5.6 Solve Absolute Value Inequalities

• You will solve absolute value inequalities.

• Essential question: How do you solve absolute value inequalities?

You will learn how to answer thisquestion by rewriting absolutevalue inequalities as compoundinequalities and solving them.

Warm-Up ExercisesSolve absolute value inequalitiesEXAMPLE 1

Solve the inequality. Graph your solution.

a. |x| 6

a. The distance between x and 0 is greater than or equal to 6. So, x –6 or x 6.

ANSWER

The solutions are all real numbers less than or equal to –6 or greater than or equal to 6.

SOLUTION

Warm-Up ExercisesSolve absolute value inequalitiesEXAMPLE 1

b. x 0.5

The distance between x and 0 is less than or equal to 0.5. So, –0.5 ≤ x ≤ 0.5.

SOLUTION

ANSWER

The solutions are all real numbers greater than or equal to –0.5 and less than or equal to 0.5.

Solve the inequality. Graph your solution.

Warm-Up ExercisesFind a base using the percent equationEXAMPLE 4GUIDED PRACTICE for Example 1

Solve the inequality. Graph your solution.

1. |x| 8

–8 x 8ANSWER

Warm-Up ExercisesFind a base using the percent equationEXAMPLE 4GUIDED PRACTICE for Example 1

2. |u| < 3.5

Solve the inequality. Graph your solution.

–3.5 < u < 3.5ANSWER

Warm-Up ExercisesFind a base using the percent equationEXAMPLE 4GUIDED PRACTICE for Example 1

3. v > 23

Solve the inequality. Graph your solution.

23

23v < – or v >ANSWER

Warm-Up ExercisesEXAMPLE 2 Solve an absolute value inequality

Solve x – 5 7. Graph your solution.

Write original inequality. | x – 5 | 7

x – 5 7 or x – 5 7 Rewrite as compound inequality.

x –2 or x 12

Add 5 to each side.

ANSWER

The solutions are all real numbers less than or equal to –2 or greater than or equal to 12. Check several solutions in the original inequality.

Warm-Up ExercisesEXAMPLE 3 Solve an absolute value inequality

Solve |–4x – 5| + 3 < 9. Graph your solution.

|–4x – 5| + 3 < 9

–6 < –4x – 5 < 6

|–4x – 5| < 6

–1 < –4x < 11

0.25 > x > –2.75

–2.75 < x < 0.25

Write original inequality.

Subtract 3 from each side.

Rewrite as compound inequality.

Add 5 to each expression.

Rewrite in the form a < x < b.

Divide each expression by –4. Reverse inequality symbol.

Warm-Up ExercisesEXAMPLE 3 Solve an absolute value inequality

ANSWER

The solutions are all real numbers greater than –2.75 and less than 0.25.

Warm-Up ExercisesGUIDED PRACTICE for Examples 2 and 3

Solve the inequality. Graph your solution.

x + 3 > 84.

x < –11 or x > 5ANSWER

Warm-Up ExercisesGUIDED PRACTICE for Examples 2 and 3

2w – 1 < 115.

Solve the inequality. Graph your solution.

ANSWER –5 < w < 6

Warm-Up ExercisesGUIDED PRACTICE for Examples 2 and 3

6. 3 5m – 6 – 8 13<–

Solve the inequality. Graph your solution.

–0.2 m 2.6ANSWER

Warm-Up ExercisesEXAMPLE 4 Solve a multi-step problem

Computers

You are buying a new computer and find 10 models in a store advertisement. The prices are $890, $750, $650, $370, $660, $670, $450, $650, $725, and $825.

Find the mean of the computer prices.

You are willing to pay the mean price with an absolute deviation of at most $100. How many of the computer prices meet your condition?

Warm-Up ExercisesEXAMPLE 4 Solve a multi-step problem

SOLUTION

STEP 1

Find the mean by dividing the sum of the prices by 10.

=664010 = 664

Mean =890 + 750 + 650 + 370 + 660 + 670 + 450 + 650 + 725 + 825

10

Warm-Up ExercisesEXAMPLE 4 Solve a multi-step problem

STEP 2

Write and solve an inequality. An absolute deviation of at most $100 from the mean, $664, is given by the inequality x – 664 ≤ 100.

Write absolute value inequality.x – 664 ≤ 100

– 100 ≤ x – 664 ≤ 100 Write as compound inequality.

564 ≤ x ≤ 764 Add 664 to each expression.

Warm-Up ExercisesEXAMPLE 4 Solve a multi-step problem

ANSWER

The prices you will consider must be at least $564 and at most $764. Six prices meet your condition: $750, $650, $660, $670, $650, and $725.

Warm-Up ExercisesGUIDED PRACTICE for Example 4

7. WHAT IF? In Example 4, suppose that you are willing to pay the mean price with an absolute

deviation of at most $75. How many of the computer prices meet this condition?

ANSWER

5 computer prices

• You will solve absolute value inequalities.

• Essential question: How do you solve absolute value inequalities?

• You can rewrite an absolutevalue inequality as a compoundinequality. Inequalities using > or≥ can be rewritten using or.Inequalities using < or ≤ can berewritten using and.

Rewrite the absolute value inequality so one side is |ax + b |. Then write the compound inequality as two inequalities joined by or for > and ≥ , or by and for < and ≤ , and solve the inequalities.

Warm-Up ExercisesDaily Homework Quiz

Solve the inequality. Graph your solution.

1. | x – 5 | > 3

ANSWER all real numbers less than 2 or greater than 8

Warm-Up ExercisesDaily Homework Quiz

2. | x + 3 | + 6 8

ANSWER all real numbers greater than or equalto –5 and less than or equal to –1

Solve the inequality.Graph your solution.

Warm-Up ExercisesDaily Homework Quiz

The clock in your car has an absolute deviation ofat most 4 minutes after 6 months. After 6 months, the clock reads 7:38. Write and solve an inequalityto find the possible times.

3.

ANSWER | x – 7:38 | 4; from 7:34 through 7:42