In Absolute Value Inequalities there are two algebraic expressions or two real numbers which is denoted with the symbol <’, ‘>’, '$\leq$' or '$\geq$' . Absolute Value Inequalities can be solved using Algebraically or by using graph. Online Absolute Value Inequality Calculator helps in solving the inequality equation.

Absolute function can be defined as:

|x| = $\left\{\begin{matrix} x&x > 0 \\ -x & x < 0 \end{matrix}\right.$

## Solving Absolute Value Inequalities

Let x be an algebraic expression and b be a real number s.t. b$\geq$ 0.

The solution of |x| < b are all values of x that lie between -b and b.

|x| < b iff -b < x < b.

The solution of |x| > b are all values of x that are less than -b or greater than b.

|x| > b iff x < -b or x > b.

Above result is also valid for $\leq$ and $\geq$.

### Solved Examples

Question 1: Solve the absolute value inequality for |4x - 3| < 9
Solution:
Given |4x - 3| < 9

This inequality is equivalent to -9 < 4x - 3 < 9

-9 + 3 < 4x - 3 + 3 < 9 + 3

-6 < 4x < 12

Divide each side by 4

$\frac{-3}{2}$ < x < 3

The solution set is:

Question 2: Solve the inequality |3x + 2| $\geq$ 6
Solution:
Given inequility |3x + 2| $\geq$ 6 is equivalent to

3x + 2 $\geq$ 6  or  3x + 2 $\leq$ -4

Case 1: 3x + 2 $\geq$ 6

Subtract 2 from both the sides

3x + 2 - 2 $\geq$ 6 - 2

3x $\geq$ 4

x $\geq$ $\frac{4}{3}$

Case 2: 3x + 2 $\leq$ -4

Subtract 2 from both the sides

3x + 2 - 2 $\leq$ -4 - 2

3x  $\leq$ -6

x  $\leq$ -2

So the solution set is x = (-$\infty$, -2] $\cup$ [$\frac{4}{3}$, $\infty$)

Graph of solution set is

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