6.5 and 6.6 Solving Absolute 6.5 and 6.6 Solving Absolute Value Equations & InequalitiesValue Equations & Inequalities

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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

6x = 18 or 6x = -12

x = 3 or x = -2

* Plug in answers to check your solutions!

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.

Solving Absolute Value Inequalities

1. ax+b < c, where c>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

2194 x

219421 x30412 x

2

153 x

-3 7 8

Solve & graph.

• Get absolute value by itself first.

• Becomes an “or” problem

11323 x

823 x

823or 823 xx63or 103 xx

2or 3

10 xx

-2 3 4

Assignment