Solving Absolute Value Equations & Inequalities You will ABSOLUTELY do this today in Mr. Peter Richard’s Very Awesome Class!

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

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

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.

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

Ex: Solve 2x + 7 -3 = 8 Get the abs value alone by adding 3! 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.

6|5x + 2| = 312 Isolate the absolute value expression by dividing by 6. 6|5x + 2| = 312 |5x + 2| = 52 Set up two equations to solve. 5x + 2 = 52 5x + 2 = -52 5x = 50 5x = -54 x = 10 or x = -10.8 Check: 6|5x + 2| = 312 6|5x + 2| = 312 6|5(10)+2| = 312 6|5(-10.8)+ 2| = 312 6|52| = 312 6|-52| = 312 312 = 312 312 = 312

3|x + 2| -7 = 14 x + 2 = 7 x + 2 = -7 x = 5 or x = -9 Isolate the absolute value expression by adding 7 and dividing by 3. 3|x + 2| -7 = 14 +7 +7 3|x + 2| = 21 3 3 |x + 2| = 7 Set up two equations to solve. x + 2 = 7 x + 2 = -7 x = 5 or x = -9 Check: 3|x + 2| - 7 = 14 3|x + 2| -7 = 14 3|5 + 2| - 7 = 14 3|-9+ 2| -7 = 14 3|7| - 7 = 14 3|-7| -7 = 14 21 - 7 = 14 21 - 7 = 14 14 = 14 14 = 14

Solving Absolute Value Inequalities ax+b < c, where c>0 Becomes an “and” problem Changes to: –c<ax+b<c ax+b > c, where c>0 Becomes an “or” problem Changes to: ax+b>c or ax+b<-c In other words, change All Signs!

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

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

Get To Work! Quiz: Page 238 # 14, 16, 20, 26, 30 Homework: # 13, 17, 27, 37, 41