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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 this question by rewriting absolute value inequalities as compound inequalities and solving them.

5.6 Solve Absolute Value Inequalities You will solve absolute value inequalities. Essential question: How do you solve absolute value inequalities? You

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Text of 5.6 Solve Absolute Value Inequalities You will solve absolute value inequalities. Essential...

You will solve absolute value inequalities.
Essential question: How do you solve absolute value inequalities?
You will learn how to answer this
question by rewriting absolute
value inequalities as compound
inequalities and solving them.
a.
SOLUTION
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.
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EXAMPLE 1
The distance between x and 0 is less than or equal to 0.5. So, –0.5 ≤ x ≤ 0.5.
SOLUTION
b.
ANSWER
The solutions are all real numbers greater than or equal to –0.5 and less than or equal to 0.5.
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EXAMPLE 4
GUIDED PRACTICE
1. |x| 8
–8 x 8
EXAMPLE 4
GUIDED PRACTICE
–3.5 < u < 3.5
EXAMPLE 4
GUIDED PRACTICE
3. v >
Write original inequality.
| x – 5 | 7
Solve x – 5 7. Graph your solution.
x – 5 7
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.
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Solve |–4x – 5| + 3 < 9. Graph your solution.
|–4x – 5| + 3 < 9
–6 < –4x – 5 < 6
Rewrite as compound inequality.
Rewrite in the form a < x < b.
Divide each expression by –4. Reverse inequality symbol.
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ANSWER
The solutions are all real numbers greater than –2.75 and less than 0.25.
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Solve the inequality. Graph your solution.
x + 3 > 8
Solve the inequality. Graph your solution.
2w – 1 < 11
Solve the inequality. Graph your solution.
<

6.
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?
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STEP 1
Find the mean by dividing the sum of the prices by 10.
= 664
890 + 750 + 650 + 370 + 660 + 670 + 450 + 650 + 725 + 825
10
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Add 664 to each expression.
Write and solve an inequality. An absolute deviation of at most $100 from the mean, $664, is given by the inequality x – 664 ≤ 100.
x – 664 ≤ 100
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.
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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
Essential question: How do you solve absolute value inequalities?
• You can rewrite an absolute
value inequality as a compound
inequality. Inequalities using > or
rewritten 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 Exercises
1. | x – 5 | > 3
all real numbers less than 2 or greater than 8
Warm-Up Exercises
ANSWER
to –5 and less than or equal to –1
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The clock in your car has an absolute deviation of
at most 4 minutes after 6 months. After 6 months,
the clock reads 7:38. Write and solve an inequality
to find the possible times.
3.
ANSWER
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