Solving Absolute Value Equations & Inequalities Solving Absolute Value Equations & Inequalities

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

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

Ex: Solve 2x = 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.

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

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

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

Graphing linear Inequalities in 2 Variables

Checking Solutions An ordered pair (x,y) is a solution if it makes the inequality true. Are the following solutions to: 3x + 2y ≥ 2 (0,0)(2,-1)(0,2) 3(0) + 2(0) ≥ 2 0 ≥ 2 Not a solution 3(2) + 2(-1) ≥ 2 4 ≥ 2 Is a solution 3(0) + 2(2) ≥ 2 4 ≥ 2 Is a solution

To sketch the graph of a linear inequality: 1.Sketch the line given by the corresponding equation ( solid if ≥ or ≤, dashed if ). This line separates the coordinate plane into 2 half-planes. In one half-plane – all of the points are solutions of the inequality. In the other half-plane - no point is a solution 2.You can decide whether the points in an entire half- plane satisfy the inequality by testing ONE point in the half-plane. 3.Shade the half-plane that has the solutions to the inequality.

The graph of an inequality is the graph of all the solutions of the inequality 3x+ 2y ≥ 2 y ≥ -3/2x + 1 (put into slope intercept to graph easier) Graph the line that is the boundary of 2 half planes Before you connect the dots check to see if the line should be solid or dashed solid if ≥ or ≤ dashed if

y ≥ -3/2x + 1 Step 1: graph the boundary (the line is solid ≥) Step 2: test a point NOT On the line (0,0) is always The easiest if it’s Not on the line!! 3(0) + 2(0) ≥ 2 0 ≥ 2 Not a solution So shade the other side of the line!!

Graph: y < 6

4x – 2y < 7