6.5 and 6.6 Solving Absolute Value Equations & Inequalities Page 322 Textbook Indicators: PFA 7,8 and 9

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

Let’s look at 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.

Example 1: 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!

Example 2: Solve 2x = 8 Get the absolute 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.

Let’s look at: x < 5 What are the possible values of x? Remember this is an AND statement. Let’s look at integer examples: x = …, 2, 3, 4 or x = -4, -3, -2, …

Let’s look at: x > 5 What are the possible values of x? Remember this is an OR statement. Let’s look at integer examples: x = 6, 7, 8, … or x =..., -8, -7, -6

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

Example 3: Solve & graph. Becomes an “and” problem The absolute value of 4x – 9 < 21 4x – x -12 x

Example 4: Solve & graph. Becomes an “or” problem The absolute value of 3x – 2 > 8 3x - 2 > 8or3x – 2 < -8 3x > 103x < -6 x > 10/3 x <

Warning!!! Do not forget to reverse the inequality sign when you multiply or divide by a negative quantity!!!

Assignment

Ex: Solve & graph. Becomes an “and” problem

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