# How to Solve an Absolute Value Inequality?

The absolute value of inequalities follows the same rules as the absolute value of numbers.

The absolute value of $$a$$ is written as $$|a|$$. For any real numbers $$a$$ and $$b$$, if $$|a| < b$$, then $$a < b$$ and $$a > -b$$ and if $$|a| > b$$, then $$a > b$$ and $$a < -b$$.

## Step by step guide to solve an absolute value inequality

To solve an absolute value inequality, follow below steps:

• Isolate the absolute value expression.
• Write the equivalent compound inequality.
• Solve the compound inequality.

### Solving Absolute Value Inequalities– Example 1:

Solve $$|x-5|<3$$.

Solution:

To solve this inequality, break it into a compound inequality: $$x-5<3$$ and $$x-5>-3$$

So, $$-3<x-5<3$$.

Add $$5$$ to each expression: $$-3+5<x-5+5<3+5 → 2<x<8$$.

### Solving Absolute Value Inequalities – Example 2:

Solve $$|x+4| ≥ 9$$.

Solution:

Split into two inequalities: $$x+4 ≥ 9$$ or $$x+4 ≤ -9$$.

Subtract $$4$$ from each sides of each inequality:

$$x+4-4 ≥ 9-4$$ → $$x ≥ 5$$

or

$$x+4-4 ≤ -9-4$$ → $$x ≤ -13$$

## Exercises for Absolute Value Inequalities

### Solve each absolute value inequalities.

• $$\color{blue}{|4x|<12}$$
• $$\color{blue}{|x-5|>9}$$
• $$\color{blue}{|3x-7|<8}$$
• $$\color{blue}{5|x-2|>20}$$
• $$\color{blue}{-3<x<3}$$
• $$\color{blue}{x< -4 \:or\: x>14}$$
• $$\color{blue}{-\frac{1}{3}<x<5}$$
• $$\color{blue}{x<-2 \:or\: x>6}$$

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