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# Graphing Absolute Value Inequalities

Absolute value inequalities have two possible solutions, and each inequality must be verified before graphing.
Graphing absolute value inequalities is nothing but drawing graph of the solutions of the absolute value inequalities.
To graph the absolute value inequalities, first we have to find the solutions of the inequalities.
The main difference between the inequalities and equalities is the range of the solutions.

In equalities, we can determine the exact solution. But, in inequalities, we find the range of the solutions.

We will see some example problems for graphing the absolute value inequalities.

Example Problems for Graphing Absolute Value Inequalities:

Given below are some of the example problems for graphing absolute value inequalities.

Example 1: Graph the absolute values of the inequalities.
|x +2| $\leq$ 8

Solution:

Given inequality is |x + 2| $\leq$ 8

To find the absolute value for the given inequality

(x + 2) $\leq$ 8 ............. (1)
-(x + 2)
$\leq$ 8 .................. (2)

Equation 1:

(x + 2) $\leq$ 8

Adding - 2 on both the sides, we get

x + 2 - 2 $\leq$ 8 - 2

x $\leq$ 6

Equation 2:

-(x + 2) $\leq$ 8

-x - 2 $\leq$ 8

Adding +2 on both the sides, we get

-x - 2 + 2 $\leq$ 8 + 2

-x $\leq$ 10

Dividing by -1 on both the sides, we get

x $\leq$ -10

So, x lies between -10 $\leq$ x $\leq$ 6

Example 2:

Find the absolute values of the inequalities. |x - 6| $\geq$ 5

Solution:

Given inequality is |x - 6| $\geq$ 5

To find the absolute value for the given inequality
(x - 6) $\geq$ 5  ................... (1)
-(x - 6) $\geq$ 5   ........................ (2)

Equation 1:

(x - 6) $\geq$ 5

Adding 6 on both the sides, we get
x - 6 + 6 $\geq$ 5 + 6
x $\geq$ 11

Equation 2:

-(x - 6) $\geq$ 5
-x + 6 $\geq$ 5

Adding -6 on both the sides, we get
-x + 6 - 6 $\geq$ 5 - 6
-x $\geq$ -1

Dividing by -1, we get
x $\leq$ 1
So, x lies between 11 $\leq$ x $\leq$ 1

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