1. 1.6 Absolute Value Inequalities and Equations Honors Algebra II

2. Absolute Value Distance from zero on the number line. Distance from zero on the number line.

3. Absolute Value Distance from zero on the number line. Distance from zero on the number line. Always positive result for absolute value. Always positive result for absolute value.

4. Absolute Value  Distance from zero on the number line.  Always positive result for absolute value.  Works as a grouping in order of operations, but it is NOT parentheses => don’t distribute!

5. Absolute Value |x| = x if x ≥ 0 |x| = -x if x < 0

6. Steps to Solve Absolute Value Equations  Isolate absolute value expression  Split into two equations  Solve both  Check both

7. Absolute Value Equations Solve: |2x – 1| = 5

8. Absolute Value Equations CHECK: |2x – 1| = 5 x = 3x = -2

9. Absolute Value Equations Solve:3 |4w – 1| - 5 = 10

10. Absolute Value Equations Check:3 |4w – 1| - 5 = 10

11. Absolute Value Equations Solve:|2x + 5| = 3x + 4

12. Absolute Value Equations Check: |2x + 5| = 3x + 4

13. Extraneous Solutions  Solutions that, though solved for correctly, do not work in the original equation.  YOU MUST CHECK YOUR ANSWERS! Absolute Value Equations

14. Absolute Value Inequalities

16. Steps to Solve Absolute Value Inequalities  Isolate absolute value expression  Split into two inequalities  Solve both  Check

17. Solve:|3x + 6| ≥ 12 Absolute Value Inequalities

18. Solve:3 |2x + 6| - 9 < 15 Absolute Value Inequalities

19. Summary Isolate Absolute Value Isolate Absolute Value If =, OR problem |x| = 5 If =, OR problem |x| = 5 x = 5 or x = -5 If >, ≥, OR problem |x| > 5 If >, ≥, OR problem |x| > 5 x 5 If <, ≤, AND problem|x| < 5 If <, ≤, AND problem|x| < 5 -5 < x < 5 -5 < x < 5 x > -5 and x -5 and x < 5

20. Tolerance: one half the difference of the maximum and minimum acceptable values. Manufacturing Application

23. More Examples: 1.) 2 |3x – 1| + 5 = 33 2.) |5z + 3| - 7 < 34 3.) -3 |2x – 3| + 1 ≤ -20