96 CHAPTER 2 Equations, Inequalities, and Problem Solving

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.

Section 2.6 Absolute Value Equations 97

If the expression inside the absolute value bars is more complicated than a single variable, we can still apply the absolute value property.

Helpful Hint

For the equation 0 X 0 = a in the box on the previous page, X can be a single variable or a

variable expression.

E X A M P L E 2 Solve: 0 5w + 3 0 = 7.

Solution Here the expression inside the absolute value bars is 5w + 3. If we think of the expression 5w + 3 as X in the absolute value property, we see that 0 X 0 = 7 is equivalent to

X = 7 or X = -7

Then substitute 5w + 3 for X, and we have

5w + 3 = 7 or 5w + 3 = -7

Solve these two equations for w.

5w + 3 = 7 or 5w + 3 = -7

5w = 4 or

5w = -10

w = 4 or 5

w = -2

Check:

To check, let w = -2 and then w = 4 in the original equation. 5

Let w = -2

Let w = 4 5

0 51 -22 + 3 0 = 7

` 5a4b + 3` = 7 5

0 -10 + 3 0 = 7

04 + 30 = 7

0 -70 = 7

070 = 7

7 = 7 True

7 = 7 True

Both solutions check, and the solutions are -2 and 4 or the solution set is e -2, 4 f.

5

5

PRACTICE

2 Solve: 2x - 3 = 5.

E X A M P L E 3 Solve: ` x - 1 ` = 11. 2

Solution ` x - 1 ` = 11 is equivalent to 2

x - 1 = 11

or

2

x - 1 = -11 2

2 a x - 1 b = 21112 or 2 a x - 1 b = 21 - 112 Clear fractions.

2

2

x - 2 = 22

or

x - 2 = -22

Apply the distributive property.

x = 24 or The solutions are 24 and -20.

x = -20

PRACTICE

3

Solve: ` x + 1 ` = 15.

5

98 CHAPTER 2 Equations, Inequalities, and Problem Solving

To apply the absolute value property, first make sure that the absolute value expression is isolated.

Helpful Hint If the equation has a single absolute value expression containing variables, isolate the absolute value expression first.

E X A M P L E 4 Solve: 0 2x 0 + 5 = 7.

Solution We want the absolute value expression alone on one side of the equation, so begin by subtracting 5 from both sides. Then apply the absolute value property.

0 2x 0 + 5 = 7

0 2x 0 = 2

Subtract 5 from both sides.

2x = 2 or 2x = -2

x = 1 or x = -1

The solutions are -1 and 1.

PRACTICE

4 Solve: 3x + 8 = 14.

E X A M P L E 5 Solve: 0 y 0 = 0.

Solution We are looking for all numbers whose distance from 0 is zero units. The only number is 0. The solution is 0.

PRACTICE

5 Solve: z = 0.

The next two examples illustrate a special case for absolute value equations. This special case occurs when an isolated absolute value is equal to a negative number.

E X A M P L E 6 Solve: 2 0 x 0 + 25 = 23.

Solution First, isolate the absolute value.

2 0 x 0 + 25 = 23 20x0 = -2 0x0 = -1

Subtract 25 from both sides. Divide both sides by 2.

The absolute value of a number is never negative, so this equation has no solution. The solution set is 5 6 or .

PRACTICE

6 Solve: 3 z + 9 = 7.

E X A M P L E 7 Solve: ` 3x + 1 ` = -2. 2

Solution Again, the absolute value of any expression is never negative, so no solution exists. The solution set is 5 6 or .

PRACTICE

7

Solve: ` 5x + 3 ` = -8.

4

Answer to Concept Check: false; answers may vary

Section 2.6 Absolute Value Equations 99

Given two absolute value expressions, we might ask, when are the absolute values of two expressions equal? To see the answer, notice that

0 2 0 = 0 2 0 , 0 -2 0 = 0 -2 0 , 0 -2 0 = 0 2 0 , and 0 2 0 = 0 -2 0

c c

c c

c c

c c

same

same

opposites

opposites

Two absolute value expressions are equal when the expressions inside the absolute value bars are equal to or are opposites of each other.

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

Solution This equation is true if the expressions inside the absolute value bars are equal to or are opposites of each other.

3x + 2 = 5x - 8 or 3x + 2 = -15x - 82

Next, solve each equation.

3x + 2 = 5x - 8 or 3x + 2 = -5x + 8

-2x + 2 = -8

or 8x + 2 = 8

-2x = -10 or

8x = 6

x=5

or

x=3

4

3 The solutions are and 5.

4

PRACTICE

8 Solve: 2x + 4 = 3x - 1 .

E X A M P L E 9 Solve: 0 x - 3 0 = 0 5 - x 0 .

Solution

x - 3 = 5 - x or

x - 3 = -15 - x2

2x - 3 = 5

or

x - 3 = -5 + x

2x = 8

or x - 3 - x = -5 + x - x

x=4

or

-3 = -5

False

Recall from Section 2.1 that when an equation simplifies to a false statement, the equation has no solution. Thus, the only solution for the original absolute value equation is 4.

PRACTICE

9 Solve: x - 2 = 8 - x .

CONCEPT CHECK

True or false? Absolute value equations always have two solutions. Explain your answer.

The following box summarizes the methods shown for solving absolute value equations.

Absolute Value Equations If a is positive, then solve X = a or X = -a.

0 X 0 = a ? If a is 0, solve X = 0. If a is negative, the equation 0 X 0 = a has no solution.

0 X 0 = 0 Y 0 Solve X = Y or X = -Y.

100 CHAPTER 2 Equations, Inequalities, and Problem Solving

Vocabulary, Readiness & Video Check

Match each absolute value equation with an equivalent statement.

1. x - 2 = 5 2. x - 2 = 0 3. x - 2 = x + 3 4. x + 3 = 5 5. x + 3 = -5

A. x - 2 = 0 B. x - 2 = x + 3 or x - 2 = -1x + 32 C. x - 2 = 5 or x - 2 = -5

D. E. x + 3 = 5 or x + 3 = -5

Martin-Gay Interactive Videos

Watch the section lecture videos and answer the following question.

OBJECTIVE

1

6. As explained in Example 3, why is a positive in the rule " 0 X 0 = a is

equivalent to X = a or X = -a"?

See Video 2.6

2.6 Exercise Set

Solve each absolute value equation. See Examples 1 through 7.

1. 0 x 0 = 7

2. 0 y 0 = 15

3. 0 3x 0 = 12.6 5. 0 2x - 5 0 = 9

4. 0 6n 0 = 12.6 6. 0 6 + 2n 0 = 4

7. ` x - 3 ` = 1 2

9. 0 z 0 + 4 = 9 11. 0 3x 0 + 5 = 14

8. ` n + 2 ` = 4 3

10. 0 x 0 + 1 = 3 12. 0 2x 0 - 6 = 4

13. 0 2x 0 = 0 15. 0 4n + 1 0 + 10 = 4 17. 0 5x - 1 0 = 0

14. 0 7z 0 = 0 16. 0 3z - 2 0 + 8 = 1 18. 0 3y + 2 0 = 0

Solve. See Examples 8 and 9.

19. 0 5x - 7 0 = 0 3x + 11 0 21. 0 z + 8 0 = 0 z - 3 0

20. 0 9y + 1 0 = 0 6y + 4 0 22. 0 2x - 5 0 = 0 2x + 5 0

MIXED PRACTICE

Solve each absolute value equation. See Examples 1 through 9.

23. 0 x 0 = 4

24. 0 x 0 = 1

25. 0 y 0 = 0

26. 0 y 0 = 8

27. 0 z 0 = -2

28. 0 y 0 = -9

29. 0 7 - 3x 0 = 7

30. 0 4m + 5 0 = 5

31. 0 6x 0 - 1 = 11

32. 0 7z 0 + 1 = 22

33. 0 4p 0 = -8

34. 0 5m 0 = -10

35. 0 x - 3 0 + 3 = 7

37. ` z + 5 ` = -7 4

39. 0 9v - 3 0 = -8 41. 0 8n + 1 0 = 0 43. 0 1 + 6c 0 - 7 = -3

45. 0 5x + 1 0 = 11 47. 0 4x - 2 0 = 0 -10 0 49. 0 5x + 1 0 = 0 4x - 7 0 51. 0 6 + 2x 0 = - 0 -7 0 53. 0 2x - 6 0 = 0 10 - 2x 0

55. ` 2x - 5 ` = 7 3

57. 2 + 0 5n 0 = 17 59. ` 2x - 1 ` = 0 -5 0

3

61. 0 2y - 3 0 = 0 9 - 4y 0 63. ` 3n + 2 ` = 0 -1 0

8

65. 0 x + 4 0 = 0 7 - x 0 67. ` 8c - 7 ` = - 0 -5 0

3

36. 0 x + 4 0 - 4 = 1

38. ` c - 1 ` = -2 5

40. 0 1 - 3b 0 = -7 42. 0 5x - 2 0 = 0 44. 0 2 + 3m 0 - 9 = -7

46. 0 8 - 6c 0 = 1 48. 0 3x + 5 0 = 0 -4 0 50. 0 3 + 6n 0 = 0 4n + 11 0 52. 0 4 - 5y 0 = - 0 -3 0 54. 0 4n + 5 0 = 0 4n + 3 0

56. ` 1 + 3n ` = 4 4

58. 8 + 0 4m 0 = 24 60. ` 5x + 2 ` = 0 -6 0

2

62. 0 5z - 1 0 = 0 7 - z 0 64. ` 2r - 6 ` = 0 -2 0

5

66. 0 8 - y 0 = 0 y + 2 0 68. ` 5d + 1 ` = - 0 -9 0

6

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