 26.

9-17. a. k < 2

b. p ! 15

c. n > 1

2

d. t ! 0

9-18. The graph should be a line with x-intercept (1.5, 0) and y-intercept (0, 3).

9-19. y = !3

5x ! 8

5

9-20. 3x2+ x !10

a. 3x ! 5 ; The product divided by one factor gives the other factor.

b. x + 2

9-21. x =5

3 and x = !

5

2

9-22. a. (2x ! 5)2

b. not possible c. 3x(x ! 4)

d. 5(x ! 4)(2x +1)

Answer Key 63

Lesson 9.2.1

9-23. a. Yes; the point is on the line and makes the equation true.

b. The point (2, –1) is a solution because it is on the line and makes the equation true. The point (0, 0) is not a solution because it is not on the line and does not make the equation true.

c. The points on the line make the equation true.

9-24. b. Any point above the line is a solution. Verify any questioned points by substituting them into the inequality. Yes; they make the inequality statement true. infinite Points below the line do not satisfy the inequality.

9-25. a. y ! "2x + 3 b. The boundary should be a dashed line.

9-26.

9-28.

a: 4 x

b:

- 3 x c:

2 x

d: 0 x

9-29. 1200 + 300x ! 2700 , so x ! 5 . Algeria can order an advertisement up to 5 inches high.

9-30. Let x = number correct for Part 1 and y = number correct for Part 2. Then x + y = 33

and 3x + 2y = 85 , and x = 19 and y = 14 . Rowan answered 19 problems correctly

on Part 1.

9-31. a. y = !2

7x ! 2

b. Yes; verify by substituting the coordinates into the equation and testing.

c. y = 7

2x ! 22

9-32. B

9-33. D

64 Algebra Connections

Lesson 9.2.2

9-34. a. Just one. If the point does not make the equation true, then the side that does not contain the point is the solution.

b. Zero conveniently eliminates the variable terms in the inequality. c. No; (–3, 2) lies on the line. She needs to test a point off the line.

9-35. a. y > 1

2x

b. Honduras, Zambia, Madagascar, and Uganda will receive aid. Rwanda is on the boundary, but the boundary is not included, so it is not a solution.

9-36.

9-37.

9-38. y < !1

3x !1

9-39. A

9-40. A

9-41. a. $16; Subtract and then make the result positive.

b. 42°

c. 360 students

d. The difference does not depend on which is greater. It represents how much greater the larger amount is than the smaller amount.

9-42. a. x = 1

3 b. x = 16 c. x = ±5 d. x > 5

9-43. No; Bernie would pass Barnaby after 40 seconds, when each was 90 meters from the starting line. Since the race was only 70 meters, that would occur after the race was over.

9-44. x = 0.6 or x = –2

Answer Key 65

Lesson 9.2.3

9-45. a. The absolute-value relation always returns the positive value of the input. b. You need it to make something positive, like finding a distance or difference, such

as those in problem 9-41 in homework.

9-46. a. Add –5 and 1 and then change the result to a positive value. Then subtract 3. The result is 1.

b. i. 2, ii. 30, iii. 13, iv. 7 c. Answers vary.

9-47. either 15 or –15; yes

9-48. a. b. Answers will vary.

9-49. She should add 1 first, since the addition is placed inside the absolute value symbol, which acts as a grouping symbol.

9-50.

9-51. 1,400,000 ! 50x > 1,200,000 , less than 4000 square miles per year

9-52. a. 1 b. 2 c. –11 d. 28

9-53. no; 3(7 ! 2) = 15 and 15 > 4

9-54.

9-55. 718 !14x = 212 + 32x , x = 11 months

9-56. b. Answers vary, but using data from games 1 and 3. y = 6x .

c. " 90 baskets

x

66 Algebra Connections

Lesson 9.3.1

9-57. a. See graphs at right. c. The point (0, 0) satisfies both inequalities.

d. There are multiple ways to justify that the region found in part (b) is the only possible solution. One way is to test points in every other region to demonstrate that they do not make both inequalities true.

9-58.

9-59. a. y = x2 + x ! 6 is part of y ! x2 + x " 6 , while y = 2

3x is not part of y > 2

3x .

b. 5 c. 2.

d. Because it lies on one of the boundary graphs. It will not tell you anything about the regions separated by the boundaries.

9-60.

9-61. y ! x + 3 , y ! 1

2x "1 , y ! 2 , y ! "

2

3x + 4 , y ! "

2

3x "1

9-62. a. 3 b. 1 c. 4 d. 2

9-63.

a. b.

c. d. no solution

9-64. All equal 1

9-65. a. (5x ! 2)(x + 3) b. 2(3t !1)(t ! 4) c. 6(x ! 2)(x + 2)

9-66. 2a + 3c = 27.75 , 3a + 2c = 32.25 , a = $8.25 , c = $3.75

9-67. B

0 1 2 3 -1 -2 -3 -4 -5 x 7 8 9 10 6 5 4 3 2 x

0 1 2 3 -1 -2 -3 -4 -5 m 0 1 2 3 -1 -2 -3 -4 -5 x

Answer Key 67

Lesson 9.3.2

9-68.

9-69. y ! "1

2x + 2000 , y ! 2

3x " 2000 , y ! "

1

5x ; The search-and-rescue teams should

search near the island of Samoa.

9-70. a. the region between and including the parallel lines

b. the region below and including the boundary line y = 2

3x

c. no solution

9-72. 9-73.

9-74. a. x ! 6 b. x > 1 c. 2 ! x < 7 d. !3 " x " !1

9-75. a. false b. false c. true d. false

9-76. C

9-77. a. r + 2.50c ! 15 b. r + c ! 25

c. No; the club cannot sell a negative number of items.

d. See graph. The solution points represent the possible sales of rulers and compasses that would allow the club to break even or make a profit while falling within the sales limit.

3 6–3–6

3

6

–3

–6

x

y

68 Algebra Connections

Lesson 9.3.3

9-78. A. y ! 500 "1.5x , B. y ! 300 " 0.25x ,

C. y > 180 ! 2

3x , D. y > 450 ! 5

2x ,

E. x < 250 , Special Assignment. x ! 100

9-79. a. 210; estimate from the graph or substitute 185 for y into the equation for Country A

( y = !3

2x + 500 ) and solve for x since that line is the upper boundary when

x = 185 .

b. 275; It is the point in the region with the largest y-value. One way to determine this is to substitute 100 for x into the equation for Country B and solve for y.

9-80. 200 of each should be requested; graph the line y = x and find its intersection with

the line for Country A.

9-81. a. Yes, they are equivalent. One way to determine this is to change both to y = mx + b form and compare slope and y-intercept.

b. Answers vary. Multiply or divide both sides of either equation to find an

equivalent equation. For example, 2x + y = 3 and 8x + 4y = 12 are both

equivalent equations.

9-82. 3280x +1500 < 50,000 , less than 14.8 pounds

9-83. a. m > 5 b. x ! "6 c. x > 7 d. no solution

9-84. b. No, it is not; it lies on both boundaries, but the boundary to y < x is not part of the solution.

9-85. x = 9 or x = 0.5

9-86. D

50 100 150 200 250 300

50

100

150

200

250

300

350

400

450

500

A

D

B

C

E

Special

Assignment

Number of Food Packages

Num

ber

of

Med

icin

e P

ackag

es