1. Solving Absolute Value Equations & Inequalities Solving Absolute Value Equations & Inequalities Isolate the absolute value

2. Absolute Value (of x) Symbol lxl The distance x is from 0 on the number line. Always positive Ex: l-3l=3 -4 -3 -2 -1 0 1 2

3. Ex: x = 5 What are the possible values of x? x = 5 or x = -5

4. To solve an absolute value equation: ax+b = c, (where c>0) To solve, set up 2 new equations, then solve each equation. ax+b = c or ax+b = -c ** make sure the absolute value is by itself before you split to solve.

5. Ex: Solve 6x-3 = 15 6x-3 = 15 or 6x-3 = -15

6. Ex: Solve 6x-3 = 15 6x-3 = 15 or 6x-3 = -15 6x = 18 or 6x = -12 x = 3 or x = -2 * Plug in answers to check your solutions!

7. Ex: Solve 2x + 7 -3 = 8 Get the abs. value part by itself first! 2x+7 = 11 Now split into 2 parts. 2x+7 = 11 or 2x+7 = -11.

8. Ex: Solve 2x + 7 -3 = 8 Get the abs. value part by itself first! 2x+7 = 11 Now split into 2 parts. 2x+7 = 11 or 2x+7 = -11 2x = 4 or 2x = -18 x = 2 or x = -9 Check the solutions.

9. Try this one:

10. Divide by 3

11. Next example:

12. Solve this example: -2 { } But they do not check

13. Last one:

14. Last one: be sure to negate the entire other side!

15. Last one: 4x = 4 X = 1

16. Hand this one in

17. Answer: 2x = -16 X = -8 BOTH ANSWERS CHECK.

18. Solving Absolute Value Inequalities 1. ax+b 0 Becomes an “and” problem Changes to: –c<ax+b<c 2. ax+b > c, where c>0 Becomes an “or” problem Changes to: ax+b>c or ax+b<-c

19. Interval notation: 2<x<9 (2,9) x 2 [-1,5] Note that

20. When it’s less than It’s and “and”

21. Ex: Solve & graph. Becomes an “and” problem -3 7 8 [-3,7.5] Interval notation:

22. Greater than Is an or

23. Solve & graph. Get absolute value by itself first. Becomes an “or” problem -2 3 4

24. Try this one:

25. Now decide that it is an “and” graph!

26. answer: