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5.6 Solve Absolute Value Inequalities

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. a.

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5.6 Solve Absolute Value Inequalities

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  1. 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.

  2. a. The distance between xand 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 –6or greater than or equal to 6. EXAMPLE 1 Solve absolute value inequalities Solve the inequality. Graph your solution. |x|  6 a. SOLUTION

  3. x 0.5 b. ANSWER The solutions are all real numbers greater than or equal to –0.5and less than or equal to 0.5. EXAMPLE 1 Solve absolute value inequalities Solve the inequality. Graph your solution. SOLUTION The distance between xand 0 is less than or equal to 0.5. So, –0.5 ≤ x ≤ 0.5.

  4. ANSWER –8x 8 EXAMPLE 4 for Example 1 Find a base using the percent equation GUIDED PRACTICE Solve the inequality. Graph your solution. 1. |x| 8

  5. –3.5<u< 3.5 ANSWER EXAMPLE 4 for Example 1 Find a base using the percent equation GUIDED PRACTICE Solve the inequality. Graph your solution. 2. |u| < 3.5

  6. 2 3 3.v > ANSWER v < – or v > 2 2 3 3 EXAMPLE 4 for Example 1 Find a base using the percent equation GUIDED PRACTICE Solve the inequality. Graph your solution.

  7. Solve x – 5  7. Graph your solution. x – 5  7 x – 5 7 or or x–2 x 12 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. EXAMPLE 2 Solve an absolute value inequality | x – 5 |  7 Write original inequality. Rewrite as compound inequality. Add 5 to each side.

  8. EXAMPLE 3 Solve an absolute value inequality Solve |–4x – 5| + 3 < 9. Graph your solution. |–4x – 5| + 3 < 9 Write original inequality. |–4x – 5| < 6 Subtract 3 from each side. –6 < –4x – 5 < 6 Rewrite as compound inequality. –1 < –4x < 11 Add 5 to each expression. 0.25 > x > –2.75 Divide each expression by –4. Reverse inequality symbol. –2.75 < x < 0.25 Rewrite in the form a < x < b.

  9. ANSWER The solutions are all real numbers greater than –2.75and less than 0.25. EXAMPLE 3 Solve an absolute value inequality

  10. 4. x + 3 > 8 or ANSWER x > 5 x <–11 for Examples 2 and 3 GUIDED PRACTICE Solve the inequality. Graph your solution.

  11. 5. 2w – 1 < 11 ANSWER –5 < w < 6 for Examples 2 and 3 GUIDED PRACTICE Solve the inequality. Graph your solution.

  12. 6. 35m – 6 – 8 13 ANSWER –0.2 m 2.6 < – for Examples 2 and 3 GUIDED PRACTICE Solve the inequality. Graph your solution.

  13. 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? EXAMPLE 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.

  14. 890 + 750 + 650 + 370 + 660 + 670 + 450 + 650 + 725 + 825 Mean= 10 6640 = 10 EXAMPLE 4 Solve a multi-step problem SOLUTION STEP 1 Find the mean by dividing the sum of the prices by 10. =664

  15. 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 EXAMPLE 4 Solve a multi-step problem STEP 2 Write absolute value inequality. – 100 ≤ x – 664 ≤ 100 Write as compound inequality. 564 ≤ x ≤ 764 Add 664 to each expression.

  16. 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. EXAMPLE 4 Solve a multi-step problem

  17. ANSWER 5 computer prices for Example 4 GUIDED PRACTICE 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?

  18. • You can rewrite an absolute value inequality as a compound inequality. Inequalities using > or ≥ can be rewritten using or. Inequalities using < or ≤ can be rewritten using and. • You will solve absolute value inequalities. • Essential question: How do you solve absolute value inequalities? 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.

  19. ANSWER all real numbers less than 2 or greater than 8 Daily Homework Quiz Solve the inequality. Graph your solution. 1. | x – 5 | > 3

  20. ANSWER all real numbers greater than or equalto –5 and less than or equal to –1 Daily Homework Quiz Solve the inequality.Graph your solution. 2. | x+ 3 | + 6  8

  21. 3. 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. | x– 7:38 |  4; from 7:34 through 7:42 ANSWER Daily Homework Quiz

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