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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