2.5 Compound Inequalities - DR. POLLONE'S WEBSITE

Section 2.5 Compound Inequalities 89

2.5 Compound Inequalities

OBJECTIVES

1 Find the Intersection of

Two Sets.

2 Solve Compound Inequalities

Containing and.

3 Find the Union of Two Sets.

4 Solve Compound Inequalities

Containing or.

Two inequalities joined by the words and or or are called compound inequalities.

Compound Inequalities x + 3 6 8 and x 7 2 2x ? 5 or -x + 10 6 7 3

OBJECTIVE

1 Finding the Intersection of Two Sets The solution set of a compound inequality formed by the word and is the intersection of the solution sets of the two inequalities. We use the symbol to represent "intersection."

Intersection of Two Sets

The intersection of two sets, A and B, is the set of all elements common to both sets. A intersect B is denoted by A B.

A B

A

B

E X A M P L E 1 If A = 5x x is an even number greater than 0 and less than 106 and B = 53, 4, 5, 66 , find A B .

Solution Let's list the elements in set A.

A = 52, 4, 6, 86

The numbers 4 and 6 are in sets A and B. The intersection is 54, 66 .

PRACTICE

1 If A = 5x x is an odd number greater than 0 and less than 106 and B = 51, 2, 3, 46, find A B.

OBJECTIVE

2 Solving Compound Inequalities Containing "and"

A value is a solution of a compound inequality formed by the word and if it is a

solution of both inequalities. For example, the solution set of the compound inequality x ... 5 and x ? 3 contains all values of x that make the inequality x ... 5 a true statement and the inequality x ? 3 a true statement. The first graph shown below is the graph of x ... 5, the second graph is the graph of x ? 3, and the third graph shows the intersection of the two graphs. The third graph is the graph of x ... 5 and x ? 3.

5x x ... 56

5xx ? 36

5x x ... 5 and x ? 36 also 5x 3 ... x ... 56 (see below)

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

1 -, 5] [3, 2 [3, 5]

Since x ? 3 is the same as 3 ... x, the compound inequality 3 ... x and x ... 5 can be written in a more compact form as 3 ... x ... 5. The solution set 5x 3 ... x ... 56 in-

cludes all numbers that are greater than or equal to 3 and at the same time less than or

equal to 5. In interval notation, the set 5x x ... 5 and x ? 36 or the set 5x 3 ... x ... 56 is

written as 33, 54.

90 CHAPTER 2 Equations, Inequalities, and Problem Solving

Helpful Hint

Don't forget that some compound inequalities containing "and" can be written in a more compact form.

Compound Inequality 2 ... x and x ... 6

Compact Form 2...x...6

Interval Notation 32, 64

Graph:

01 234567

E X A M P L E 2 Solve: x - 7 6 2 and 2x + 1 6 9

Solution First we solve each inequality separately.

x - 7 6 2 and 2x + 1 6 9

x 6 9 and

2x 6 8

x 6 9 and

x64

Now we can graph the two intervals on two number lines and find their intersection. Their intersection is shown on the third number line.

5x x 6 96

3 4 5 6 7 8 9 10 1 - , 92

5x x 6 46 5x 0 x 6 9 and x 6 46 = 5x 0 x 6 46 The solution set is 1 -, 42 .

3 4 5 6 7 8 9 10 3 4 5 6 7 8 9 10

1 -, 42 1 -, 42

PRACTICE

2 Solve: x + 3 6 8 and 2x - 1 6 3. Write the solution set in interval notation.

E X A M P L E 3 Solve: 2x ? 0 and 4x - 1 ... -9.

Solution

First we solve each inequality separately.

2x ? 0 and 4x - 1 ... -9

x ? 0 and

4x ... -8

x ? 0 and

x ... -2

Now we can graph the two intervals and find their intersection.

5x x ? 06

3 2 1 0 1 2 3 4

[0, 2

5x x ... -26

3 2 1 0 1 2 3 4

1 -, -2]

5x x ? 0 and x ... -26 =

There is no number that is greater than or equal to 0 and less than or equal to -2. The solution set is .

PRACTICE

3 Solve: 4x ... 0 and 3x + 2 7 8. Write the solution set in interval notation.

Helpful Hint

Example 3 shows that some compound inequalities have no solution. Also, some have all real numbers as solutions.

Section 2.5 Compound Inequalities 91

To solve a compound inequality written in a compact form, such as 2 6 4 - x 6 7, we get x alone in the "middle part." Since a compound inequality is really two inequalities in one statement, we must perform the same operations on all three parts of the inequality. For example:

2 6 4 - x 6 7 means 2 6 4 - x and 4 - x 6 7,

Helpful Hint

Don't forget to reverse both inequality symbols.

E X A M P L E 4 Solve: 2 6 4 - x 6 7

Solution To get x alone, we first subtract 4 from all three parts.

264-x67

2-464-x-467-4

-2 6 -x 6 3

-2 -1

7

-x -1

7

3 -1

2 7 x 7 -3

Subtract 4 from all three parts. Simplify.

Divide all three parts by - 1 and reverse the inequality symbols.

This is equivalent to -3 6 x 6 2. The solution set in interval notation is 1 -3, 22 , and its graph is shown.

4 3 2 1 0 1 2 3

PRACTICE

4 Solve: 3 6 5 - x 6 9. Write the solution set in interval notation.

E X A M P L E 5 Solve: -1 ... 2x + 5 ... 2. 3

Solution First, clear the inequality of fractions by multiplying all three parts by the LCD 3.

-1 ... 2x + 5 ... 2 3

31 -12 ... 3a 2x + 5b ... 3122 3

-3 ... 2x + 15 ... 6 -3 - 15 ... 2x + 15 - 15 ... 6 - 15

-18 ... 2x ... -9 -18 ... 2x ... -9

2 22 -9 ... x ... - 9

2 The graph of the solution is shown.

Multiply all three parts by the LCD 3. Use the distributive property and multiply. Subtract 15 from all three parts. Simplify. Divide all three parts by 2.

Simplify.

t

109 8 7 6 5 4 3

The solution set in interval notation is c -9, - 9 d . 2

PRACTICE

5

Solve: -4 ... x - 1 ... 3. Write the solution set in interval notation.

2

92 CHAPTER 2 Equations, Inequalities, and Problem Solving

Helpful Hint

The word either in this definition means "one or the other or both."

OBJECTIVE

3 Finding the Union of Two Sets

The solution set of a compound inequality formed by the word or is the union of the solution sets of the two inequalities. We use the symbol to denote "union."

Union of Two Sets

The union of two sets, A and B, is the set of elements that belong to either of the sets. A union B is denoted by A B.

A

B

A B

E X A M P L E 6 If A = 5x x is an even number greater than 0 and less than 106 and B = 53, 4, 5, 66, find A B .

Solution Recall from Example 1 that A = 52, 4, 6, 86 . The numbers that are in either set or both sets are 52, 3, 4, 5, 6, 86. This set is the union.

PRACTICE

6 If A = 5x x is an odd number greater than 0 and less than 106 and B = 52, 3, 4, 5, 66, find A B.

OBJECTIVE

4 Solving Compound Inequalities Containing "or"

A value is a solution of a compound inequality formed by the word or if it is a solution of either inequality. For example, the solution set of the compound inequality x ... 1 or x ? 3 contains all numbers that make the inequality x ... 1 a true statement or the inequality x ? 3 a true statement.

5xx ... 16

1 0 1 2 3 4 5 6

1 -, 1]

5xx ? 36 5x x ... 1 or x ? 36

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

[3, 2 1 -, 1] [3, 2

In interval notation, the set 5x x ... 1 or x ? 36 is written as 1 -, 1] [3, 2.

E X A M P L E 7 Solve: 5x - 3 ... 10 or x + 1 ? 5.

Solution First we solve each inequality separately.

5x - 3 ... 10 or x + 1 ? 5

5x ... 13 or

x?4

x ... 13 or 5

x?4

Now we can graph each interval and find their union.

e x ` x ... 13 f 5

{ 1 0 1 2 3 4 5 6

5xx ? 46

1 0 1 2 3 4 5 6

e x ` x ... 13 or x ? 4 f 5

{ 1 0 1 2 3 4 5 6

a -, 13 d 5

[4, 2 a -, 13 d [4, 2

5

Section 2.5 Compound Inequalities 93

The solution set is a -, 13 d [4, 2. 5

PRACTICE

7 Solve: 8x + 5 ... 8 or x - 1 ? 2. Write the solution set in interval notation.

E X A M P L E 8 Solve: -2x - 5 6 -3 or 6x 6 0.

Solution First we solve each inequality separately.

-2x - 5 6 -3 or 6x 6 0 -2x 6 2 or x 6 0 x 7 -1 or x 6 0

Now we can graph each interval and find their union.

5x x 7 -16

5x x 6 06 5x x 7 -1 or x 6 06 = all real numbers The solution set is 1 -, 2 .

4 3 2 1 0 1 2 3 4 3 2 1 0 1 2 3 4 3 2 1 0 1 2 3

1 -1, 2 1 -, 02 1 -, 2

PRACTICE

8 Solve: -3x - 2 7 -8 or 5x 7 0. Write the solution set in interval notation.

Answer to Concept Check: b is not correct

CONCEPT CHECK

Which of the following is not a correct way to represent the set of all numbers

between -3 and 5? a. 5x -3 6 x 6 56 c. 1 -3, 52

b. -3 6 x or x 6 5 d. x 7 -3 and x 6 5

Vocabulary, Readiness & Video Check

Use the choices below to fill in each blank.

or

and

compound

1. Two inequalities joined by the words "and" or "or" are called

2. The word

means intersection.

3. The word

means union.

4. The symbol

represents intersection.

5. The symbol

represents union.

6. The symbol

is the empty set.

inequalities.

94 CHAPTER 2 Equations, Inequalities, and Problem Solving

Martin-Gay Interactive Videos See Video 2.5

Watch the section lecture video and answer the following questions.

OBJECTIVE

1

OBJECTIVE

2

OBJECTIVE

3

OBJECTIVE

4

7. Based on Example 1 and the lecture before, complete the following

statement. For an element to be in the intersection of sets A and B,

the element must be in set A

in set B.

8. In Example 2, how can using three number lines help us find the

solution to this "and" compound inequality?

9. Based on Example 4 and the lecture before, complete the following

statement. For an element to be in the union of sets A and B, the ele-

ment must be in set A

in set B.

10. In Example 5, how can using three number lines help us find the

solution to this "or" compound inequality?

2.5 Exercise Set

MIXED PRACTICE

If A = 5 x 0 x is an even integer 6, B = 5 x 0 x is an odd integer 6, C = 52 , 3 , 4 , 56, and D = 54 , 5 , 6 , 76, list the elements of each set.

See Examples 1 and 6.

1. C D 3. A D 5. A B 7. B D 9. B C 11. A C

2. C D 4. A D 6. A B 8. B D 10. B C 12. A C

Solve each compound inequality. Graph the solution set and write it in interval notation. See Examples 2 and 3.

13. x 6 1 and x 7 -3 15. x ... -3 and x ? -2 17. x 6 - 1 and x 6 1 18. x ? - 4 and x 7 1

14. x ... 0 and x ? -2 16. x 6 2 and x 7 4

Solve each compound inequality. Write solutions in interval notation. See Examples 2 and 3.

19. x + 1 ? 7 and 3x - 1 ? 5 20. x + 2 ? 3 and 5x - 1 ? 9 21. 4x + 2 ... -10 and 2x ... 0 22. 2x + 4 7 0 and 4x 7 0 23. -2x 6 -8 and x - 5 6 5 24. -7x ... -21 and x - 20 ... - 15

Solve each compound inequality. See Examples 4 and 5.

25. 5 6 x - 6 6 11

26. -2 ... x + 3 ... 0

27. -2 ... 3x - 5 ... 7

28. 1 6 4 + 2x 6 7

29.

1

...

2 x

+

3

...

4

3

30.

-2

6

1 x

-

5

6

1

2

31. -5 ... - 3x + 1 ... 2 4

32. - 4 ... - 2x + 5 ... 1 3

Solve each compound inequality. Graph the solution set and write it in interval notation. See Examples 7 and 8.

33. x 6 4 or x 6 5 34. x ? - 2 or x ... 2 35. x ... - 4 or x ? 1 36. x 6 0 or x 6 1 37. x 7 0 or x 6 3 38. x ? - 3 or x ... -4

Solve each compound inequality. Write solutions in interval notation. See Examples 7 and 8.

39. -2x ... - 4 or 5x - 20 ? 5 40. -5x ... 10 or 3x - 5 ? 1 41. x + 4 6 0 or 6x 7 -12 42. x + 9 6 0 or 4x 7 -12 43. 31x - 12 6 12 or x + 7 7 10 44. 51x - 12 ? - 5 or 5 + x ... 11

MIXED PRACTICE

Solve each compound inequality. Write solutions in interval notation. See Examples 1 through 8.

45. x 6 2 and x 7 - 1

3

2

46. x 6 5 and x 6 1 7

47. x 6 2 or x 7 - 1

3

2

48. x 6 5 or x 6 1 7

49. 0 ... 2x - 3 ... 9

50. 3 6 5x + 1 6 11

51. 1 6 x - 3 6 2

2

4

52. 2 6 x + 1 6 4

3

2

53. x + 3 ? 3 and x + 3 ... 2

54. 2x - 1 ? 3 and - x 7 2

55.

3x

?

5

or

-

5 x

-

6

7

1

8

56.

3 x

+

1

...

0 or

- 2x

6

-4

8

57. 0 6 5 - 2x 6 5 3

58. - 2 6 - 2x - 1 6 2 3

59. -6 6 31x - 22 ... 8

60. -5 6 21x + 42 6 8

61. - x + 5 7 6 and 1 + 2x ... - 5

62. 5x ... 0 and - x + 5 6 8

63. 3x + 2 ... 5 or 7x 7 29

64. - x 6 7 or 3x + 1 6 - 20

65. 5 - x 7 7 and 2x + 3 ? 13

66. -2x 6 - 6 or 1 - x 7 - 2

67. - 1 ... 4x - 1 6 5

2

6

6

68. - 1 ... 3x - 1 6 1

2

10

2

69. 1 6 8 - 3x 6 4

15

15

5

70. - 1 6 6 - x 6 - 1

4 12

6

71. 0.3 6 0.2x - 0.9 6 1.5

72. -0.7 ... 0.4x + 0.8 6 0.5

REVIEW AND PREVIEW

Evaluate the following. See Sections 1.2 and 1.3.

73. 0 -7 0 - 0 19 0

74. 0 -7 - 19 0

75. -1 -62 - 0 -10 0

76. 0 -4 0 - 1 -42 + 0 -20 0

Find by inspection all values for x that make each equation true.

77. 0 x 0 = 7

78. 0 x 0 = 5

79. 0 x 0 = 0

80. 0 x 0 = -2

Section 2.5 Compound Inequalities 95

CONCEPT EXTENSIONS Use the graph to answer Exercises 81 and 82.

United States ? Single-Family Homes Housing Starts vs. Housing Completions

2000

Number of Single-Family Homes (in thousands)

1800 Started

1600

1400

Completed

1200

1000

800

600

400

1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 Year

Source: U.S. Census Bureau

81. For which years were the number of single-family housing starts greater than 1500 and the number of single-family home completions greater than 1500?

82. For which years were the number of single-family housing starts less than 1000 or the number of single-family housing completions greater than 1500?

83. In your own words, describe how to find the union of two sets.

84. In your own words, describe how to find the intersection of two sets.

Solve each compound inequality for x. See the example below. To solve x - 6 6 3x 6 2x + 5 , notice that this inequality contains a variable not only in the middle but also on the left and the right. When this occurs, we solve by rewriting the inequality using the word and.

x - 6 6 3x and 3x 6 2x + 5 - 6 6 2x and x 6 5 -3 6 x x 7 -3 and x 6 5

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

4 3 2 1 0 1 2 3 4 5 6 x65

4 3 2 1 0 1 2 3 4 5 6 -3 6 x 6 5 or 1 - 3 , 52

85. 2x - 3 6 3x + 1 6 4x - 5 86. x + 3 6 2x + 1 6 4x + 6 87. -31x - 22 ... 3 - 2x ... 10 - 3x 88. 7x - 1 ... 7 + 5x ... 311 + 2x2

96 CHAPTER 2 Equations, Inequalities, and Problem Solving

89. 5x - 8 6 212 + x2 6 - 211 + 2x2 90. 1 + 2x 6 312 + x2 6 1 + 4x

The formula for converting Fahrenheit temperatures to Celsius

temperatures is C = 51F - 322 . Use this formula for Exercises

91 and 92.

9

91. During a recent year, the temperatures in Chicago ranged from - 29C to 35?C. Use a compound inequality to convert these temperatures to Fahrenheit temperatures.

92. In Oslo, the average temperature ranges from - 10 to 18? Celsius. Use a compound inequality to convert these temperatures to the Fahrenheit scale.

Solve.

93. Christian D'Angelo has scores of 68, 65, 75, and 78 on his algebra tests. Use a compound inequality to find the scores he can make on his final exam to receive a C in the course. The final exam counts as two tests, and a C is received if the final course average is from 70 to 79.

94. Wendy Wood has scores of 80, 90, 82, and 75 on her chemistry tests. Use a compound inequality to find the range of scores she can make on her final exam to receive a B in the course. The final exam counts as two tests, and a B is received if the final course average is from 80 to 89.

2.6 Absolute Value Equations

OBJECTIVE

1 Solve Absolute Value

Equations.

OBJECTIVE

1 Solving Absolute Value Equations

In Chapter 1, we defined the absolute value of a number as its distance from 0 on a number line.

2 units

3 units

3 2 1 0 1 2 3 4

3 2 1 0 1 2 3 4

-2 = 2 and 3 = 3

In this section, we concentrate on solving equations containing the absolute value of a variable or a variable expression. Examples of absolute value equations are

0 x 0 = 3 -5 = 0 2y + 7 0 0 z - 6.7 0 = 0 3z + 1.2 0

Since distance and absolute value are so closely related, absolute value equations

and inequalities (see Section 2.7) are extremely useful in solving distance-type problems

such as calculating the possible error in a measurement.

For the absolute value equation 0 x 0 = 3, its solution set will contain all numbers

whose distance from 0 is 3 units. Two numbers are 3 units away from 0 on the number line: 3 and - 3 .

3 units 3 units

4 3 2 1 0 1 2 3 4

Thus, the solution set of the equation 0 x 0 = 3 is 53, -36 . This suggests the following:

Solving Equations of the Form 0 X 0 a If a is a positive number, then 0 X 0 = a is equivalent to X = a or X = -a.

E X A M P L E 1 Solve: 0 p 0 = 2.

Solution Since 2 is positive, 0 p 0 = 2 is equivalent to p = 2 or p = -2.

To check, let p = 2 and then p = -2 in the original equation.

0 p 0 = 2 Original equation 0 2 0 = 2 Let p = 2.

0 p 0 = 2 Original equation 0 - 2 0 = 2 Let p = -2.

2 = 2 True

2 = 2 True

The solutions are 2 and -2 or the solution set is 52, -26 .

PRACTICE

1 Solve: q = 3.

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