CHAPTER Inequalities 2 x Solutions Key - 9th grade...

InequalitiesSolutions Key

Are You reAdY?

1. B 2. E

3. F 4. D

5. C 6. b - a = 6 - 2 = 4

7. ab = (2)(6) = 12

8. b ÷ a = 6 ÷ 2 = 3

9. a + b = 2 + 6 = 8

10. 10 < 21

11. 5.27 > 5.23 12. 20% = 0.2

13. 1 _ 3 < 2 _

5 14. 6x + x

= 7x

15. -8a + 3a = -5a

16. 9 x 2 - 15 x 2 = -6 x 2

17. 2.1x + 4.3x = 6.4x

18. 2(x + 3) = 2(x) + 2(3) = 2x + 6

19. (3 - d)5 = (3)5 + (-d)5 = 15 - 5d

20. 4(r - 1) = 4(r) + 4(-1) = 4r - 4

21. 3(4 + m) = 3(4) + 3(m) = 12 + 3m

22. s - 3 = 8 ____ + 3 ___ +3 s = 11

23. -7x = 21

-7x _ -7

= 21 _ -7

x = -3

24. y + 11 = 2 _____ - 11 ____ -11 y = -9

25. h _ 2 = 6

(2) h _ 2 = (2)6

h = 12

26. t + 2 = -2 ____ - 2 ___ -2 t = -4

27. 6x = 42

6x _ 6 = 42 _

6

x = 7

28. r - 8 = -13 ____ + 8 ____ +8 r = -5

29. y _

3 = -12

(3) y _

3 = (3)(-12)

y = -36

GrAphInG And wrItInG InequAlItIes

CheCk it out!

1. all real numbers greater than 4

2a. 2 3 4 2 1 __

2 3 1 __

2

b. 2 2 - 4 ≥ w 4 - 4 ≥ w 0 ≥ w w ≤ 0

0 1 2 3 -1 -2 -3

c. 0 -1 -2 -3 -4 -5 -6

3. x < 2.5

4. Let d represent the amount the employee can earn per hour.

d ≥ 8.25 2 4 6 8

8.25

10 12 0

think and disCuss

1. Both graphs include the real numbers greater than 2. The graph of x ≥ 2 also includes 2.

2. Inequality Graph

x > 1

x ≤ -3

0 1 2 5 3 4 -1

-4 -3 -2 1 -1 0 -5

exerCisesguided practice

1. A solution of an inequality makes the inequality true when substituted for the variable.

2. all real numbers greater than or equal to 11

3. all real numbers greater than -3

4. all real numbers less than 4


6. 0 -6 -5 -4 -3 -2 -1

7. 3 4 5 3 1 __

2 4 1 __

2

8. (4 - 2 ) 3 > m 2 3 > m 8 > m m < 8

2 4 6 8 10 12 0

9. p ≥ √ 17 + 8 p ≥ √ 25 p ≥ 5

1 2 3 4 5 6 0

10. a ≤ -2 11. b > -8 1 _ 2

12. c < 5.5 13. d < -7

14. e ≥ 3 15. f ≤ 14

16. Let m represent the number of members present. m ≥ 20, where m is a whole number

18 19 20 21 22 23 24

17. Let r represent the athlete’s heart rate. r < 140, where r is positive

120 160

140

20080400 240

41 Holt McDougal Algebra 1

xCHAPTER

x-1


2CHAPTER

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practice and problem Solving




21. all real numbers less than or equal to 12

22. 5 6 7 8 9

23. 0 -1 -2 - 1 __

2 -1 1 __

2

24. d > 4(5 - 8) d > 4(-3) d > -12

0 -9 -6 -3 -12

25. t ≤ 3 2 - 2 2 t ≤ 9 - 4 t ≤ 5

1 0 2 3 6 4 5

26. u ≥ 5 27. v < -11

28. w > -3.5 29. x > -3.3

30. y < 4 31. z ≥ 9

32. Let s represent the speed allowed in miles per hour. s ≤ 25, where s is nonnegative

50 10 15 20 25 30

33. Let y represent the number of years of experience. y ≥ 5

1 0 2 3 6 4 5

34. x is greater than 7. 35. h is less than -5.

36. d is less than or equal to 23.

37. r is greater than or equal to -2.

38. g > 19

18 17 19 20 21

39. p ≤ 17

16 15 17 18 19

40. e > 10

5 0 10 15 20

41. f > 0

1 2 3 0 -3 -2 -1

42. Let t represent the temperature on Earth in °F. t ≤ 135.9

50

135.9

100 150 200 0

43. Let p represent the profits. p < 10,000

10,000 0

44. Let h represent the height in inches. h ≥ 46

45 46 47 48 44

45. Let e represent the elevation in feet. e ≤ 5000

2500 5000 0

46. Possible answer: x represents the distance in miles between two locations.

47. Possible answer: x represents the age in years of a child at a childcare center where x is positive.

48. Possible answer: x represents the hour on an analog clock when x is a natural number.

49. Possible answer: x represents the number of millions of albums sold by a popular band.

50. A 51. D

52. B 53. C

54. A is incorrect; it should be drawn with an empty circle.

55a. Let s represent the amount she can spend. s ≤ 125 - 90 s ≤ 35, where s is nonnegative

b. 50 15 25 35

c. s ≤ 35 - 15 s ≤ 20 where s

is nonnegative

56. 1 0 2 3 6 4 5

57. Look for a solid or empty circle. A solid circle tells you to use ≤ or ≥, and an empty circle tells you to use < or >. Then look at the direction of the arrow. An arrow pointing left tells you to use < or ≤, and an arrow pointing right tells you to use > or ≥.

58a. less than or equal to b. greater than or equal to

teSt prep

59. D; Since 5 - 2(5) = 5 - 10 = -5 and -5 ≱ -3, 5 is not a solution of the inequality 5 - 2x ≥ -3.

60. F; Since 3 - 1 = 2 and 2 ≮ 2, 1 is not a solution of the inequality 3 - x < 2.

61. C; Try t = -5. Since 1 - (-5) = 6 and -2 ≤ 6, t = -5 is a solution so it should be shaded. Try t = 0. Since 1 - 0 = 1 and -2 ≤ 1, t = 0 is a solution so it should also be shaded. Therefore C is correct.

challenge and extend

62. all real numbers

63. any nonzero numbers such that x and y are the same sign and |x| < |y|

64. any numbers such that y is less than or equal to x

65. < 66. >

67. any number between 0.35 and 1.27

68. yes; infinitely many

69. Draw an empty circle at 5. Then draw arrows going left and right from 5.

solvInG InequAlItIes bY AddInG or subtrActInG

CheCk it out!

1a. s + 1 ≤ 10 ____ - 1 ___ -1 s + 0 ≤ 9 s ≤ 9

3 0 6 9 12

b. 2 1 _ 2 > -3 + t

___ +3 ______ +3

5 1 __ 2 > 0 + t

t < 5 1 __ 2

4 5 6 4 1 __ 2

5 1 __ 2


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c. q - 3.5 < 7.5 ______ + 3.5 ____ +3.5 q - 0 < 11 q < 11

10 11 12 13 9

2. Let m represent the number of milligrams of iron. 11 + m ≤ 15 ________ -11 ____ -11 m ≤ 4, where m is nonnegative Sarah can consume 4 mg or less without exceeding

the RDA.

3. Let p represent the number of more pounds he needs to break the school record.

250 + p > 282 ________ -250 _____ -250 p > 32 Josh needs to bench press more than 32 additional

pounds to break the school record.

think and disCuss

1. Substitute the endpoint into the related equation. Then substitute a value in the solution region into the original inequality.

_____________ d - 3 = -6 ______________ d - 3 > -6 (-3) - 3 -6 (0) - 3 > -6 -6 -6 3 -3 > -6 3

2. You can add or subtract the same number on both sides of an equation or inequality, and the statement will still be true.

3.

11

Properties of Inequality

12 13 14 15

Addition: a - 4 < 9 + 4 + 4

a < 13

4 5 6 7 3

Subtraction: a + 5 > 10 - 5 - 5

a > 5


1. 12 6

2 4 6 12 8 10 0

2. w + 3 ≥ 4 _____ - 3 ___ -3 w ≥ 1

1 2 3 0 -3 -2 -1

3. -5 + x ≤ -20 ______ +5 ____ +5 x ≤ -15

-20 -15 -10 -5 0

4. z - 2 > -11 ____ + 2 ____ +2 z > -9

-12 -9 -6 -3 0

5. 102 + t ≤ 104 ________ -102 _____ -102 t ≤ 2 where t is nonnegative

1 2 30-3 -2 -1

The temperature can increase no more than 2°F.

6. Let d represent the additional amount you need to spend for the restaurant to deliver.

17.95 + d ≥ 25.00 __________ -17.95 ______ -17.95 d ≥ 7.05 You must spend at least $7.05 more for the

restaurant to deliver.


7. a - 3 ≥ 2 _____ + 3 ___ +3 a ≥ 5

1 2 3 4 5 6 0

8. 2.5 > q - 0.8 ____ +0.8 ______ + 0.8 3.3 > q q < 3.3

1 2 3

3.3

4 0

9. -45 + x < -30 _______ +45 ____ +45 x < 15

5 0 10 15 20

10. r + 1 _ 4

≤ 3 _ 4

_____

- 1 _ 4

___

- 1 __ 4

r ≤ 1 _ 2

1 0 -1 1 __ 2

- 1 __ 2

11. 1400 + 243 + w ≤ 2000 1643 + w ≤ 2000 __________ -1643 ______ -1643 w ≤ 357 However, weight is nonnegative.

0 357 The crate cannot weigh more than 357 pounds.

12. 7 + g ≤ 15 ______ -7 ___ -7 g ≤ 8 where g is nonnegative Mindy can add no more than 8 gallons to the tank.

13. x - 10 > 32 _____ + 10 ____ +10 x > 42

41 40 42 43 44

14. n + 6 ≤ 4 _____ - 6 ___ -6 n ≤ -2

1 2 3 0 -3 -2 -1

15. r - 13 ≤ 15 _____ + 13 ____ +13 r ≤ 28

27 26 28 29 30

16. x + 4 ≤ 2 ____ - 4 ___ -4 x ≤ -2

1 2 3 0 -3 -2 -1

17. -12 + q > 39 _______ +12 ____ +12 q > 51

50 49 51 52 53

18. x + 3 _ 5

< 7

_____

- 3 _ 5

___

- 3 __ 5

x < 6 2 _ 5

7 6 6 1 __

5 6 2 __

5 6 3 __

5 6 4 __

5

19. 4.8 ≥ p + 4 ___ -4 _____ - 4 0.8 ≥ p p ≤ 0.8

1.6 0 0.8

20. -12 ≤ x - 12 ____ +12 ______ + 12 0 ≤ x x ≥ 0

1 2 3 0 -3 -2 -1

43 Holt McDougal Algebra 1 43 Holt McDougal Algebra 1

CS10_A1_MESK710372_C02.indd 43 3/30/11 10:34:45 PM

21. 4 < 206 + c _____ -206 ________ -206 -202 < c c > -202

-203 -202 -201 -200

22. y - 1 _ 3 > 2 _

3

_____

+ 1 _ 3

___ + 1 _

3

y > 1

1 2 3 0 -3 -2 -1

23. x + 1.4 ≥ 1.4 ______ - 1.4 ____ -1.4 x ≥ 0

1 2 3 0 -3 -2 -1

24a. 8

b. 22 c. 0

25. Let d represent the number of days he can still wear his contact lenses.

21 + d ≤ 30 _______ -21 ____ -21 d ≤ 9 where d is nonnegative

3 6 9 12 0

Alex can wear his contact lenses for up to 9 more days.

26. 1 ≤ x - 2 ___ +2 _____ + 2 3 ≤ x x ≥ 3; C

27. 8 > x - (-5) 8 > x + 5 ___ -5 _____ - 5 3 > x x < 3; B

28. x + 6 > 9 ____ - 6 ___ -6 x > 3; A

29. -4 ≥ x - 7 ___ +7 _____ + 7 3 ≥ x x ≤ 3; D

30. It is a reasonable answer. If you round each number to the nearest whole number and then solve the inequality 12 + x < 22, the solution is x < 10.

31. 936 + 4254 + p ≤ 45,611 5190 + p ≤ 45,611 _________ -5190 ______ -5190 p ≤ 40,421 where p is nonnegative There can be at most 40,421 people in the other

types of seats.

32. Possible answer: If a scale is unbalanced and the same amount is added to or subtracted from both sides, the scale should maintain the same amount of imbalance.

33. When you isolate the variable in each inequality, you get x ≥ 2 and x ≥ 2.

34. Both inequalities have all numbers greater than 1 as solutions. x + 2 ≥ 3 also includes 1. The graph of

x + 2 > 3 has an empty circle at 1, but the graph of x + 2 ≥ 3 has a solid circle at 1.

35a. 411 + 411 = 822 miles

b. 822 + m ≤ 1000

c. 822 + m ≤ 1000 _________ -822 _____ -822 m ≤ 178, where m is nonnegative.

50 100 150

178

200 0

Possible answer: Washington is about 800 miles away. Daryl is willing to drive 1000 miles. So he can drive up to 1000 - 800 = 200 miles in Washington. Therefore 178 is reasonable.

teSt prep

36. A; If you round each number and then solve the inequality 5 + p < 20, the solution is p < 15.

37. F; If x is the number of bottles Dave has, then x + 3 represents the number of bottles Sam and

Dave have, which is at most 12. Therefore, situation F represents x + 3 ≤ 12.

38. A; Solving gives p < -2 which is graph A.

39. J; The solutions of n + 12 ≤ 26 are n ≤ 14. However the solutions of n - 12 ≤ 14 are n ≤ 2. So J does not have the same soltuions of n + 12 ≤ 26.


40. 6 9 ___ 10

≥ 4 4 __ 5 + x

69 _ 10

≥ 48 _ 10

+ x

_____

- 48 ___ 10

________

- 48 _ 10

21 ___ 10

≥ x

x ≤ 2 1 ___ 10

2 2 4 __ 10

2 3 __ 10

2 2 __ 10

2 1 __ 10

41. r - 1 2 _ 5 ≤ 3 7 ___

10

r - 14 _ 10

≤ 37 _ 10

______

+ 14 _ 10

____

+ 14 _ 10

r ≤ 51 _ 10

r ≤ 5 1 ___ 10

5 5 4 __ 10

5 3 __ 10

5 2 __ 10

5 1 __ 10

42. 6 2 _ 3 + m > 7 1 _

6

40 _ 6 + m > 43 _

6

________

- 40 _ 6

____ - 40 _

6

m > 3 _ 6

m > 1 _ 2

1 0 -1 1 __

2 - 1 __

2

43. Sometimes; when b is positive, the inequality is true.

44. Always; adding the same number to both sides keeps the statement true.

45. Always; a and c are the two greater numbers, so their sum will be greater than that of the two lesser numbers.

46. b and c are equal.


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solvInG InequAlItIes bY multIplYInG or dIvIdInG

CheCk it out!

1a. 4k > 24

4k _ 4 > 24 _

4

k > 6

2 4 6 8 10 12 0

b. -50 ≥ 5q

-50 _ 5 ≥

5q _

5

-10 ≥ q q ≤ -10

-10 -5 0

c. 3 _ 4 g > 27

4 _ 3 ( 3 _

4 g) > 4 _

3 (27)

g > 36

35 36 37 38 34

2a. 10 ≥ -x -1(10) ≤ -1(-x) -10 ≤ x x ≥ -10

-10 -5 0

b. 4.25 > -0.25h

4.25 _ -0.25

< -0.25h _ -0.25

-17 < h h > -17

-18 -17 -16 -15 -19

3. Let g represent the number of 10-ounce servings. 10g ≤ 128

10g

_ 10

≤ 128 _ 10

g ≤ 12.8 Since only a whole number of glasses can be filled,

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, or 12 10-ounce servings can be filled.

think and disCuss

1. They are alike because you can multiply or divide by a postive number on both sides and the inequality or equation will still be true. They are different because you have to reverse the inequality symbol if you multiply or divide by a negative number.

2. Solving Inequalities by Using Multiplication and Division

Divide

Multiply

x ≤ 16

-3x > 9

x < -3

<

x ≤ -10

By a positive number By a negative number

2x < 8

x < 4

< 2x __ 2

8 __ 2

≤ 4 x __ 4

(4) ≤ 4(4) x __ 4

-3x __ -3

9 __ -3

≥ 2 x __ -5

(-5) ≤ 2(-5) x __ -5


1. 3b > 27

3b _ 3 > 27 _

3

b > 9

3 6 9 12 0

2. -40 ≥ 8b

-40 _ 8 ≥ 8b _

b

-5 ≥ b b ≤ -5

0 -20 -15 -10 -5

3. d _ 3 > 6

3 ( d _ 3 ) > 3(6)

d > 18

0 6 12 18 24 30

4. 24d ≤ 6

24d _ 24

≤ 6 _ 24

d ≤ 1 _ 4

1 0 1 __ 4

2 __ 4

3 __ 4

5. 1.1m ≤ 1.21

1.1m _ 1.1

≤ 1.21 _ 1.1

m ≤ 1.1

1.3 1.4 1.2 1 1.1

6. 2 _ 3

k > 6

3 _ 2

( 2 _ 3

k) > 3 _ 2

(6)

k > 9

3 6 9 12 0

7. 9s > -18

9s _ 9 > -18 _

9

s > -2

1 2 3 0 -3 -2 -1

8. 4 _ 5

≥ r _ 2

2 ( 4 _ 5

) ≥ 2 ( r _ 2

)

8 _ 5

≥ r

r ≤ 8 _ 5

2 1 6 __ 5

7 __ 5

8 __ 5

9 __ 5

9. -2x < -10

-2x _ -2

> -10 _ -2

x > 5

1 2 3 4 5 6 0

10. b _ -2

≥8

-2 ( b _ -2

) ≤ -2(8)

b ≤ -16

-18 -17 -16 -15 -14

11. -3.5n < 1.4

-3.5n _ -3.5

> 1.4 _ -3.5

n > -0.4

-0.6 -0.5 -0.4 -0.3 -0.2

12. 4 > -8g

4 _ -8

< -8g

_ -8

- 1 _ 2

< g

g > - 1 _ 2

1 0 -1 1 __ 2

- 1 __ 2

13. d _ -6

< 1 _ 2

(-6) d _ -6

> (-6) 1 _ 2

d > -3

0 -6 -5 -4 -3 -2 -1

14. -10h ≥ -6

-10h _ -10

≤ -6 _ -10

h ≤ 0.6

0 0.2 0.4 0.6 0.8

15. 12 > t _ -6

-6(12) < -6 ( t _ -6

)

-72 < t t > -72

-73 -72 -71 -70 -69

16. - 1 _ 2

m ≥ -7

-2 (- 1 _ 2

m) ≤ -2(-7)

m ≤14

12 13 14 15 16


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17. Let n represent the number of nights he can stay. 80n ≤ 550

80n _ 80

≤ 550 _ 80

n ≤ 6.875 Tom can stay only a whole number of nights. So

Tom can stay 0, 1, 2, 3, 4, 5, or 6 nights.


18. 10 < 2t

10 _ 2 < 2t _

2

5 < t t > 5

1 2 3 4 5 6 0

19. 1 _ 3 j ≤ 4

3 ( 1 _ 3 j) ≤ 3(4)

j ≤ 12

0 4 8 12 16

20. -80 < 8c

-80 _ 8 < 8c _

8

-10 < c c > -10

0 -20 -15 -10 -5

21. 21 > 3d

21 _ 3 > 3d _

3

7 > d d < 7

5 6 7 8 9

22. w _ 4 ≥ -2

4 ( w _ 4 ) ≥ 4(-2)

w ≥ -8

-12 -8 -4 0

23. h _ 4 ≤ 2 _

7

4 ( h _ 4 ) ≤ 4 ( 2 _

7 )

h ≤ 8 _ 7

2 1 0

8 __ 7

24. 6y < 4.2

6y

_ 6 < 4.2 _

6

y < 0.7

0.6 0.7 0.8 0.9 1

25. 12c ≤ -144

12c _ 12

≤ -144 _ 12

c ≤ -12

-14 -13 -12 -11 -10

26. 4 _ 5 x ≥ 2 _

5

5 _ 4 ( 4 _

5 x) ≥ 5 _

4 ( 2 _

5 )

x ≥ 1 _ 2

0 1 2 1 __ 2

1 1 __ 2

27. 6b ≥ 3 _ 5

1 _ 6 (6b) ≥ 1 _

6 ( 3 _

5 )

b ≥ 1 _ 10

0 1 __ 10

2 __ 10

3 __ 10

4 __ 10

28. -25 > 10p

-25 _ 10

> 10p

_ 10

-2.5 > p p < -2.5

-4 -3.5 -3 -2.5 -2

29. b _ 8 ≤ -2

8 ( b _ 8 ) ≤ 8(-2)

b ≤ -16

-32 -24 -16 -8 0

30. -9a > 81

-9a _ -9

< 81 _ -9

a < -9

-12 -9 -6 -3 0

31. 1 _ 2 < r _

-3

-3 ( 1 _ 2 ) > -3 ( r _

-3 )

- 3 _ 2 > r

r < - 3 _ 2

0 -1 -2 - 1 __

2 - 3 __

2

32. -6p > 0.6

-6p

_ -6

< 0.6 _ -6

p < -0.1

0 -0.4 -0.3 -0.2 -0.1

33. y _

-4 > -1 _

2

-4 ( y _

-4 ) < -4 (- 1 _

2 )

y < 2

1 2 3 0 -3 -2 -1

34. - 1 _ 6 f < 5

-6 (- 1 _ 6 f) > -6(5)

f > -30

0 -40 -30 -20 -10

35. -2.25t < -9

-2.25t _ -2.25

> -9 _ -2.25

t > 4

1 2 3 4 5 6 0

36. 24 ≤ -10w

24 _ -10

≥ -10w _ -10

-2.4 ≥ w w ≤ -2.4

-2.4 -2 -2.6 -2.2 -2.8

37. -11z > 121

-11z _ -11

< 121 _ -11

z < -11

-13 -12 -11 -10 -9

38. 3 _ 5 < f _

-5

-5 ( 3 _ 5 ) > -5 ( f _

-5 )

-3 > f f < -3

0 -6 -5 -4 -3 -2 -1

39. -k ≥ 7 -1(-k) ≤ -1(7) k ≤ -7

-9 -8 -7 -6 -5

40. -2.2b < -7.7

-2.2b _ -2.2

> -7.7 _ -2.2

b > 3.5

3 3.5 4 4.5 5

41. 16 ≥ - 4 _ 3 p

- 3 _ 4 (16) ≤ - 3 _

4 (- 4 _

3 p)

-12 ≤ p p ≥ -12

0 -16 -12 -8 -4

42. Let r represent the number of pieces of rope. 18r ≤ 54

18r _ 18

≤ 54 _ 18

r ≤ 3 Roz can only cut a whole number of pieces, so she

can cut 0, 1, 2, or 3 pieces of rope.

43. -8x < 24

-8x _ -8

> 24 _ -8

x > -3

0 -6 -5 -4 -3 -2 -1

44. 3t ≤ 24

3t _ 3 ≤ 24 _

3

t ≤ 8

0 2 4 6 8 10 12


CS10_A1_MESK710372_C02.indd 46 3/30/11 10:35:02 PM

45. 1 _ 4 x < 5

4 ( 1 _ 4 x) < 4(5)

x < 20

0 10 20 30 40

46. 4 _ 5 p ≥ -24

5 _ 4 ( 4 _

5 p) ≥ 5 _

4 (-24)

p ≥ -30

0 -40 -30 -20 -10

47. 54 ≤ -9p

54 _ -9

≥ -9p

_ -9

-6 ≥ p p ≤ -6

0 -8 -6 -4 -2

48. 3t > - 1 _ 2

1 _ 3 (3t) > 1 _

3 (- 1 _

2 )

t > - 1 _ 6

0 - 4 __ 6

- 3 __ 6

- 2 __ 6

- 1 __ 6

49. - 3 _ 4 b > - 3 _

2

- 4 _ 3 (- 3 _

4 b) < - 4 _

3 (- 3 _

2 )

b < 2

1 2 3 0 -3 -2 -1

50. 216 > 3.6r

216 _ 3.6

> 3.6r _ 3.6

60 > r r < 60

0 20 40 60 80

51. 7x ≥ 21

7x _ 7 ≥ 21 _

7

x ≥ 3

0 1 2 3 4 5 6

52. h _ -6

≥ 5

-6 ( h _ -6

) ≤ -6(5)

h ≤ -30

0 -40 -30 -20 -10

53. - 4 _ 5 b ≤ -16

- 5 _ 4 (- 4 _

5 b) ≥ - 5 _

4 (-16)

b ≥ 20

0 10 20 30 40

54. 10 ≤ t _ 4

4(10) ≤ 4 ( t _ 4 )

40 ≤ t t ≥ 40

0 20 40 60 80

55. You reverse the symbol only when you mulitply or divide both sides of the inequality by the same negative number.

56. A ≤ 21 ℓw ≤ 21 ℓ(3.5) ≤ 21

3.5ℓ _ 3.5

≤ 21 _ 3.5

ℓ ≤ 6 where ℓ is positive The length of the rectangle is at most 6 inches.

57. -0.5t ≥ 1.5

-0.5t _ -0.5

≤ 1.5 _ -0.5

t ≤ -3; C

58. 1 _ 9 t ≤ -3

9 ( 1 _ 9 t) ≤ 9(-3)

t ≤ -27; D

59. -13.5 ≤ -4.5t

-13.5 _ -4.5

≥ -4.5t _ -4.5

3 ≥ t t ≤ 3; A

60. t _ -6

≤ - 1 _ 2

-6 ( t _ -6

) ≥ -6 (- 1 _ 2 )

t ≥ 3; B

61. Let b represent the number of bags. The shelter needs at least 10 lb of cat chow per

week. Since there are 52 weeks in a year, the shelter needs at least 52 × 10 = 520 lb of cat chow per year.

20b ≥ 520

20b _ 20

≥ 520 _ 20

b ≥ 26 The shelter needs at least 26 bags of cat chow per

year.

62. Let p represent the possible number of points a student could earn.

0.90p ≤ 567

0.90p

_ 0.90

≤ 567 _ 0.9

p ≤ 630 where p is nonnegative A student could earn no more than 630 points.

63. Multiplying both sides of an inequality by zero makes both sides equal zero, so there is no longer an inequality to solve.

64. A is incorrect. Both sides are divided by a positive number, so the inequality symbol should not be reversed.

65. Let g represent the number of guests. 12.5g ≤ 800

12.5g

_ 12.5

≤ 800 _ 12.5

g ≤ 64 where g is nonnegative Jan can invite up to 64 guests.

66a. 75n ≤ 250

b. 75n ≤ 250

75n ____ 75

≤ 250 ____ 75

n ≤ 3.33 The club can reserve

up to 3 rooms.

2 1 0 3 4

c. 65n ≤ 250

65n ____ 65

≤ 250 ____ 65

n ≤ 3.85 No, they still can

reserve only 3 or fewer rooms.

teSt prep

67. B; - 2 _ 3 y > 4 has the solution y < -6 but

y _

2 < -12

has the solution y < -24.

68. G; The graph represents x ≥ -4 but the solution to -5x ≥ 20 is x ≤ -4.

69. B; Since s represents the number of stamps, the total cost of the stamps is 0.39s and this must be less than or equal to 4.


CS10_A1_MESK710372_C02.indd 47 3/30/11 10:35:05 PM

70. Answers may vary. 2x > 8 2x _

2 > 8 _

2 Divide both sides by 2.

x > 4

1 _ 4 x > 1

4 ( 1 _ 4 x) > 4(1) Multiply both sides by 4.

x > 4 3x > 12 3x _

3 > 12 _

3 Divide both sides by 3.

x > 4


71. 2 1 _ 3 ≤ - 5 __

6 g

- 6 _ 5 ( 7 _

3 ) ≥ - 6 _

5 (- 5 _

6 g)

- 14 ___ 5 ≥ g

g ≤ - 14 ___ 5

72. 2x _ 3 < 8.25

3 _ 2 ( 2 _

3 x) < 3 _

2 (8.25)

x < 12.375

73. 2 5 _ 8 m > 7 _

10

21 _ 8 m > 7 _

10

8 _ 21

( 21 _ 8 m) > 8 _

21 ( 7 _

10 )

m > 4 _ 15

74. 3 3 _ 5 f ≥ 14 2 _

5

18 _ 5 f ≥ 72 _

5

5 _ 18

( 18 _ 5 f) ≥ 5 _

18 ( 72 _

5 )

f ≥ 4

75. Round to 4x ≤ 20. 4x ≤ 20

4x _ 4 ≤ 20 _

4

x ≤ 5So, the greatest possible integer solution is x = 5

76. yes; 2 < 3 and 3 < 4 → 2 < 4

77. no; 0 < 1 but 1 ≮ 0

reAdY to Go on? section A quiz

1. all real numbers greater than -2




5. 1 2 3 0 -3 -2 -1

6. 0 1 2 1 __

2 1 1 __

2

7. g < √ 8 + 1 g < √ 9 g < 3

6 5 4 3 2 1 0

8. h ≥ 2 3 h ≥ 8

8 12 16 4 0

9. x ≥ -3 10. y < 5

11. z ≤ -1.5 12. t ≥ 5, where t is a whole number

3 4 5 76210

13. a < 13, where a is nonnegative

15 13 10 5 0

14. m ≤ 250, where m is nonnegative

150 50 250 0

15. k + 5 ≤ 7 ____ - 5 ___ -5 k ≤ 2

1 2 3 0 -3 -2 -1

16. 4 > p - 3 ___ +3 _____ + 3 7 > p p < 7

8 9 7 6 5

17. r - 8 ≥ -12 ____ + 8 ____ +8 r ≥ -4

0 -6 -5 -4 -3 -2 -1

18. -3 + p < -6 ______ +3 ___ +3 p < -3

0 -6 -5 -4 -3 -2 -1

19. Let b represent the number of gift baskets. 36 + b ≥ 50 _______ -36 ____ -36 b ≥ 14 Allie must sell at least 14 more baskets.

20. Let m represent the amount of money. 7.50 + m ≤ 12.00 _________ -7.50 _____ -7.50 m ≤ 4.50 Dante can spend $4.50 at most.

21. -4x < 8

-4x _ -4

> 8 _ -4

x > -2

1 2 3 0 -3 -2 -1

22. d _ 3 ≥ -3

3 ( d _ 3 ) ≥ 3(-3)

d ≥ -9

0 -12 -9 -6 -3

23. 3 _ 4 t ≤ 12

4 _ 3 ( 3 _

4 t) ≤ 4 _

3 (12)

t ≤ 16

0 8 16 24 32

24. 8 > -16c

8 _ -16

< -16c _ -16

- 1 __ 2 < c

c > - 1 __ 2

0 -1 -2 - 1 __ 2

-1 1 __ 2

25. Let r represent the number of ribbons. 14r ≤ 80

14r _ 14

≤ 80 _ 14

r ≤ 5.7 Since only a whole number of ribbons can be cut,

Riley can cut 0, 1, 2, 3, 4, or 5 strips of ribbon.


x-4


CS10_A1_MESK710372_C02.indd 48 3/30/11 10:35:10 PM

solvInG two-step And multI-step InequAlItIes

CheCk it out!

1a. -12 ≥ 3x + 6 ____ -6 ______ - 6 -18 ≥ 3x

-18 _ 3 ≥ 3x _

3

-6 ≥ x x ≤ -6

0 -8 -6 -4 -2

b. x + 5 _ -2

> 3

-2 ( x + 5 _ -2

) < -2(3)

x + 5 < -6 _____ - 5 ___ -5 x < -11

-13 -12 -11 -10 -9

c. 1 - 2n _ 3

≥ 7

3 ( 1 - 2n _ 3 ) ≥ 3(7)

1 - 2n ≥ 21 _______ -1 ___ -1 -2n ≥ 20

-2n _ -2

≤ 20 _ -2

n ≤ -10

-12 -10 -11 -8 -9

2a. 2m + 5 > 5 2 2m + 5 > 25 ______ - 5 ___ -5 2m > 20

2m _ 2 > 20 _

2

m > 10

8 9 10 11 12

b. 3 + 2(x + 4) > 3 3 + 2(x) + 2(4) > 3 3 + 2x + 8 > 3 2x + 11 > 3 _______ - 11 ____ -11 2x > -8

2x _ 2 > -8 _

2

x > -4

0 -6 -5 -4 -3 -2 -1

c. 5 _ 8 < 3 _

8 x - 1 _

4

8 ( 5 _ 8 ) < 8 ( 3 _

8 x - 1 _

4 )

8 ( 5 _ 8 ) < 8 ( 3 _

8 x) + 8 (- 1 _

4 )

5 < 3x - 2 ___ +2 ______ +2 7 < 3x

7 _ 3 < 3x _

3

2 1 _ 3 < x

x > 2 1 _ 3

0 1 2 3 4

2 1 __ 3

3. Let g represent the grade on the second test.

95 + g

_ 2 ≥ 90

2 ( 95 + g

_ 2 ) ≥ 2(90)

95 + g ≥ 180 _______ -95 ____ -95 g ≥ 85 Jim must score at least 85 on the second test.

think and disCuss

1. minimum value of v

2. 1. Mulitply both sides by 3, and then subtract 5 from both sides.

2. Divide the left side to get x __ 3 + 5 __

3 , subtract 5 __

3 from

both sides, and then multiply both sides by 3.

3.

How are they alike? To solve multi-step equations or inequalities,

follow the order of operations to simplify the expressions on both sides of the equal sign or inequality symbol,

and then undo each operation.

How are they different? When solving multi-step inequalities, you must

reverse the inequality symbol if you multiply or divide both sides by a negative number. There are many

solutions of an inequality but usually only one solution of an equation.

Solving Multi-Step Equations and Inequalities


1. 2m + 1 > 13 ______ - 1 ___ -1 2m > 12

2m _ 2 > 12 _

2

m > 6

0 2 4 6 8

2. 2d + 21 ≤ 11 _______ - 21 ____ -21 2d ≤ -10

2d _ 2

≤ -10 _ 2

d ≤ -5

0 -6 -5 -4 -3 -2 -1

3. 6 ≤ -2x + 2 ___ -2 _______ - 2 4 ≤ -2x

4 _ -2

≥ -2x _ -2

-2 ≥ x x ≤ -2

1 2 3 0 -3 -2 -1

4. 4c - 7 > 5 _____ + 7 ___ +7 4c > 12

4c _ 4

> 12 _ 4

c > 3

4 5 6 3 2 1 0

5. 4 + x _ 3

> -4

3 ( 4 + x _ 3 ) > 3(-4)

4 + x > -12 ______ -4 ____ -4 x > -16

0 -32 -24 -16 -8

6. 1 < 0.2x - 0.7 ____ +0.7 _________ + 0.7 1.7 < 0.2x

1.7 ___ 0.2

< 0.2x _ 0.2

8.5 < x x > 8.5

9

8.5

12 6 3 0

7. 3 - 2x _ 3

≤ 7

3 ( 3 - 2x _ 3 ) ≤ 3(7)

3 - 2x ≤ 21 _______ -3 ___ -3 -2x ≤ 18

-2x _ -2

≥ 18 _ -2

x ≥ -9

0 -12 -9 -6 -3

8. 2x + 5 ≥ 2 _____ - 5 ___ -5 2x ≥ -3

2x _ 2

≥ -3 _ 2

x ≥ - 3 _ 2

0 -1 -2 - 1 __ 2

- 3 __ 2


x-4


2-4

CS10_A1_MESK710372_C02.indd 49 3/30/11 10:35:15 PM

9. 4(x + 2) > 6 4(x) + 4(2) > 6 4x + 8 > 6 ______ - 8 ___ -8 4x > -2

4x _ 4 > -2 _

4

x > - 1 _ 2

0 -1 -2 - 1 __ 2

-1 1 __ 2

10. 1 _ 4 x + 2 _

3 < 3 _

4

______

- 2 _ 3

___ - 2 _

3

1 _ 4 x < 1 _

12

4 ( 1 _ 4 x) < 4 ( 1 _

12 )

x < 1 _ 3

0 1 1 __ 3

2 __ 3

11. 4 - x + 6 2 ≥ 21 4 - x + 36 ≥ 21 40 - x ≥ 21 _______ -40 ____ -40 -x ≥ -19 -1(-x) ≤ -1(-19) x ≤ 19

20 21 19 18 17

12. 4 - x > 3(4 - 2) 4 - x > 3(2) 4 - x > 6 ______ -4 ___ -4 -x > 2 -1(-x) < -1(2) x < -2

1 2 3 0 -3 -2 -1

13. 0.2(x - 10) > -1.8 0.2(x) + 0.2(-10) > -1.8 0.2x - 2 > -1.8 _______ + 2 ____ +2 0.2x > 0.2

0.2x _ 0.2

> 0.2 _ 0.2

x > 1

1 2 3 0 -3 -2 -1

14. 3(j + 41) ≤ 35 3(j) + 3(41) ≤ 35 3j + 123 ≤ 35 _______ - 123 _____ -123 3j ≤ -88

3j

_ 3 ≤ -88 _

3

j ≤ -29 1 _ 3

-30 -29 -28

-29 1 __ 3

15. Let x represent the amount of sales. 300 + 0.1x > 1200 ___________ -300 _____ -300 0.1x > 900

0.1x _ 0.1

> 900 _ 0.1

x > 9000 The sales representative will make more money with

the first plan with sales of more than $9000.


16. 4r - 9 > 7 _____ + 9 ___ +9 4r > 16

4r _ 4 > 16 _

4

r > 4

4 5 6 3 2 1 0

17. 3 ≤ 5 - 2x ___ -5 _______ -5 -2 ≤ -2x

-2 _ -2

≥ -2x _ -2

1 ≥ x x ≤ 1

1 2 3 0 -3 -2 -1

18. w + 3 _ 2

> 6

2 ( w + 3 _ 2 ) > 2(6)

w + 3 > 12 _____ - 3 ___ -3 w > 9

9 12 6 3 0

19. 11w + 99 < 77 ________ - 99 ____ -99 11w < -22

11w _ 11

< -22 _ 11

w < -2

1 2 3 0 -3 -2 -1

20. 9 ≥ 1 _ 2 v + 3

___ -3 ______ - 3

6 ≥ 1 _ 2 v

2(6) ≥ 2 ( 1 _ 2 v)

12 ≥ v v ≤ 12

12 16 8 4 0

21. -4x - 8 > 16 _______ + 8 ___ +8 -4x > 24

-4x _ -4

< 24 _ -4

x < -6

0 -8 -6 -4 -2

22. 8 - 2 _ 3 z ≤ 2

________ -8 ___ -8

- 2 _ 3 z ≤ -6

- 3 _ 2 (- 2 _

3 z) ≥ - 3 _

2 (-6)

z ≥ 9

9 12 6 3 0

23. f + 2 1 _ 2 < -2

______

- 2 1 _ 2

____ -2 1 _

2

f < -4.5

-5 -4.5 -4 -3.5 -3

24. 3n - 8 _ 5

≥ 2

5 ( 3n - 8 _ 5 ) ≥ 5(2)

3n - 8 ≥ 10 ______ + 8 ___ +8 3n ≥ 18

3n _ 3 ≥ 18 _

3

n ≥ 6

6 8 4 2 0

25. -5 > -5 - 3w ___ +5 ________ +5 0 > -3w

0 _ -3

< -3w _ -3

0 < w w > 0

1 2 3 0 -3 -2 -1


CS10_A1_MESK710372_C02.indd 50 3/30/11 10:35:20 PM

26. 10 > 5 - 3p

_ 2

2(10) > 2 ( 5 - 3p

_ 2 )

20 > 5 - 3p ___ -5 _______ -5 15 > -3p

15 _ -3

< -3p

_ -3

-5 -5

0 -6 -5 -4 -3 -2 -1

27. 2v + 1 > 2 1 _ 3

_____ - 1 ___ -1

2v > 4 _ 3

1 _ 2 (2v) > 1 _

2 ( 4 _

3 )

v > 2 _ 3

0 1 1 __ 3

2 __ 3

28. 4(x + 3) > -24 4(x) + 4(3) > -24 4x + 12 > -24 _______ - 12 ____ -12 4x > -36

4x _ 4 > -36 _

4

x > -9

0 -12 -9 -6 -3

29. 4 > x - 3(x + 2) 4 > x - 3(x) - 3(2) 4 > x - 3x - 6 4 > -2x - 6 ___ +6 _______ + 6 10 > -2x

10 _ -2

< -2x _ -2

-5 < x x > -5

0 -6 -5 -4 -3 -2 -1

30. -18 ≥ 33 - 3h ____ -33 ________ -33 -51 ≥ -3h

-51 _ -3

≤ -3h _ -3

17 ≤ h h ≥ 17

18 19 17 16 15

31. -2 > 7x - 2(x - 4) -2 > 7x - 2(x) - 2(-4) -2 > 7x - 2x + 8 -2 > 5x + 8 ___ -8 ______ - 8 -10 > 5x

-10 _ 5 > 5x _

5

-2 > x x < -2

1 2 3 0 -3 -2 -1

32. 9 - (9 ) 2 > 10x - x 9 - 81 > 10x - x -72 > 9x

-72 _ 9 > 9x _

9

-8 > x x < -8

0 -16 -12 -8 -4

33. 2a - (-3 ) 2 ≥ 13 2a - 9 ≥ 13 ______ + 9 ___ +9 2a ≥ 22

2a _ 2 ≥ 22 _

2

a ≥ 11

12 13 11 10 9

34. 6 - x _ 3 + 1 > 2 _

3

7 - x _ 3 > 2 _

3

_______ -7 ___ -7

- x _ 3 > - 19 _

3

-3 (- x _ 3 ) < -3 (- 19 _

3 )

x < 19

20 21 19 18 17

35. 12(x - 3) + 2x > 6 12(x) + 12(-3) + 2x > 6 12x - 36 + 2x > 6 14x - 36 > 6 _______ + 36 ____ +36 14x > 42

14x _ 14

> 42 _ 14

x > 3

4 5 6 3 2 1 0

36. 15 ≥ 19 + 2(q - 18) 15 ≥ 19 + 2(q) + 2(-18) 15 ≥ 19 + 2q - 36 15 ≥ 2q - 17 ____ +17 _______ + 17 32 ≥ 2q

32 _ 2 ≥

2q _

2

16 ≥ q q ≤ 16

24 32 16 8 0

37. Let x represent the number of minutes. 29.99 < 19.99 + 0.35x ______ -19.99 _____________ -19.99 10.00 < 0.35x

10.00 _ 0.35

< 0.35x _ 0.35

28.6 < x x > 28.6 The second company’s plan costs more starting at

29 minutes.

38. -12 > -4x - 8 ____ +8 _______ + 8 -4 > -4x

-4 _ -4

< -4x _ -4

1 < x x > 1

1 2 3 0 -3 -2 -1

39. 5x + 4 ≤ 14 _____ - 4 ___ -4 5x ≤ 10

5x _ 5

≤ 10 _ 5

x ≤ 2

1 2 3 0 -3 -2 -1


CS10_A1_MESK710372_C02.indd 51 3/30/11 10:35:26 PM

40. 2 _ 3 x - 5 > 7

______ + 5 ___ +5

2 _ 3 x > 12

3 _ 2 ( 2 _

3 x) > 3 _

2 (12)

x > 18

19 20 18 17 16

41. x - 3x > 2 - 10 -2x > -8

-2x _ -2

< -8 _ -2

x < 4

4 5 6 3 2 1 0

42. 5 - x - 2 > 3 3 - x > 3 ______ -3 ___ -3 -x > 0 -1(-x) < -1(0) x < 0

1 2 3 0 -3 -2 -1

43. 3 < 2x - 5(x + 3) 3 < 2x - 5(x) - 5(3) 3 < 2x - 5x - 15 3 < -3x - 15 ____ +15 ________ + 15 18 < -3x

18 _ -3

> -3x _ -3

-6 > x x < -6

0 -8 -6 -4 -2

44. 1 _ 6 - 2 _

3 m ≥ 1 _

4

_________

- 1 _ 6

___ - 1 _

6

- 2 _ 3

m ≥ 1 _ 12

- 3 _ 2 (- 2 _

3 m) ≤ - 3 _

2 ( 1 _

12 )

m ≤ - 1 _ 8

0 - 1 __ 2 - 3 __

8 - 1 __

4 - 1 __

8

45. 4 - (r - 2) > 3 - 5 4 - (r) - (-2) > 3 - 5 4 - r + 2 > 3 - 5 6 - r > -2 ______ -6 ___ -6 -r > -8 -1(-r) < -1(-8) r < 8

0 4 8 12 16

46. 0.3 - 0.5n + 1 ≥ 0.4 1.3 - 0.5n ≥ 0.4 __________ -1.3 ____ -1.3 -0.5n ≥ -0.9

-0.5n _ -0.5

≤ -0.9 _ -0.5

n ≤ 1.8

0.5 0 1 1.5

1.8

2

47. 6 2 > 4(x + 2) 6 2 > 4(x) + 4(2) 6 2 > 4x + 8 36 > 4x + 8 ___ -8 ______ - 8 28 > 4x

28 _ 4 > 4x _

4

7 > x x < 7

6 5 7 8 9

48. -4 - 2n + 4n > 7 - 2 2 -4 - 2n + 4n > 7 - 4 2n - 4 > 3 ______ + 4 ___ +4 2n > 7

2n _ 2 > 7 _

2

n > 3.5

2.5 2 3 3.5 4

49. 1 _ 4 (p - 10) ≥ 6 - 4

1 _ 4 (p - 10) ≥ 2

4 ( 1 _ 4 (p - 10)) ≥ 4(2)

p - 10 ≥ 8 ______ + 10 ____ +10 p ≥ 18

6 0 12 18 24 30

50a. -5 b. -1

c. -5 d. 5

e. -15 f. 1

51. 1 _ 2 x + 9 < 33

______ - 9 ___ -9

1 _ 2 x < 24

2 ( 1 _ 2 x) < 2(24)

x < 48

16 0 32 48 64

52. 6 ≤ 4 - 2x ___ -4 _______ -4 2 ≤ -2x

2 _ -2

≥ -2x _ -2

-1 ≥ x x ≤ -1

0 1 2 3 -1 -2 -3

53. 4(x + 12) ≤ 16 4(x) + 4(12) ≤ 16 4x + 48 ≤ 16 _______ - 48 ____ -48 4x ≤ -32

4x _ 4 ≤ -32 _

4

x ≤ -8

0 -16 -12 -8 -4

54. 1 _ 2 x + 2 _

3 x < 14

7 _ 6 x < 14

6 _ 7 ( 7 _

6 x) < 6 _

7 (14)

x < 12

4 0 8 12 16

55. 4x - 9 ≥ 7 _____ + 9 ___ +9 4x ≥ 16

4x _ 4 ≥ 16 _

4

x ≥ 4; B

56. -6 ≥ 3(x - 2) -6 ≥ 3(x) + 3(-2) -6 ≥ 3x - 6 ___ +6 ______ + 6 0 ≥ 3x

0 _ 3 ≥ 3x _

3

0 ≥ x x ≤ 0; D

57. -2x - 6 ≥ -4 + 2 -2x - 6 ≥ -2 _______ + 6 ___ +6 -2x ≥ 4

-2x _ -2

≤ 4 _ -2

x ≤ -2; A

58. 1 _ 2 - 1 _

3 x ≤ ( 2 _

3 + 1 _

3 )

2

1 _ 2 - 1 _

3 x ≤ 1 2

1 _ 2 - 1 _

3 x ≤ 1

________

- 1 _ 2

___ - 1 _

2

- 1 _ 3 x ≤ 1 _

2

-3 (- 1 _ 3 x) ≥ -3 ( 1 _

2 )

x ≥ - 3 _ 2

; C


CS10_A1_MESK710372_C02.indd 52 3/30/11 10:35:32 PM

59. Let m represent the number of months. 225 + 400 < 275 + 15m 625 < 275 + 15m _____ -275 ___________ -275 350 < 15m

350 _ 15

< 15m _ 15

23.

3 < m m > 23.

3

A consumer will pay less for the machine at Easy Electronics for 24 months or more.

60a. 1 _ 2 (5)(2x + 3) < 55 b. 5 _

2 (2x + 3) < 55

5 _ 2 (2x) + 5 _

2 (3) < 55

5x + 15 _ 2 <

55

_______

- 15 _ 2

____ - 15 _

2

5x < 47.5

5x _ 5 < 47.5 _

5

x < 9.5

c. h = 2x + 3 h ≤ 2(9.5) + 3 h ≤ 19 + 3 h ≤ 22 The height of the triangle is less than or equal to 22 in.

61a. number Process Cost

1 350 + 3 353

2 350 + 3(2) 356

3 350 + 3(3) 359

10 350 + 3(10) 380

n 350 + 3(n) 350 + 3(n)

b. C = 350 + 3n

c. C ≤ 500 350 + 3n ≤ 500 _________ -350 _____ -350 3n ≤ 150

3n _ 3 ≤ 150 _

3

n ≤ 50 They can make 50 or fewer CDs.

62. 4r - 4.9 > 14.95 _______ + 4.9 ______ +4.9 4r > 19.85

4r _ 4 > 19.85 _

4

r > 4.9625 r = 5

63. 1. Divide both sides by 2, and then subtract 3 from both sides. 2. Distribute 2 on the left side, subtract 6 from both sides, then divide both sides by 2.

teSt prep

64. A; Substituting 6 for y gives the inequality 18 > 2x + 4. Solving for x gives 7 > x or x < 7.

65. G; If x represents the number of tickets, the price of all the tickets is 1.25x. The total cost is

5 + 6 + 1.25x and this must be less than or equal to 30.

66. B; Since the > sign is used, and since only situation B involves less than, B must be correct.

67. 59 Let x represent the number of points scored in the

second game. (x - 8) + x + 42 > 150 2x + 34 > 150 _______ - 34 ____ -34 2x > 116

2x _ 2 > 116 _

2

x > 58 The team must have scored at least 59 points.


68. 3(x + 2) - 6x + 6 ≤ 0 3(x) + 3(2) - 6x + 6 ≤ 0 3x + 6 - 6x + 6 ≤ 0 12 - 3x ≤ 0 ________ -12 ____ -12 -3x ≤ -12

-3x _ -3

≥ -12 _ -3

x ≥ 4

3 2 1 0 4 5 6

69. -18 > -(2x + 9) - 4 + x -18 > -(2x) - (9) - 4 + x -18 > -2x - 9 - 4 + x -18 > -x - 13 ____ +13 _______ + 13 -5 > -x -1(-5) < -1(-x) 5 < x x > 5

3 4 5 0 1 2 6

70. 2 + x _ 2 - (x - 1) > 1

2 _ 2 + x _

2 - (x) - (-1) > 1

1 + x _ 2 - x + 1 > 1

2 - 1 _ 2 x > 1

________ -2 ___ -2

- 1 _ 2 x > -1

-2 (- 1 _ 2 x) < -2(-1)

x < 2

0 1 2 3 -1 -2 -3


CS10_A1_MESK710372_C02.indd 53 3/30/11 10:35:34 PM

71. x > 0

0 1 2 3 -1 -2 -3

72. x < 0

0 1 2 3 -1 -2 -3

73. x ≥ 0

0 1 2 3 -1 -2 -3

74. x ≤ 0

0

75. -3x > 0

-3x ____ -3

< 0 ___ -3

x < 0

0 1 2 3 -1 -2 -3

76. -x > 2 -1(-x) < -1(2) x < -2

0 1 2 3 -1 -2 -3

solvInG InequAlItIes wIth vArIAbles on both sIdes

CheCk it out!

1a. 4x ≥ 7x + 6 ____ -4x _______ -4x 0 ≥ 3x + 6 ___ -6 ______ - 6 -6 ≥ 3x

-6 _ 3 ≥ 3x _

3

-2 ≥ x x ≤ -2

0 1 2 3 -1 -2 -3

b. 5t + 1 < -2t - 6 _______ +2t _______ +2t 7t + 1 < -6 _____ - 1 ___ -1 7t < -7

7t _ 7 < -7 _

7

t < -1

0 1 2 3 -1 -2 -3

2. Let f represent the number of flyers. 24 + 0.10f < 0.25f _________ - 0.10f ______ -0.10f 24 < 0.15f

24 _ 0.15

< 0.15f _ 0.15

160 < f f > 160 The cost at A-Plus Advertising is less than the cost

at Print and More if more than 160 flyers are printed.

3a. 5(2 - r) ≥ 3(r - 2) 5(2) + 5(-r) ≥ 3(r) + 3(-2) 10 - 5r ≥ 3r - 6 _______ + 5r _______ +5r 10 ≥ 8r - 6 ___ +6 _____ + 6 16 ≥ 8r

16 _ 8 ≥ 8r _

8

2 ≥ r r ≤ 2

0 1 2 3 -1 -2 -3

b. 0.5x - 0.3 + 1.9x < 0.3x + 6 2.4x - 0.3 < 0.3x + 6 __________ -0.3x _________ -0.3x 2.1x - 0.3 < 6 _________ + 0.3 ____ +0.3 2.1x < 6.3

2.1x _ 2.1

< 6.3 _ 2.1

x < 3

3 2 1 0 4 5 6

4a. 4(y - 1) ≥ 4y + 2 4(y) + 4(-1) ≥ 4y + 2 4y - 4 ≥ 4y + 2 _______ -4y _______ -4y -4 ≥ 2 7 no solutions

b. x - 2 < x + 1 ______ -x ______ -x -2 < 1 3 all real numbers

think and disCuss

1. Subtract 5c from both sides of the inequality so that all variable terms are on the right side. Then subtract 2 from both sides so that all constant terms are on the left side.

2.

All real numbersx + 2 < x + 5

No solutionsx + 5 > x + 9

Solutions ofInequalities with

Variables on Both Sides


1. 2x > 4x - 6 ____ -4x _______ -4x -2x > -6

-2x _ -2

< -6 _ -2

x < 3

3 2 1 0 4 5 6

2. 7y + 1 ≤ y - 5 ______ -y ______ -y 6y + 1 ≤ -5 ______ - 1 ___ -1 6y ≤ -6

6y

_ 6 ≤ -6 _

6

y ≤ -1

0 1 2 3 -1 -2 -3

3. 27x + 33 > 58x - 29 _________ -27x _________ -27x 33 > 31x - 29 ____ +29 ________ + 29 62 > 31x

62 _ 31

> 31x _ 31

2 > x x < 2

0 1 2 3 -1 -2 -3


x-5


2-5

CS10_A1_MESK710372_C02.indd 54 3/30/11 10:35:39 PM

4. -3r < 10 - r ____ +r ______ + r -2r < 10

-2r _ -2

> 10 _ -2

r > -5

0 -4 -3 -2 -1 -5 -6

5. 5c - 4 > 8c + 2 _______ -5c _______ -5c -4 > 3c + 2 ___ -2 ______ - 2 -6 > 3c

-6 _ 3 > 3c _

3

-2 > c c < -2

0 1 2 3 -1 -2 -3

6. 4.5x - 3.8 ≥ 1.5x - 2.3 __________ -1.5x __________ -1.5x 3.0x - 3.8 ≥ -2.3 _________ + 3.8 ____ +3.8 3.0x ≥ 1.5

3.0x _ 3.0

≥ 1.5 _ 3.0

x ≥ 1 _ 2

0 1 2 1 __

2 1 1 __

2

7. Let p represent the number of pizzas. 100 + 4p < 7p ________ - 4p ____ -4p 100 < 3p

100 _ 3 <

3p _

3

33.3 33.3 They will have to sell at least 34 pizzas to make a

profit.

8. 5(4 + x) ≤ 3(2 + x) 5(4) + 5(x) ≤ 3(2) + 3(x) 20 + 5x ≤ 6 + 3x _______ - 3x ______ - 3x 20 + 2x ≤ 6 ________ -20 ____ -20 2x ≤ -14

2x _ 2 ≤ -14 _

2

x ≤ -7

-9 -8 -7 -6 -5

9. -4(3 - p) > 5(p + 1) -4(3) - 4(-p) > 5(p) + 5(1) -12 + 4p > 5p + 5 ________ - 4p _______ -4p -12 > p + 5 ____ -5 _____ - 5 -17 > p p < -17

-19 -18 -17 -16 -15

10. 2(6 - x) < 4x 2(6) + 2(-x) < 4x 12 - 2x < 4x ______ + 2x ____ +2x 12 < 6x

12 _ 6 < 6x _

6

2 < x x > 2

0 1 2 3 -1 -2 -3

11. 4x > 3(7 - x) 4x > 3(7) + 3(-x) 4x > 21 - 3x ____ +3x _______ + 3x 7x > 21

7x _ 7

> 21 _ 7

x > 3

3 2 1 0 4 5 6

12. 1 _ 2 f + 3 _

4 ≥ 1 _

4 f

________

- 1 _ 2 f

____ - 1 _

2 f

3 _ 4 ≥ - 1 _

4 f

-4 ( 3 _ 4 ) ≤ -4 (- 1 _

4 f)

-3 ≤ f f ≥ -3

0 -4 -3 -2 -1 -5 -6

13. -36.72 + 5.65t < 0.25t ____________ - 5.65t ______ -5.65t -36.72 < -5.40t

-36.72 _ -5.40

> -5.40t _ -5.40

6.8 > t t < 6.8

6 6.2 6.4 6.6 6.8

14. 2(x - 2) ≤ -2(1 - x) 2(x) + 2(-2) ≤ -2(1) - 2(-x) 2x - 4 ≤ -2 + 2x _______ -2x _______ - 2x -4 ≤ -2 3 all real numbers

15. 4(y + 1) < 4y + 2 4(y) + 4(1) < 4y + 2 4y + 4 < 4y + 2 _______ -4y _______ -4y 4 < 2 7 no solutions

16. 4v + 1 < 4v - 7 _______ -4v _______ -4v 1 < -7 7 no solutions

17. b - 4 ≥ b - 6 ______ -b ______ -b -4 ≥ -6 3 all real numbers

18. 3(x - 5) > 3x 3(x) + 3(-5) > 3x 3x - 15 > 3x ________ -3x ____ -3x -15 > 0 7 no solutions

19. 2k + 7 ≥ 2(k + 14) 2k + 7 ≥ 2(k) + 2(14) 2k + 7 ≥ 2k + 28 _______ -2k ________ -2k 7 ≥ 28 7 no solutions


CS10_A1_MESK710372_C02.indd 55 3/30/11 10:35:43 PM

20. 3x ≤ 5x + 8 ____ -5x _______ -5x -2x ≤ 8

-2x _ -2

≥ 8 _ -2

x ≥ -4

0 -4 -3 -2 -1 -5 -6

21. 9y + 3 > 4y - 7 _______ -4y _______ -4y 5y + 3 > -7 ______ - 3 ___ -3 5y > -10

5y

_ 5 > -10 _

5

y > -2

0 1 2 3 -1 -2 -3

22. 1.5x - 1.2 < 3.1x - 2.8 __________ -1.5x __________ -1.5x -1.2 < 1.6x - 2.8 ____ +2.8 _________ + 2.8 1.6 < 1.6x

1.6 _ 1.6

< 1.6x _ 1.6

1 < x x > 1

0 1 2 3 -1 -2 -3

23. 7 + 4b ≥ 3b ______ - 3b ____ -3b 7 + b ≥ 0 ______ -7 ___ -7 b ≥ -7

-9 -8 -7 -6 -5

24. 7 - 5t < 4t - 2 _____ + 5t _______ +5t 7 < 9t - 2 ___ +2 _____ + 2 9 < 9t

9 _ 9 < 9t _

9

1 < t t > 1

0 1 2 3 -1 -2 -3

25. 2.8m - 5.2 > 0.8m + 4.8 ___________ -0.8m ___________ -0.8m 2.0m - 5.2 > 4.8 _________ + 5.2 ____ +5.2 2.0m > 10.0

2.0m _ 2.0

> 10.0 _ 2.0

m > 5

3 2 1 0 4 5 6

26. ℓw > 1 _ 2 bh

12(x + 2) > 1 _ 2 (x + 16)(10)

12(x + 2) > 5(x + 16) 12(x) + 12(2) > 5(x) + 5(16) 12x + 24 > 5x + 80 ________ -5x ________ -5x 7x + 24 > 80 _______ - 24 ____ -24 7x > 56

7x _ 7 > 56 _

7

x > 8

27. 4(2 - x) ≤ 5(x - 2) 4(2) + 4(-x) ≤ 5(x) + 5(-2) 8 - 4x ≤ 5x - 10 ______ + 4x ________ +4x 8 ≤ 9x - 10 ____ +10 _______ + 10 18 ≤ 9x

18 _ 9 ≤ 9x _

9

2 ≤ x x ≥ 2

0 1 2 3 -1 -2 -3

28. -3(n + 4) < 6(1 - n) -3(n) - 3(4) < 6(1) + 6(-n) -3n - 12 < 6 - 6n ________ +6n ______ +6n 3n - 12 < 6 _______ + 12 ____ +12 3n < 18

3n _ 3 < 18 _

3

n < 6

2 0 4 6 8

29. 9(w + 2) ≤ 12w 9(w) + 9(2) ≤ 12w 9w + 18 ≤ 12w _________ -9w ____ -9w 18 ≤ 3w

18 _ 3 ≤ 3w _

3

6 ≤ w w ≥ 6

2 0 4 6 8

30. 4.5 + 1.3t > 3.8t - 3 ________ - 1.3t ________ -1.3t 4.5 > 2.5t - 3 ____ +3 _______ + 3 7.5 > 2.5t

7.5 _ 2.5

> 2.5t _ 2.5

3 > t t < 3

3 2 1 0 4 5 6

31. 1 _ 2 r + 2 _

3 ≥ 1 _

3 r

________

- 1 _ 2 r

____ - 1 _

2 r

2 _ 3 ≥ - 1 _

6 r

-6 ( 2 _ 3 ) ≤ -6 (- 1 _

6 r)

-4 ≤ r r ≥ -4

0 -4 -3 -2 -1 -5 -6

32. 2(4 - n) < 3n - 7 2(4) + 2(-n) < 3n - 7 8 - 2n < 3n - 7 ______ + 2n _______ +2n 8 < 5n - 7 ___ +7 ______ + 7 15 < 5n

15 _ 5 < 5n _

5

3 < n n > 3

3 0 1 2 4 5 6

33. 3(2 - x) < -3(x - 1) 3(2) + 3(-x) < -3(x) - 3(-1) 6 - 3x < -3x + 3 ______ + 3x _______ +3x 6 < 3 7 no solutions

34. 7 - y > 5 - y _____ + y _____ + y 7 > 5 3 all real numbers


CS10_A1_MESK710372_C02.indd 56 3/30/11 10:35:47 PM

35. 3(10 + z) ≤ 3z + 36 3(10) + 3(z) ≤ 3z + 36 30 + 3z ≤ 3z + 36 _______ - 3z ________ -3z 30 ≤ 36 3 all real numbers

36. -5(k - 1) ≥ 5(2 - k) -5(k) - 5(-1) ≥ 5(2) + 5(-k) -5k + 5 ≥ 10 - 5k _______ +5k _______ + 5k 5 ≥ 10 7 no solutions

37. 4(x - 1) ≤ 4x 4(x) + 4(-1) ≤ 4x 4x - 4 ≤ 4x _______ -4x ____ -4x -4 ≤ 0 3 all real numbers

38. 3(v - 9) ≥ 15 + 3v 3(v) + 3(-9) ≥ 15 + 3v 3v - 27 ≥ 15 + 3v ________ -3v _______ - 3v -27 ≥ 15 7 no solutions

39. 3t - 12 > 5t + 2 ________ -3t _______ -3t -12 > 2t + 2 ____ -2 _____ - 2 -14 > 2t

-14 _ 2 > 2t _

2

-7 > t t < -7

-9 -8 -7 -6 -5

40. -5(y + 3) - 6 < y + 3 -5(y) - 5(3) - 6 < y + 3 -5y - 15 - 6 < y + 3 -5y - 21 < y + 3 ________ +5y _______ +5y -21 < 6y + 3 ____ -3 ______ - 3 -24 < 6y

-24 _ 6 <

6y _

6

-4 < y y > -4

0 -4 -3 -2 -1 -5 -6

41. 3x + 9 - 5x < x 9 - 2x < x ______ + 2x ____ +2x 9 < 3x

9 _ 3 < 3x _

3

3 < x x > 3

6 5 4 3 2 1 0

42. 18 + 9p > 12p - 31 _______ - 9p ________ -9p 18 > 3p - 31 ____ +31 _______ + 31 49 > 3p

49 _ 3 >

3p _

3

16 1 _ 3 > p

p < 16 1 _ 3

16 1 __ 3 16 2 __

3 16 17

43. 2(x - 5) < -3x 2(x) + 2(-5) < -3x 2x - 10 < -3x ________ -2x ____ -2x -10 < -5x

-10 _ -5

> -5x _ -5

2 > x x < 2

1 2 3 0 -3 -2 -1

44. - 2 _ 5

x ≤ 4 _ 5

- 3 _ 5

x

____

+ 3 _ 5

x _______

+ 3 _ 5

x

1 _ 5

x ≤ 4 _ 5

5 ( 1 _ 5

x) ≤ 5 ( 4 _ 5

)

x ≤ 4

4 5 6 3 2 1 0

45. -2(x - 7) - 4 - x < 8x + 32 -2(x) - 2(-7) - 4 - x < 8x + 32 -2x + 14 - 4 - x < 8x + 32 10 - 3x < 8x + 32 _______ + 3x _________ +3x 10 < 11x + 32 ____ -32 _______ - 32 -22 < 11x

-22 _ 11

< 11x _ 11

-2 < x x > -2

1 2 3 0 -3 -2 -1

46. -3(2r - 4) ≥ 2(5 - 3r) -3(2r) - 3(-4) ≥ 2(5) + 2(-3r) -6r + 12 ≥10 - 6r ________ +6r _______ + 6r 12 ≥ 10 3 all real numbers

0 -6 -5 -4 -3 -2 -1

47. -7x - 10 + 5x ≥ 3(x + 4) + 8 -7x - 10 + 5x ≥ 3(x) + 3(4) + 8 -7x - 10 + 5x ≥ 3x + 12 + 8 -2x - 10 ≥ 3x + 20 ________ +2x ________ +2x -10 ≥ 5x + 20 ____ -20 _______ - 20 -30 ≥ 5x

-30 _ 5 ≥ 5x _

5

-6 ≥ x x ≤ -6

0 -8 -6 -4 -2


CS10_A1_MESK710372_C02.indd 57 3/30/11 10:35:51 PM

48. - 1 _ 3 (n + 8) + 1 _

3 n ≤ 1 - n

- 1 _ 3 (n) - 1 _

3 (8) + 1 _

3 n ≤ 1 - n

- 1 _ 3 n - 8 _

3 + 1 _

3 n ≤ 1 - n

- 8 _ 3 ≤ 1 - n

___ -1 ______ -1

- 11 _ 3 ≤ -n

-1 (- 11 _ 3 ) ≥ -1(-n)

11 _ 3 ≥ n

n ≤ 11 _ 3

4 20 7 __

3 8 __

3 11 __

3 10 __ 3 3

49. Let s represent the number of seconds. 100 + 8s > 180 + 5t _______ - 5s _______ - 5t 100 + 3s > 180 _________ -100 _____ -100 3s > 80

3s ___ 3 > 80 ___

3

s > 26.67 The red kite will be higher than the blue kite in 27 s.

50a. decreased by 4 b. 1192 - 4(x - 1)

c. increase by 20 d. 921 + 20(x - 1)

e. 1192 - 4(x - 1) < 921 + 20(x - 1) 1192 - 4x + 4 < 921 + 20x - 20 ____________ -921 +20 ______________ -921 +20 271- 4x + 24 < 20x 195 - 4x < 20x ________ +4x ____ +4x 195 < 24x

195 ____ 24

< 24x ____ 24

12.29 < x, where x is a whole number

1411 12 13 15 16

51a. 400 + 4.50n b. 12n

c. 400 + 4.50n < 12.00n __________ - 4.50n ______ -4.50n 400 < 7.50n

400 _ 7.50

< 7.50n _ 7.50

53 1 _ 3 < n

n > 53 1 _ 3

They must sell 54 CDs or more to make a profit.

52. 2x + 4 > 2 _ 3 x

_______ -2x ____ -2x

4 > - 4 _ 3 x

- 3 _ 4 (4) < - 3 _

4 (- 4 _

3 x)

-3 < x x > -3

53. 5x - 10 < 6x - 8 ________ -5x _______ -5x -10 < x - 8 ____ +8 _____ + 8 -2 < x x > -2

54. x + 20 < 4x - 1 _______ -x ______ -x 20 < 3x - 1 ___ +1 ______ + 1 21 < 3x

21 _ 3 < 3x _

3

7 < x x > 7

55. 3 _ 4 x ≥ x - 5

___ -x ______ -x

- 1 _ 4 x ≥ -5

-4 (- 1 _ 4 x) ≤ -4(-5)

x ≤ 20

56. Let v represent the number of videos. 2.99v ≥ 19.99 + 1.99v ______ -1.99v ____________ - 1.99v v ≥ 19.99 You would need to rent 20 videos or more for Video

View to be less expensive than Movie Place.

57. No; the sum of the measures x and x - 1 would need to be greater than 2x. x + (x - 1) > 2x. 2x - 1 > 2x has no solutions.

58. The steps are identical except when you multiply or divide both sides of an inequality by a negative number, when you must reverse the inequality symbol.

59. x can never be greater than itself plus 1.

60. B is incorrect. The student should have added 4x to both sides to undo the subtraction.

teSt prep

61. D; Isolating a and b gives b < 0, so b must be negative.

62. J; Multiplying both sides by -1 causes the inequality sign to change and gives a > -b.

63. A; Since 7 (2 - (-2)) = 7(4) = 28 > -16 = 4(-4) = 4 ((-2)-2) , -2 is a solution.

64. J; Dividing both sides by -3 gives the inequality x > 2 which is shown in graph J.

65. Possible answer: Parking lot A starts with 4 cars and 7 more cars park each hour. Parking lot B starts with 13 cars and 4 more park each hour. The inequality helps you find that parking lot A has more cars than parking lot B after 3 hours.


CS10_A1_MESK710372_C02.indd 58 3/30/11 10:35:53 PM

66. 2 1 _ 2 + 2x ≥ 5 1 _

2 + 2 1 _

2 x

_______ - 2x _________ - 2x

2 1 _ 2 ≥ 5 1 _

2 + 1 _

2 x

____

-5 1 _ 2

__________ -5 1 _

2

-3 ≥ 1 _ 2 x

2(-3) ≥ 2 ( 1 _ 2 x)

-6 ≥ x x ≤ -6

67. 1.6x - 20.7 > 6.3x - (-2.2x) 1.6x - 20.7 > 6.3x + 2.2x 1.6x - 20.7 > 8.5x ___________ -1.6x _____ -1.6x -20.7 > 6.9x

-20.7 _ 6.9

> 6.9x _ 6.9

-3 > x x < -3

68. 1.3x - 7.5x < 8.5x - 29.4 -6.2x < 8.5x - 29.4 _____ -8.5x ___________ -8.5x -14.7x < -29.4

-14.7x _ -14.7

> -29.4 _ -14.7

x > 2

69. -4w + -8 - 37 _ 9 ≤ 75 - 3 _

9 + 3w

-4w + -45 _ 9 ≤

72 _ 9 + 3w

-4w - 5 ≤ 8 + 3w ________ +4w ______ +4w -5 ≤ 8 + 7w ___ -8 ________ -8 -13 ≤ 7w

-13 _ 7 ≤ 7w _

7

-1 6 _ 7 ≤ w

w ≥ -1 6 _ 7

70. Check students’ work: the number in the square should be less than the number in the circle.

71. Check students’ work: the number in the square should be greater than the number in the circle.

72. Check students’ work: if the same number is added to both sides and x is negative, the inequality is an identity.

solvInG compound InequAlItIes

CheCk it out!

1. Let c be the free chlorine in a pool. 1.0 ≤ c ≤ 3.0

4 5 6 3 2 1 0

2a. -9 < x - 10 < -5 ____ +10 ______ + 10 ____ +10 1 < x < 5

4 5 6 3 2 1 0

b. -4 ≤ 3n + 5 < 11 ___ -5 ______ - 5 ___ -5 -9 ≤ 3n < 6

- 9 _ 3

≤ 3n _ 3

< 6 _ 3

-3 ≤ n < 2

0 1 2 3 -3 -2 -1

3a. 2 + r < 12 OR r + 5 > 19 ______ -2 ___ -2 _____ - 5 ___ -5 r < 10 OR r > 14

10 8 6 4 12 14 16

b. 7x ≥ 21 OR 2x < -2

7x _ 7 ≥ 21 _

7 2x _

2 < -2 _

2

x ≥ 3 OR x < -1

0 1 2 3 -1 -2 -3

4a. -9 < y < -2 b. x ≤ -3 OR x ≥ 2

think and disCuss

1. y > 4 can be written as 4 < y. Combine the ineqalities 4 < y and y ≤ 12 to write

4 < y ≤ 12.

2.

Ax > 5

Bx < 10

12

15 21

6 7

4 3

2 8

x > 5 AND x < 10 x > 5 OR x < 10

6 7 8 1 3 4 6 7 8 12 15 21


1. intersection 2. Let t represent the temperature.

70 ≤ t ≤ 95

85 80 75 70 90 95


x-6


2-6

CS10_A1_MESK710372_C02.indd 59 3/30/11 10:35:56 PM

3. -3 < x + 2 < 7 ___ -2 _____ - 2 ___ -2 -5 < x < 5

0 5 10 -10 -5

4. 5 ≤ 4x + 1 ≤ 13 ___ -1 ______ - 1 ___ -1 4 ≤ 4x ≤ 12

4 _ 4 ≤ 4x _

4 ≤ 12 _

4

1 ≤ x ≤ 3

3 2 1 0 4 5 6

5. 2 < x + 2 < 5 ___ -2 _____ - 2 ___ -2 0 < x < 3

3 2 1 0 4 5 6

6. 11 < 2x + 3 < 21 ___ -3 ______ - 3 ___ -3 8 < 2x < 18

8 _ 2 < 2x _

2 < 18 _

2

4 < x < 9

7 6 5 4 8 9

7. x + 2 < -6 OR x + 2 > 6 ____ - 2 ___ -2 _____ - 2 ___ -2 x < -8 OR x > 4

0 2 4 -4 -2 -6 -8

8. r - 1 < 0 OR r - 1 > 4 ____ + 1 ___ +1 ____ + 1 ___ +1 r < 1 OR r > 5

3 2 1 0 4 5 6

9. n + 2 < 3 OR n + 3 > 7 _____ - 2 ___ -2 _____ - 3 ___ -3 n < 1 OR n > 4

3 2 1 0 4 5 6

10. x - 1 < -1 OR x - 5 > -1 ____ + 1 ___ +1 _____ + 5 ___ +5 x < 0 OR x > 4

3 2 1 0 4 5 6

11. -5 ≤ a ≤ -3 12. b ≤ -3 OR b > 3

13. c < 1 OR c ≥ 9 14. 4 ≤ d < 8


15. Let k be the distance from the surface of the Earth. 16 ≤ k ≤ 50

0 20 30 40 50 10

16

16. -1 < x + 1 < 1 ___ -1 _____ - 1 ___ -1 -2 < x < 0

0 1 2 3 -1 -2 -3

17. 1 ≤ 2n - 5 ≤ 7 ___ +5 ______ + 5 ___ +5 6 ≤ 2n ≤ 12

6 _ 2 ≤ 2n _

2 ≤ 12 _

2

3 ≤ n ≤ 6

3 2 1 0 4 5 6

18. -2 < x - 2 < 2 ___ +2 _____ + 2 ___ +2 0 < x < 4

3 2 1 0 4 5 6

19. 5 < 3x - 1 < 17 ___ +1 ______ + 1 ___ +1 6 < 3x < 18

6 _ 3 < 3x _

3 < 18 _

3

2 < x < 6

3 2 1 0 4 5 6

20. x - 4 < -7 OR x + 3 > 4 ____ + 4 ___ +4 _____ - 3 ___ -3 x < -3 OR x > 1

0 1 2 -1 -2 -3 -4

21. 2x + 1 < 1 OR x + 5 > 8 _____ - 1 ___ -1 _____ - 5 ___ -5 2x < 0 OR x > 3

2x _ 2 < 0 _

2

x < 0 OR x > 3

0 3 6 -6 -3

22. x + 1 < 2 OR x + 5 > 8 ____ - 1 ___ -1 _____ - 5 ___ -5 x < 1 OR x > 3

3 2 1 0 4 5 6

23. x + 3 < 0 OR x - 2 > 0 ____ - 3 ___ -3 _____ + 2 ___ +2 x < -3 OR x > 2

0 1 2 3 -1 -2 -3

24. p < 0 OR p > 5 25. q < 0 OR q ≥ 2

26. -6 < r ≤ 5 27. -2 < s < 1

28. Let f represent the frequencies in Hz. 82.4 ≤ f ≤ 659.2

659.2 82.4

29a. 225 + 80n gives the cost of the studio and technicians. They will spend between $200 and $550.

b. 200 ≤ 225 + 80n ≤ 550 _____ -225 __________ -225 _____ -225 -25 ≤ 80n ≤ 325

-25 _ 80

≤ 80n _ 80

≤ 325 _ 80

-0.3125 ≤ n ≤ 4.0625 n cannot be negative since time cannot be

negative.

c. Right now they can use the studio for 4.0625 hours. 6 h - 4.0625 h = 1.9375 h

1.9375 h · $80

____ 1 h

= $155

They need an additional $155 to use the studio for 6 hours.

30. -6 < x < 6

0 2 4 6 -2 -4 -6

31. 1 ≤ x ≤ 2

0 1 2 3 -1 -2 -3


CS10_A1_MESK710372_C02.indd 60 3/30/11 10:36:03 PM

32. 0 < x < 15

5 0 10 15 20

33. -10 ≤ x ≤ 10

0 10 20 -20 -10

34. 55 - 3 ≤ s ≤ 55 + 3 52 ≤ s ≤ 58

56 54 52 58

35. t < 0 OR t > 100

36. 5 ≤ 4b - 3 ≤ 9 ___ +3 ______ + 3 ___ +3 8 ≤ 4b ≤ 12

8 _ 4 ≤ 4b _

4 ≤ 12 _

4

2 ≤ b ≤ 3

3 2 1 0 4 5 6

37. -3 < x - 1 < 4 ___ +1 _____ + 1 ___ +1 -2 < x < 5

0 2 4

5

6 -2

38. r + 2 < -2 OR r - 2 > 2 ____ - 2 ___ -2 ____ + 2 ___ +2 r < -4 OR r > 4

0 4 8 -8 -4

39. 2a - 5 < -5 OR 3a - 2 > 1 ______ + 5 ___ +5 ______ + 2 ___ +2 2a < 0 OR 3a > 3

2a _ 2 < 0 _

2 3a _

3 > 3 _

3

a < 0 OR a > 1

0 1 2 3 -1 -2 -3

40. x - 4 ≥ 5 AND x - 4 ≤ 5 ____ + 4 ___ +4 _____ + 4 ___ +4 x ≥ 9 AND x ≤ 9

8 9 10 11 7

41. n - 4 < -2 OR n + 1 > 6 _____ + 4 ___ +4 _____ - 1 ___ -1 n < 2 OR n > 5

3 2 1 0 4 5 6

42. Let w represent the original weight of the ball. 14 ≤ w + 1.5 ≤ 16 ____ -1.5 ______ - 1.5 ____ -1.5 12.5 ≤ w ≤ 14.5

43. Let m represent the additional miles needed. 207 ≤ 200 + m ≤ 260 _____ -200 _________ -200 _____ -200 7 ≤ m ≤ 60

44. Possible answer: Margaret is expecting between 25 and 35 guests; 25 ≤ g ≤ 35 where g is a natural number.

45. Both graphs include numbers less than 3. The graph of x < 3 AND x < 7 does not include numbers greater than or equal to 3. The graph of x < 3 OR x < 7 also includes all numbers less than 7.

46. An inequality with OR always has solutions because the two simple inequalities do not have to be true at the same time. A compound inequality with no solutions must involve AND.

teSt prep

47. D; Solving both inequalities gives x < -1 OR x > 3, which represnts all real numbers greater than 3 or less than -1.

48. H; Solving gives x < 5 AND x > -1, which is graph H.

49. B; The open circle at 2 and arrow pointing left means that < is used. Therefore, the answer must be B or D. The closed circle at 5 and the arrow pointing right means that ≥ must be used so the answer is B.

50. H; Since 2 + 1 = 3 ≥ 3 AND 2 + 1 = 3 ≤ 3, x = 2 is a solution to both inequalities and therefore a solution to the compound statement.


51. 2c - 10 < 5 - 3c < 7c 2c - 10 < 5 - 3c AND 5 - 3c < 7c ________ +3c ______ + 3c ______ + 3c ____ +3c 5c - 10 < 5 AND 5 < 10c _______ + 10 ____ +10 5c < 15 AND 5 < 10c

5c _ 5 < 15 _

5 5 _

10 < 10c _

10

c < 3 AND 0.5 < c 0.5 < c < 3

2 1.5 1 0.5 2.5 3

52. 5p - 10 < p + 6 < 3p 5p - 10 < p + 6 AND p + 6 < 3p _______ -p ______ -p ______ -p ___ -p 4p - 10 < 6 AND 6 < 2p _______ + 10 ____ +10 4p < 16 AND 6 < 2p

4p

_ 4 < 16 _

4 6 _

2 <

2p _

2

p < 4 AND 3 < p 3 < p < 4

3 2 1 0 4 5 6

53. 2s ≤ 18 - s OR 5s ≥ s + 36 ___ +s ______ + s ___ -s _______ -s 3s ≤ 18 OR 4s ≥ 36

3s _ 3 ≤ 18 _

3 4s _

4 ≥ 36 _

4

s ≤ 6 OR s ≥ 9

3 0 6 9 12

54. 9 - x ≥ 5x OR 20 - 3x ≤ 17 _____ + x ___ +x ________ -20 ____ -20 9 ≥ 6x OR -3x ≤ -3

9 _ 6 ≥ 6x _

6 -3x _

-3 ≥ -3 _

-3

1 1 __ 2 ≥ x OR x ≥ 1

1 ≤ x ≤ 1 1 __ 2

0 1 2 1 1 __ 2

1 __ 2

55. -1 ≤ x ≤ 3


CS10_A1_MESK710372_C02.indd 61 3/30/11 10:36:07 PM

56. Solve for x. x + 2 ≥ a AND x - 7 ≤ b ____ - 2 ___ -2 _____ + 7 ___ +7 x ≥ a - 2 AND x ≤ b + 7 a - 2 ≤ x ≤ b + 7 For x = 1 to be the only solution, it must be true that

1 ≤ x ≤ 1. Therefore: a - 2 = 1 AND b + 7 = 1 _____ + 2 ___ +2 _____ - 7 ___ -7 a = 3 AND b = -6 So for x = 1 to be the only solution, a must be equal

to 2 and b must be equal to -6.

solvInG Absolute-vAlue InequAlItIes

CheCk it out!

1a. 2 ⎜x⎟ ≤ 6

2 ⎜x⎟

____ 2 ≤ 6 __

2

⎜x⎟ ≤ 3-3 ≥ x ≤ 3

0 1 2 3 4 5-1-2-3-4-5

1b. ⎜x + 3⎟ - 4.5 ≤ 7.5 ___________ + 4.5 _____ + 4.5 ⎜x + 3⎟ ≤ 12x + 3 ≤ 12 AND x + 3 ≥ -12 _____ + 3 ___ - 3 _____ - 3 ____ - 3 x ≤ 9 AND x ≥ -15-15 ≤ x ≤ 9

0 5 10 15-5-10-15-20

9

2a. |x| + 10 ≥ 12 ______ - 10 ____ -10 |x| ≥ 2 x ≤ -2 OR x ≥ 2

0 1 2 3-2 -1-3

2b. ⎜x + 2 1 _ 2

⎟ + 1 _ 2 ≥ 4

___________

- 1 _ 2

___ - 1 _

2

⎜x + 2 1 _ 2

⎟ ≥ 3 1 _ 2

x + 2 1 _ 2 ≤ -3 1 _

2 OR x + 2 1 _

2 ≥ 3 1 _

2

______

- 2 1 _ 2

____ -2 1 _

2

______ -2 1 _

2

____ -2 1 _

2

x ≤ -6 OR x ≥ 1

0

1

2 4-6 -4 -2-8

3. Let acceptable pressure be represented by p. Then, ⎜p - 125⎟ ≤ 75p - 125 ≤ 75 AND p - 125 ≥ -75 _______ + 125 _____ + 125 _______ + 125 _____ + 125 p ≤ 200 AND p ≥ 5050 ≤ p ≤ 200The range of acceptable pressures is 50 ≤ p ≤ 200

125100755025 150 175 200 225 4a. ⎜x⎟ - 9 ≥ -11

___ + 9 ___ + 9 ⎜x⎟ ≥ -2This is true for all real numbers; all real numbers are solutions.

b. 4 ⎜x - 3.5 ⎟ ≤ -8

4 ⎜x - 3.5⎟

_________ 4 ≤ -8 ___

4

⎜x - 3.5⎟ ≤ -2The is false for all real numbers; there are no solutions.

think and disCuss

1. The former has solution x ≤ 3 (all real numbers to the left of and including 3), whereas the latter has solution x < 3 (all real numbers to the left of, but not including 3).

2. Absolute-Value Inequalities

OR |x| > 4

x > 4 or x < -4

AND |x| < 5

-5 < x < 5

exCerCisesguided practice 1. |x| - 5 ≤ -2 _____ + 5 ___ +5 |x| ≤ 3 x ≥ -3 AND x ≤ 3 -3 ≤ x ≤ 3

10 2 3-2 -1-3

2. |x + 1| - 7.8 < 6.2 __________ + 7.8 ____ +7.8 |x + 1| < 14.0 x + 1 > -14 AND x + 1 < 14 ____ - 1 ____ -1 _____ - 1 ___ -1 x > -15 AND x < 13 -15 < x < 13

0 15

13

-15


x-7


2-7

CS10_A1_MESK710372_C02.indd 62 3/30/11 10:36:11 PM

3. |3x| + 2 < 8 ______ - 2 ___ -2 |3x| < 6 3x > -6 AND 3x < 6

3x _ 3 > -6 _

3 3x _

3 < 6 _

3

x > -2 AND x < 2 -2 < x < 2

0 1 2 3-1-2-3

4. 4 ⎜x⎟ ≤ 20

4 ⎜x⎟

____ 4 ≤ 20 ___

4

⎜x⎟ ≤ 5-5 ≤ x ≤ 5

0 1 2 3 4 5 -1 -2 -3 -4 -5

5. ⎜x - 5⎟ + 1 < 2 __________ - 1 ___ - 1 ⎜x - 5⎟ < 1x - 5 < 1 AND x - 5 > -1 _____ + 5 ___ + 5 _____ + 5 ___ + 5 x < 6 AND x > 44 < x < 6

3 2 4 5 6 7 8

6. ⎜x + 1 __ 2 ⎟ - 1 __

2 ≤ 3 1 __

2

__________

+ 1 __ 2

____ + 1 __

2

⎜x + 1 __ 2 ⎟ ≤ 4

x + 1 __ 2 ≤ 4 AND x + 1 __

2 ≥ -4

_____

- 1 __ 2

____ - 1 __

2

______ - 1 __

2

____ - 1 __

2

x ≤ 3 1 __ 2 AND x ≥ -4 1 __

2

-4 1 __ 2 ≤ x ≤ 3 1 __

2

0 1 2 3 4 5 -1 -2 -3 -4 -5

3 1 __ 2 - 4 1 __

2

7. |x| - 6 > 16 _____ + 6 ___ +6 |x| > 22 x < -22 OR x > 22

0 11 22-22 -11

8. |x| + 2.9 > 8.6 _______ - 2.9 ____ -2.9 |x| > 5.7 x < -5.7 OR x > 5.7

0 5

5.7-5.7

10-10 -5

9. 2 ⎜x⎟ ≥ 8

2 ⎜x⎟

____ 2 ≥ 8 __

2

⎜x⎟ ≥ 4x ≤ -4 OR x ≥ 4

0 1 2 3 4 5 -1 -2 -3 -4 -5

10. ⎜x + 2⎟ > 7x + 2 < -7 OR x + 2 > 7 _____ - 2 ___ - 2 _____ - 2 ___ - 2 x < -9 OR x > 5

0 5 10 -5 -10

-9

11. ⎜x - 3⎟ + 2 ≥ 4 ___________ - 2 ___ - 2 ⎜x - 3⎟ ≥ 2x - 3 ≥ 2 OR x - 3 ≤ -2 _____ + 3 ___ + 3 _____ + 3 ___ + 3 x ≥ 5 OR x ≤ 1

1 0 2 3 4 5 6

12. |x + 5| - 4 1 _ 2 ≥ 7 1 _

2

__________

+ 4 1 _ 2

____ +4 1 _

2

|x + 5| ≥ 12 x + 5 ≤ -12 OR x + 5 ≥ 12 ____ - 5 ____ -5 _____ - 5 ___ -5 x ≤ -17 OR x ≥ 7

0 10

-17 7

20-20 -10

13. Let x represent an acceptable level of fat intake (in grams) per day. Then, ⎜x - 55⎟ ≤ 25x - 55 ≤ 25 AND x - 55 ≥ -25 ______ + 55 ____ + 55 ______ + 55 ____ + 55 x ≤ 80 AND x ≥ 3030 ≤ x ≤ 80Therefore an acceptable range of fat intake is 30 grams ≤ x ≤ 80 grams.

30 20 40 50 60 70 80

14. ⎜x⎟ + 8 ≤ 2 ______ - 8 ___ - 8 ⎜x⎟ ≤ -6False for all real numbers; there are no solutions.

15. ⎜x + 3⎟ < -5False for all real numbers; there are no solutions.

16. ⎜x + 4⎟ ≥ -8True for all real numbers; therefore solution set is all real numbers.

17. ⎜x - 5⎟ + 1 __ 3 > -1

__________

- 1 __ 3

_____ - 1 __

3

⎜x - 5⎟ > -1 1 __ 3

True for all real numbers; therefore solution set is all real numbers.

18. ⎜3x⎟ + 7 > 2 _______ - 7 ___ - 7 ⎜3x⎟ > -5True for all real numbers; therefore solution set is all real numbers.

19. ⎜x - 7⎟ + 3.5 ≤ 2 ___________ - 3.5 _____ - 3.5 ⎜x - 7⎟ ≤ -1.5False for all real numbers; there are no solutions.


CS10_A1_MESK710372_C02.indd 63 3/30/11 10:36:17 PM

20. ⎜x⎟ + 6 ≤ 10 ______ - 6 ___ - 6 ⎜x⎟ ≤ 4x ≤ 4 AND x ≥ -4-4 ≤ x ≤ 4

0 1 2 3 4 5 -1 -2 -3 -4 -5

21. ⎜x - 3⎟ < 1x - 3 < 1 AND x - 3 > -1 _____ + 3 ___ + 3 _____ + 3 ___ + 3 x < 4 AND x > 22 < x < 4

2 1 3 4 5 6

22. ⎜x - 2⎟ - 8 ≤ -3 __________ + 8 ___ + 8 ⎜x - 2⎟ ≤ 5x - 2 ≤ 5 AND x - 2 ≥ -5 _____ + 2 ___ + 2 _____ + 2 ___ + 2 x ≤ 7 AND x ≥ -3-3 ≤ x ≤ 7

0 1 2 3 4 5 6 7 -1 -2 -3

23. ⎜5x⎟ < 155x < 15 AND 5x > -15

5x ___ 5 < 15 ___

5 AND 5x ___

5 > - 15 ___

5

x < 3 AND x > -3-3 < x < 3

0 1 2 3 4 5 -1 -2 -3 -4 -5

24. ⎜x - 2.4⎟ + 4 ≤ 6.4 ___________ - 4 ___ - 4 ⎜x - 2.4⎟ ≤ 2.4x - 2.4 ≤ 2.4 AND x - 2.4 ≥ -2.4 ______ + 2.4 _____ + 2.4 ______ + 2.4 _____ + 2.4 x ≤ 4.8 AND x ≥ 00 ≤ x ≤ 4.8

0 1 2 3 4 5

4.8

25. 4 + ⎜x + 3⎟ < 7 ____________ - 4 ___ - 4 ⎜x + 3⎟ < 3x + 3 < 3 AND x + 3 > -3 _____ - 3 ____ - 3 _____ - 3 ___ - 3 x < 0 AND x > -6-6 < x < 0

0 2 -8 -6 -4 -2

26. ⎜x - 1⎟ > 2x - 1 > 2 OR x - 1 < -2 _____ + 1 ____ + 1 _____ + 1 ____ + 1 x > 3 OR x < -1

0 1 2 3 4 5 -1 -2 -3 -4 -5

27. 6 ⎜x⎟ ≥ 60

6 ⎜x⎟

____ 6 ≥ 60 ___

6

⎜x⎟ ≥ 10x ≤ -10 OR x ≥ 10

0 10 20 -10 -20 28. ⎜x - 4⎟ + 3 > 8

__________ - 3 ___ - 3 ⎜x - 4⎟ > 5x - 4 > 5 OR x - 4 < -5 _____ + 4 ___ + 4 _____ + 4 ___ + 4 x > 9 OR x < -1

0 5 10 -5 -10

-1 9

29. 2 ⎜x+ 2⎟ ≥ 16

2 ⎜x + 2⎟

_______ 2 ≥ 16 ___

2

⎜x + 2⎟ ≥ 8x + 2 ≥ 8 OR x + 2 ≤ -8 _____ - 2 ___ - 2 _____ - 2 ___ - 2 x ≥ 6 OR x ≤ -10

0 5 10 -5 -10

6

30. 3 + ⎜x - 4⎟ > 4 ___________ - 3 ___ - 3 ⎜x - 4⎟ > 1x - 4 > 1 OR x - 4 < -1 _____ + 4 ___ + 4 _____ + 4 ___ + 4 x > 5 OR x < 3

2 1 3 4 5 6 7

31. ⎜x - 1 __ 2 ⎟ + 9 > 10 1 __

2

__________ - 9 ___ - 9

⎜x - 1 __ 2 ⎟ > 1 1 __

2

x - 1 __ 2 > 1 1 __

2 OR x - 1 __

2 < -1 1 __

2

_____

+ 1 __ 2

____ + 1 __

2

_____ + 1 __

2

____ + 1 __

2

x > 2 OR x < -1

0 1 2 3 4 5 -1 -2 -3 -4 -5


CS10_A1_MESK710372_C02.indd 64 3/30/11 10:36:22 PM

32. Let the actual temperature be represented by t, then ⎜t - 175⎟ ≤ 12t - 175 ≤ 12 AND t - 175 ≥ -12 _______ + 175 _____ + 175 _______ + 175 _____ + 175 t ≤ 187 AND t ≥ 163163 ≤ t ≤ 187The range of possible temperatures is 163 to 187.

160 165 170 175 180 185 190

187 163

33. 12 + ⎜x⎟ ≤ 10

_________ _______ -12 ____ - 12 ⎜x⎟ ≤ -2

False for all real numbers; there are no solutions.

34. ⎜x + 3 __ 5 ⎟ - 2 > -4

__________ + 2 ___ + 2

⎜x + 3 __ 5 ⎟ > -2

True for all real numbers; therefore solution set is the set of real numbers.

35. ⎜x + 1⎟ + 5 ≥ 4 __________ - 5 ___ - 5 ⎜x + 1⎟ ≥ -1True for all real numbers; therefore solution set is the set of real numbers.

36. ⎜4x⎟ - 3 < -6 _______ + 3 ___ + 3 ⎜4x⎟ < -3False for all real numbers; there are no solutions.

37. 3 ⎜x - 4⎟ ≤ -9

3 ⎜x - 4⎟

_______ 3 ≤ -9 ___

3

⎜x - 4⎟ ≤ -3False for all real numbers; there are no solutions.

38. ⎜2x⎟ + 9 ≥ 9 - 9 - 9 ⎜2x⎟ ≥ 0True for all real numbers; therefore solution set is the set of real numbers.

39. Always true since the absolute value is always greater than or equal to zero, which is greater than all negative numbers.

40. Never true since always positive and is therefore always greater than zero.

41. This is sometimes true as there are times when the solution may be Ø or some other set such as x > 4.

42. |x| ≤ 15 x ≥ -15 AND x ≤ 15 -15 ≤ x ≤ 15

0 15 30-30 -15

43. |x - 2| ≤ 3 x - 2 ≥ -3 AND x - 2 ≤ 3 ____ + 2 ___ +2 _____ + 2 ___ +2 x ≥ -1 AND x ≤ 5 -1 ≤ x ≤ 5

210 3 4 5-1

44. |x - 8| ≥ 2 x - 8 ≤ -2 OR x - 8 ≥ 2 ____ + 8 ___ +8 _____ + 8 ___ +8 x ≤ 6 OR x ≥ 10

8765 9 10 11

45. |a| ≤ 2 46. |b| > 3

47. |c| ≥ 6 1 _ 2 48. |d| < 7

49a. Middle frequency = 20000 + 20

__________ 2

= 20020

______ 2

= 10010 Hz.

49b. An inequality representing the range of frequencies is:

20 ≤ ƒ ≤ 20000 or ⎜ƒ - 10010⎟ ≤ 9990 where ƒ represents frequency.

50. Let t represent temperature. Then

sea Bass rainBoW trout euroPean eeL

12 ≤ t ≤ 30 4 ≤ t ≤ 26 12 ≤ t ≤ 34

21 - 9 ≤ t ≤ 21 + 9

15 - 11 ≤ t ≤ 15 + 11

13 - 11 ≤ t ≤ 23 + 11

-9 ≤ t - 21 ≤ 9

-11 ≤ t - 15 ≤ 11

-11 ≤ t - 23 ≤ 11

⎜t - 21⎟ ≤ 9 ⎜t - 15⎟ ≤ 11 ⎜t - 23⎟ ≤ 11

51. Let n represent the number, then n > 23 + 12 OR n < 23 - 12 ____ - 23 ________ - 23 _____ - 23 _______ - 23 -23 + n > 12 OR -23 + n < -12 ⎜n - 23⎟ > 12

52a. This is given by the difference between the values p and 8.75 or ⎜p - 8.75⎟ .

b. We require this value to be less than or equal to $1.25, therefore ⎜p - 8.75⎟ ≤ 1.25.

c. ⎜p - 8.75⎟ ≤ 1.25p - 8.75 ≤ 1.25 OR p - 8.75 ≥ -1.25 ________ + 8.75 ______ + 8.75 ________ + 8.75 ______ + 8.75 p ≤ 10.00 OR p ≥ 7.50$7.50 ≤ p ≤ $10.00

53. ⎜x⎟ + 1 < k ______ - 1 ___ - 1 ⎜x⎟ < k - 1For no solutions, k - 1 must be less than or equal to zero, that is, k -1 ≤ 0k ≤ 1.


CS10_A1_MESK710372_C02.indd 65 3/30/11 10:36:24 PM

54. For the values within a range, use “less than”. The distance from the number to 6 must be less than 5 units. So the absolute value of the difference of a number and 6 is less than 5. |x - (-6)| < 5.

|x + 6| < 5.

teSt prep

55. 3 + ⎜x + 4⎟ < 6 _________ - 3 ___ - 3 ⎜x + 4⎟ < 3x + 4 < 3 AND x + 4 > -3 _____ - 4 ___ - 4 ______ - 4 ___ - 4 x < -1 AND x > -7-7 < x < -1Therefore the best answer is B.

56. x ≤ 75 + 2 AND x ≥ 75 - 2 ____ - 75 _______ - 75 ____ - 75 _______ - 75 x - 75 ≤ 2 AND x - 75 ≥ -2 ⎜x - 75⎟ ≤ 2Therefore the best answer is F.

57. ⎜w - 156⎟ ≤ 3w - 156 ≤ 3 AND w - 156 ≥ -3 _______ + 156 _____ + 156 _______ + 156 _____ + 156 w ≤ 159 AND w ≥ 153153 ≤ w ≤ 159Clearly, only B is not true.


58. x ≤ 4.2 AND x ≥ -2.4 x ≤ 0.9 + 3.3 AND x ≥ 0.9 - 3.3 ________ -0.9 ________ -0.9 ______ -0.9 ________ -0.9 x - 0.9 ≤ 3.3 AND x - 0.9 ≥ -3.3 ⎜x - 0.9⎟ ≤ 3.3

59. x > 3 1 __ 2 OR x < -1 1 __

2

x > 1 + 2 1 __ 2 OR x < 1 - 2 1 __

2

____ - 1 ______ - 1 ___ - 1 _______ - 1

x - 1 > 2 1 __ 2 OR x - 1 < -2 1 __

2

⎜x - 1⎟ > 2 1 __ 2

60. statements reason

1. ⎜2x - 6⎟ + 5 ≤ 7 Given

2. ⎜2x - 6⎟ ≤ 2 Subtraction Property of Inequality

3. 2x - 6 ≥ -2 AND 2x - 6 ≤ 2

Definition of absolute value

4. 2x ≥ 4 AND 2x ≤ 8 Addition Property of Inequality

5. x ≥ 2 AND x ≤ 4 Division Property of Inequality

reAdY to Go on? section b quiz

1. 2x + 3 < 9 _____ - 3 ___ -3 2x < 6

2x _ 2 < 6 _

2

x < 3

3 4 5 6 0 1 2

2. 3t - 2 > 10 _____ + 2 ___ +2 3t > 12

3t _ 3 > 12 _

3

t > 4

3 4 5 6 0 1 2

3. 7 ≥ 1 - 6r ___ -1 _______ -1 6 ≥ -6r

6 _ -6

≤ -6r _ -6

-1 ≤ r r ≥ -1

0 1 2 3 -3 -2 -1

4. 2(x - 3) > -1 2(x) + 2(-3) > -1 2x - 6 > -1 ______ + 6 ___ +6 2x > 5

2x _ 2 > 5 _

2

x > 2.5

5. 1 __ 3 a + 1 __

2 > 2 __

3

- 1 __ 2 - 1 __

2

1 __ 3 a > 1 __

6

3 1 __ 3 a > 3 1 __

6

a > 1 __ 2

6. 15 < 5(m - 7) 15 < 5(m) + 5(-7) 15 < 5m - 35 ____ +35 _______ + 35 50 < 5m

50 _ 5 < 5m _

5

10 < m m > 10

7. 2 + (-6) > 0.8p -4 > 0.8p

-4 ___ 0.8

> 0.8p

____ 0.8

-5 > p p < -5

8. Let s represent her mark on the second test.

s + 88 _ 2 ≥ 92

2 ( s + 88 _ 2 ) ≥ 2(92)

s + 88 ≥ 184 ______ - 88 ____ -88 s ≥ 96 Mindy must make a score of 96 or higher.

9. 5x < 3x + 8 ____ -3x _______ -3x 2x < 8

2x _ 2 < 8 _

2

x < 4

3 4 5 6 0 1 2

10. 6p - 3 > 9p _______ -6p ____ -6p -3 > 3p

-3 _ 3 >

3p _

3

-1 > p p < -1

0 1 2 3 -3 -2 -1


CS10_A1_MESK710372_C02.indd 66 3/30/11 10:36:27 PM

11. r - 8 ≥ 3r - 12 ______ -r _______ -r -8 ≥ 2r - 12 ____ +12 ______ + 12 4 ≥ 2r

4 _ 2 ≥ 2r _

2

2 ≥ r r ≤ 2

0 1 2 3 -3 -2 -1

12. 3(y + 6) > 2(y + 4) 3(y) + 3(6) > 2(y) + 2(4) 3y + 18 > 2y + 8 ________ -2y _______ -2y y + 18 > 8 ______ - 18 ____ -18 y > -10

13. 4(5 - g) ≥ g 4(5) + 4(-g) ≥ g 20 - 4g ≥ g _______ + 4g ____ +4g 20 ≥ 5g

20 _ 5 ≥

5g _

5

4 ≥ g g ≤ 4

14. 4x < 4(x - 1) 4x < 4(x) + 4(-1) 4x < 4x - 4 ____ -4x _______ -4x 0 < -4 7 no solutions

15. 3(1 - x) ≥ -3(x + 2) 3(1) + 3(-x) ≥ -3(x) - 3(2) 3 - 3x ≥ -3x - 6 ______ + 3x _______ +3x 3 ≥ -6 3 all real numbers

16. Let m represent the number of months. 100 + 18m < 145 + 15m _________ - 15m _________ - 15m 100 + 3m < 145 __________ -100 _____ -100 3m < 45

3m _ 3 < 45 _

3

m < 15 Gil will have a larger bank balance than Phillip for

15 months.

17. -2 ≤ x + 3 < 9 ___ -3 _____ - 3 ___ -3 -5 ≤ x < 6

0 -2 2 4 6 -4 -6

-5

18. m + 2 < -1 OR m - 2 > 6 _____ - 2 ___ -2 _____ + 2 ___ +2 m < -3 OR m > 8

2 0 4 6 8 -2

-3

-4

19. -3 ≥ x - 1 > 2 ____ + 1 ______ + 1 ___ + 1 -2 ≥ x > 3x ≤ -2 AND x > 3No solution.

20. -2 > r + 2 OR r + 4 < 5 ___ -2 _____ - 2 _____ - 4 ___ -4 -4 > r OR r < 1 r < 1

0 1 2 3 -1 -2 -3

21. Let t represent the temperature of the medicine. 32 < t < 70 22. ⎜x⎟ + 9 ≤ 12

_____ - 9 ___ - 9 ⎜x⎟ ≤ 3-3 ≤ x ≤ 3

3 -3

23. ⎜x + 7⎟ - 15 < 6 ___________ + 15 ____ + 15 ⎜x + 7⎟ < 21x + 7 < 21 AND x + 7 > -21 _____ - 7 ___ - 7 ______ - 7 ____ - 7 x < 14 AND x > -28-28 < x < 14

14 -28

24. 4.5 ⎜x ⎟ ≥ 31.5

4.5 ⎜x⎟

_____ 4.5

≥ 31.5 ____ 4.5

⎜x⎟ ≥ 7x ≤ -7 OR x ≥ 7

7 -7

25. ⎜x - 2⎟ ≤ 14x - 2 ≤ 14 AND x - 2 ≥ -14 _____ + 2 ___ + 2 _____ + 2 ____ + 2 x ≤ 16 AND x ≥ -12-12 ≤ x ≤ 16

26. ⎜x⎟ - 9.2 < -5.7 + 9.2 + 9.2 ⎜x⎟ < 3.5-3.5 < x < 3.5

27. 1 __ 2 + 2 ⎜x⎟ > -4

________

- 1 __ 2

____ - 1 __

2

2 ⎜x⎟ > - 9 __ 2

2 ⎜x⎟

____ 2 > - 9 __

2 ÷ 2

⎜x⎟ > - 9 __ 4

⎜x⎟ is always greater than - 9 __ 4 . So x is all real

numbers.


CS10_A1_MESK710372_C02.indd 67 3/30/11 10:36:31 PM

28. 7 + 3 ⎜x⎟ > 13 _______ - 7 ___ - 7 3 ⎜x⎟ > 6

3 ⎜x⎟

____ 3 > 6 __

3

⎜x ⎟ > 2

x < -2 OR x > 2

29. ⎜d - 110⎟ ≤ 22d - 110 ≤ 22 AND d - 110 ≥ -22 _______ + 110 _____ + 110 ________ + 110 _____ + 110 d ≤ 132 AND d ≥ 8888 ≤ d ≤ 132

132 88

studY GuIde: revIew

1. inequality 2. union

3. compound inequality 4. intersection

5. solution of an inequality

GraPhinG and WritinG inequaLities

6. 0 3 6 -6 -3

7. 0 1 2 3 4 5 6

8. 0 1 2 3 -1 -2 -3

9. 9.5 9 10 10.5 11

10. 2(3 - 5) < k 2(-2) < k -4 < k k > -4

0 -4 -3 -2 -1 -5 -6

11. 3 2 1 0 4 5 6

12. a < 2 13. k ≥ -3.5

14. q < -10

15. Let t represent the temperature. t ≥ 72

24 0 48 72 96

16. Let s represent the number of students present. 0 ≤ s ≤ 12, where s is a natural number

9 6 3 0 12 15 18

17. Let m represent the number of minutes to complete the lab.

0 ≤ m < 30 10 0 20 30 40

soLvinG inequaLities By addinG or suBtraCtinG

18. t + 3 < 10 ____ - 3 ___ -3 t < 7

5 6 7 8 9

19. k - 7 ≤ -5 ____ + 7 ___ +7 k ≤ 2

0 1 2 3 -1-2 -3

20. -1 < m + 4 ___ -4 _____ - 4 -5 < m m > -5

0 -4 -3 -2 -1 -5 -6

21. x + 2.3 ≥ 6.8 ______ - 2.3 ____ -2.3 x ≥ 4.5

4.5 4 5 5.5 6

22. w - 3 < 6.5 _____ + 3 _____ +3 w < 9.5

8.5 8 9 9.5 10

23. 4 > a - 1 ___ +1 _____ + 1 5 > a a < 5

3 2 1 0 4 5 6

24. h - 1 _ 4 < 3 _

4

_____

+ 1 _ 4

___ + 1 _

4

h < 1

0 1 2 3 -1 -2 -3

25. 5 > 7 + v ___ -7 ______ -7 -2 > v v < -2

0 1 2 3 -1 -2 -3

26. Let m represent the number of miles left to run. 4.5 + m ≥ 10 ________ -4.5 ____ -4.5 m ≥ 5.5 Tammy must run 5.5 mi or more.

27. Let d represent the amount Rob can spend. 32 + d ≤ 50 _______ -32 ____ -32 d ≤ 18 Rob can spend $18 or less.

soLvinG inequaLities By muLtiPLyinG or dividinG

28. 3a ≤ 15

3a _ 3 ≤ 15 _

3

a ≤ 5

3 2 1 0 4 5 6

29. -18 < 6t

-18 _ 6 < 6t _

6

-3 < t t > -3

0 -4 -3 -2 -1 -5 -6

30. p _

4 > 2

4 ( p _

4 ) > 4(2)

p > 8

6 4 2 0 8 10 12

31. 2 _ 5 x ≤ -10

5 _ 2 ( 2 _

5 x) ≤ 5 _

2 (-10)

x ≤ -25

-35 -30 -25 -20 -15

32. -3n < -18

-3n _ -3

> -18 _ -3

n > 6

6 4 2 0 8 10 12

33. g _

-2 > 6

-2 ( g _

-2 ) < -2(6)

g < -12

0 -16 -12 -8 -4


x-4

x-1

x-2

x-3


2-2

2-1

2-3

CS10_A1_MESK710372_C02.indd 68 3/30/11 10:36:38 PM

34. -2k < 14

-2k _ -2

> 14 _ -2

k > -7

-9 -8 -7 -6 -5

35. -3 > 1 _ 3 r

3(-3) > 3 ( 1 _ 3 r)

-9 > r r < -9

0 -12 -9 -6 -3

36. 27 < -9h

27 _ -9

> -9h _ -9

-3 > h h < -3

0 -4 -3 -2 -1 -5 -6

37. -0.4g > -1

-0.4g

_ -0.4

< -1 _ -0.4

g < 2.5

1.5 1 2 2.5 3

38. Let n represent the number of notebooks. 1.39n ≤ 10

1.39n _ 1.39

≤ 10 _ 1.39

n ≤ 7.19 Only a whole number of notebooks can be

purchased, so 0, 1, 2, 3, 4, 5, 6, or 7 notebooks can be purchased.

39. Let n represent the number of lanyards. 0.75n ≥ 250

0.75n _ 0.75

≥ 250 _ 0.75

n ≥ 333 1 _ 3

They must sell at least 334 lanyards.

soLvinG tWo-steP and muLti-steP inequaLities

40. 3x + 4 < 19 _____ - 4 ___ -4 3x < 15

3x _ 3 < 15 _

3

x < 5

3 2 1 0 4 5 6

41. 7 ≤ 2t - 5 ___ +5 _____ + 5 12 ≤ 2t

12 _ 2 ≤ 2t _

2

6 ≤ t t ≥ 6

6 4 2 0 8 10 12

42. m + 3 ______ 2 > -4

2 × m +3 _____ 2

> 2 × (-4)

m + 3 > -8 _______ - 3 ____ - 3 m > -11

-8 -12 -11 -10 -9

43. 2(x + 5) < 8 2(x) + 2(5) < 8 2x + 10 < 8 _______ - 10 ____ -10 2x < -2

2x _ 2 < -2 _

2

x < -1

0 1 2 3 -1 -2 -3

44. -4(2 - 5) > (-3 ) 2 - h -4(-3) > (-3 ) 2 - h 12 > 9 - h ___ -9 _______ -9 3 > -h -1(3) < -1(-h) -3 < h h > -3

0 -4 -3 -2 -1 -5 -6

45. 1 _ 5

x + 1 _ 2

> 4 _ 5

______

- 1 _ 2

___

- 1 _ 2

1 _ 5

x > 3 _ 10

5 ( 1 _ 5

x) > 5 ( 3 _ 10

)

x > 1 1 _ 2

0 1 2 1 __ 2

1 1 __ 2

46. 0.5(b - 2) ≤ 4 0.5(b) + 0.5(-2) ≤ 4 0.5b - 1 ≤ 4 _______ + 1 ___ +1 0.5b ≤ 5

0.5b _ 0.5

≤ 5 _ 0.5

b ≤ 10

6 8 10 0 2 4 12

47. 1 _ 3

y - 1 _ 2

> 2 _ 3

______

+ 1 _ 2

___

+ 1 _ 2

1 _ 3

y > 7 _ 6

3 ( 1 _ 3

y) > 3 ( 7 _ 6

)

y > 3 1 _ 2

3 4 5 4 1 __ 2

3 1 __ 2

48. 6 - 0.2n < 9

_________ -6 ___ -6 -0.2n < 3

-0.2n _ -0.2

> 3 _ -0.2

n > -15

0 -20 -15 -10 -5

49. Let m represent the number of movies per month. 55 + 4m < 110 _________ -55 ____ -55 4m < 55

4m _ 4 < 55 _

4

m < 13.75 Only a whole number of movies can be watched, so

if 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, or 13 movies are watched, Carl’s Cable Company is cheaper than Teleview.

soLvinG inequaLities With variaBLes on Both sides

50. 5 + 2m < -3m ______ - 2m ____ -2m 5 < -5m

5 _ -5

> -5m _ -5

-1 > m m < -1

0 1 2 3 -1 -2 -3

51. y ≤ 6 + 4y ____ -4y ______ - 4y -3y ≤ 6

-3y

_ -3

≥ 6 _ -3

y ≥ -2

0 1 2 3 -1 -2 -3


x-4


2-5

2-4

CS10_A1_MESK710372_C02.indd 69 3/30/11 10:36:44 PM

52. 4c - 7 > 9c + 8 _______ -4c _______ -4c -7 > 5c + 8 ___ -8 ______ - 8 -15 > 5c

-15 _ 5 > 5c _

5

-3 > c c < -3

0 -4 -3 -2 -1 -5 -6

53. -3(2 - q) ≥ 6(q + 1) -3(2) - 3(-q) ≥ 6(q) + 6(1) -6 + 3q ≥ 6q + 6 _______ - 3q ________ -3q -6 ≥ 3q + 6 ___ -6 ______ - 6 -12 ≥ 3q

-12 _ 3 ≥

3q _

3

-4 ≥ q q ≤ -4

0 -4 -3 -2 -1 -5 -6

54. 2(5 - x) < 3x 2(5) + 2(-x) < 3x 10 - 2x < 3x ______ + 2x ____ +2x 10 < 5x

10 _ 5 < 5x _

5

2 < x x > 2

0 1 2 3 -1 -2 -3

55. 3.5t - 1.8 < 1.6t + 3.9 __________ -1.6t __________ -1.6t 1.9t - 1.8 < 3.9 ________ + 1.8 ____ +1.8 1.9t < 5.7

1.9t _ 1.9

< 5.7 _ 1.9

t < 3

3 4 5 0 1 2 6

56. d - 2 < d - 4 ______ -d ______ -d -2 < -4 7 no solutions

57. 2(1 - x) > -2(1 + x) 2(1) + 2(-x) > -2(1) - 2(x) 2 - 2x > -2 - 2x ______ + 2x _______ + 2x 2 > -2 3 all real numbers

58. 4(1 - p) < 4(2 + p) 4(1) + 4(-p) < 4(2) + 4(p) 4 - 4p < 8 + 4p ______ + 4p ______ + 4p 4 < 8 + 8p ___ -8 _______ -8 -4 < 8p

-4 _ 8 <

8p _

8

- 1 _ 2 - 1 _ 2

0 1 1 __ 2 -1 - 1 __

2

59. 3w + 1 > 3(w - 1) 3w + 1 > 3(w) + 3(-1) 3w + 1 > 3w - 3 ________ -3w ________ -3w 1 > -3 3 all real numbers

60. 5(4 - k) < 5k 5(4) + 5(-k) < 5k 20 - 5k < 5k _______ + 5k ____ +5k 20 < 10k

20 _ 10

< 10k _ 10

2 < k k > 2

0 1 2 3 -1 -2 -3

61. 3(c + 1) > 3c + 5 3(c) + 3(1) > 3c + 5 3c + 3 > 3c + 5 _______ -3c _______ -3c 3 > 5 7 no solutions

62. Let m represent the number of months. 210 + 16m > 175 + 20m _________ - 16m _________ - 16m 210 > 175 + 4m _____ -175 ___________ -175 35 > 4m

35 _ 4 > 4m _

4

8.75 > m m < 8.75 Hanna’s acount will be greater than Faith’s account

for 8 months.

soLvinG ComPound inequaLities

63. -4 < t + 6 < 10 ___ -6 _____ - 6 ___ -6 -10 < t < 4

-12 -8

-10

-4 0 4

64. -8 < k - 2 ≤ 5 ___ +2 _____ + 2 ___ +2 -6 < k ≤ 7

0 4

7

8 -8 -4

-6

65. -3 + r > 4 OR r + 1 < -1 ______ +3 ___ +3 ____ - 1 ___ -1 r > 7 OR r < -2

0 2 4 6

7

8 -2 -4


x-5

x-6


2-6

CS10_A1_MESK710372_C02.indd 70 3/30/11 10:36:48 PM

66. 2 > n + 3 > 5 ___ - 3 _____ - 3 ___ - 3 -1 > n > 2n < -1 AND n > 2no solutions

67. 12 ≥ p + 7 > 5- 7 - 7 - 75 ≥ p > - 2-2 s - 4 ___ -9 _____ - 9 ___ +4 _____ + 4 -6 < s OR 5 > s s > -6 OR s < 5 all real numbers

0

69. Let t represent the day’s temperature. 68 ≤ t ≤ 84

70. Let n represent the heart rate for a 16-year-old. 0.5 × (220 - 16) ≤ n ≤ 0.9 × (220 - 16) 0.5 × 204 ≤ n ≤ 0.9 × 204 102 ≤ n ≤ 183.6 71. ⎜x⎟ - 7 ≤ 15

_____ + 7 ___ + 7 ⎜x⎟ ≤ 22-22 ≤ x ≤ 22

11 0 22 -11 -22

72. ⎜x + 4 ⎟ > 8 x + 4 > 8 OR x + 4 < -8 _____ - 4 ____ - 4 _____ - 4 ____ - 4 x > 4 OR x < -12

4 12 -4 -12 ]

73. 6 ⎜x⎟ ≤ 24

6 ⎜x⎟

____ 6 ≤ 24 ___

6

⎜x⎟ ≤ 4-4 ≤ x ≤ 4

4 2 6 0 -2 -4 74. ⎜x + 9⎟ + 11 < 20

___________ - 11 _____ - 11 ⎜x + 9⎟ < 9 x + 9 < 9 AND x + 9 > -9 _____ - 9 ___ - 9 _____ - 9 ____ - 9 x < 0 AND x > -18

0 -9 -6 -3 -12 -15 -18

75. 3 ⎜x⎟ ≥ 9

3 ⎜x⎟

____ 3 ≥ 9 __

3

⎜x⎟ ≥ 3x ≤ -3 OR x ≥ 3

3 1 5 -1 -3 -5 76. 4 ⎜2x⎟ < 24

4 ⎜2x⎟

_____ 4 < 24 ___

4

⎜2x⎟ < 62x < 6 AND 2x > -6

2x ___ 2 < 6 __

2 AND 2x ___

2 > - 6 __

2

x < 3 AND x > -3-3 < x < 3

3 -3 -2 -1 0 1 2

77. ⎜x⎟ - 5.4 > 8.5 ________ + 5.4 _____ + 5.4 ⎜x⎟ > 13.9x < -13.9 OR x > 13.9

78. ⎜5.2 + x⎟ < 7.3 5.2 + x < 7.3 AND 5.2 + x > -7.3 ________ - 5.2 _____ - 5.2 ________ - 5.2 _____ - 5.2 x < 2.1 AND x > -12.5-12.5 < x < 2.1

79. ⎜x - 7⎟ + 10 ≥ 12 ___________ - 10 ____ - 10 ⎜x - 7⎟ ≥ 2x - 7 ≥ 2 OR x - 7 ≤ -2 _____ + 7 ___ + 7 _____ + 7 ___ + 7 x ≥ 9 OR x ≤ 5

80. 14 ⎜x⎟ - 15 ≥ 41 _________ + 15 ____ + 15 14 ⎜x⎟ ≥ 56

14 ⎜x⎟

_____ 14

≥ 56 ___ 14

⎜x⎟ ≥ 4x ≤ -4 OR x ≥ 4

81. ⎜x - 1 __ 2 ⎟ + 4 ≤ 5 __

2

___________ - 4 ___ - 4

⎜x - 1 __ 2 ⎟ ≤ -1 1 __

2

no solutions 82. ⎜x + 5.5⎟ - 6.4 ≤ 4.9

_____________ + 6.4 _____ + 6.4 ⎜x + 5.5⎟ ≤ 11.3x + 5.5 ≤ 11.3 AND x + 5.5 ≥ -11.3 ______ - 5.5 _____ - 5.5 _______ - 5.5 ______ - 5.5 x ≤ 5.8 AND x ≤ -16.8-16.8 ≤ x ≤ 5.8

83. Let actual depth be d, then ⎜d - 72⎟ ≤ 4 d - 72 ≤ 4 AND d - 72 ≥ -4 ______ + 72 ____ + 72 ______ + 72 ____ + 72 d ≤ 76 AND d ≥ 6868 ≤ d ≤ 76


CS10_A1_MESK710372_C02.indd 71 3/30/11 10:36:52 PM

chApter test

1. all real numbers greater than or equal to -6


3. all real numbers less than or equal to -2


5. 0 -4 -3 -2 -1 -5 -6

6. 2 2.5 3 3.5 4

7. y ≤ - √ 25 y ≤ -5

0 -4 -5 -2 -1 -3 -6

8. 3 - (4 + 7) ≥ h 3 - (11) ≥ h -8 ≥ h h ≤ -8

0 -16 -12 -8 -4

9. d < 1 10. p ≥ -4.5

11. Let m represent the number of minutes. m ≤ 9

3 0 6 9 12

12. d - 5 > -7 _____ + 5 ___ +5 d > -2

0 1 2 3 -1 -2 -3

13. f + 4 < -3 ____ - 4 ___ -4 f < -7

-9 -8 -7 -6 -5

14. 4.5 ≥ s + 3.2 ____ -3.2 ______ - 3.2 1.3 ≥ s s ≤ 1.3

0.5 0 1

1.3

1.5 2

15. g + (-2) ≤ 9 g - 2 ≤ 9 _____ + 2 ___ +2 g ≤ 11

10 9 11 12 13

16. Let h represent the number of hours. 48 + h ≥ 75 _______ -48 ____ -48 h ≥ 27 Samir needs at least 27 more hours.


CS10_A1_MESK710372_C02.indd 72 3/30/11 10:36:55 PM

CHAPTER Inequalities 2 x Solutions Key - 9th grade...

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