1. Chapter 2.7 – Absolute Value Inequalities

2. Objectives Solve absolute value inequalities of the form /x/ < a Solve absolute value inequalities of the form /x/ > a

3. Example 1 Solve: ￨ x ￨ 3 [ ] The solution set is {-3, 3} -5 0-2-4-3 4 3 2 1 5

4. Solving Absolute Value expression of the form ￨ x ￨ < a If a is a positive number, then ￨ x ￨ < a is equivalent to – a < x< a

5. Example 2: Solve for m: ￨ m – 6 ￨ < 2 Step 1: -a < x < a Replace x with m - 6 and a with 2 -2 < m – 6 < 2

6. Example 2 continued Solve the compound inequality -2 < m – 6 < 2 -2 + 6 < m – 6 + 6 < 2 + 6 4 < m < 8 The solution set is (4, 8) and its graph is,

7. Give it a try! Solve ￨ x-2 ￨ 1

8. HINT MUST ISOLATE the absolute value expression Before using an absolute value inequality property, you MUST ISOLATE the absolute value expression on one side of the inequality!

9. Example 3: Solve for x: ￨ 5x + 1 ￨ + 110

10. Give it a try! Solve: ￨ 2x - 5 ￨ + 29

11. Example 4 Solve for x: The absolute value of a number is always nonnegative and can never be less than – 13. This inequality has NO solution! The solution set is { } or 0

12. Give it a try! Solve:

13. Let ￨ x ￨ 3 The solution set includes all numbers who distance from 0 is 3 or more units. The graph of the solution set contains 3 and all points to the right of 3 on the number line or – 3 and all points to the left of – 3 on the number line.. The solution is ( -, -3] U [3, ) Form of ￨ x ￨ > a

14. Solving Absolute Value Inequalities of the form ￨ x ￨ > a If a is a positive number, the ￨ x ￨ > a is equivalent to x a

15. Example 5 Solve for y: Step 1: Rewrite the inequalities without absolute value bars y – 3 7 Step 2: Solve the compound inequality y 10

16. Example 5 Continued Step 3: Graph the inequalities Step 4: Write the solution set (-∞, -4) U (10, ∞)

17. Give it a try! Solve:

18. Example 6: Isolate the Absolute Value expression! Solve: Step 1: Isolate the Absolute Value Expression

19. Example 6 - Continued ***Remember*** The absolute value of any number is always a nonnegative and thus is always great than -2. The inequality and the original inequality are true for all values of x. The solution set is {x/ x is a real number} or (-∞,∞)

20. Example 7: Isolate the Absolute Value Expression! Solve: Step 1: Isolate the Absolute Value Expression

21. Example 7 - Continued Step 2: Write the absolute value inequality as an equivalent compound inequality.

22. Example 7 continued Step 3: Solve each inequality

23. Example 7 continued Step 4: Write the solution set and graph (-∞, -3] U [9, ∞)

24. Give it a try! Solve:

25. Example 8 - Zero Solve for x: equal **Remember – the absolute value of any expression will never be less than 0, but it may be equal to 0. Thus solve the equation by setting it equal to zero!

26. Example 8 Continued

27. You give try! Solve:

28. Chart on Page 113 Copy the chart from 113 into your notes!