Sec 7.6: Solving Absolute Value Equations & Inequalities

1. Sec 7.6: Solving Absolute Value Equations & Inequalities

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 6x = 18 or 6x = -12 x = 3 or x = -2 * Plug in answers to check your solutions!

6. 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.

7. Special Case |3d - 9| + 6 = 0 |3d - 9| = -6 There is no need to go any further with this problem! • Absolute value is never negative. • Therefore, the solution is the empty set!

8. Solving Absolute Value Inequalities • ax+b < c, where c>0 Becomes an “and” problem Changes to: ax+b<c and ax+b>-c ALSO: -c<ax+b<c • ax+b > c, where c>0 Becomes an “or” problem Changes to: ax+b>c or ax+b<-c

9. Absolute Value Inequalities • “Less thAN” becomes an AND |2x - 3| < 9 2x -3 < 9 and 2x – 3 > -9 • “GreatOR than” becomes and OR |2x - 3| > 9 2x -3 > 9 or 2x – 3 < -9

10. Ex: Solve & graph. • Becomes an “and” problem -3 7 8

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

12. Special Cases |3d - 9| + 6 < 0 |3d - 9| < -6 There is no need to go any further with this problem! • Absolute value is never negative and cannot be less than a negative. • Therefore, the solution is the empty set!

13. Special Cases |3d - 9| + 6 > 0 |3d - 9| > -6 There is no need to go any further with this problem! • Absolute value is always positive and will always be greater than a negative. • Therefore, the solution is the ALL REALS!

14. Assignment