# How to Graph Absolute Value Inequalities?

To graph an absolute value inequality, first, you need to solve the absolute value inequality.

The absolute value of a number describes its distance from zero. To graph absolute value inequality you must first solve the inequality and then represent absolute value inequalities on a number line.

## Related Topics

## Step-by-step guide to graph absolute value inequalities

- Isolate the absolute value expression.
- Write the equivalent compound inequality.
- Find two values.
- Draw a number line and Place dots or open dots on the two points corresponding to the solutions. if the symbol is (\(≥\) or \(≤\)) then you fill in the dot if the symbol is (\(>\) or \(<\)) then you do not fill in the dot.
- Sketch a line and show all numbers the variable can be.

**Graphing Absolute Value Inequalities** **– Example 1:**

Solve and graph \(|3x-3|<9\).

**Solution:**

Use absolute properties: if \(|a|<b\), \(b > 0\) then: \( -b<a<b → -9 < 3x-3 < 9\)

Add \(3\) to each side of inequality: \(-9+3 < 3x-3+3 < 9+3 → -6 < 3x < 12\)

Divide each side of inequality by \(3\): \( -2 < x < 4\)

Graph it on a number line.

**Graphing Absolute Value Inequalities** **– Example 2:**

Solve and graph \(|x+2| ≥ 3\).

**Solution:**

Split into two inequalities: \(x+2 ≥ 3\) or \(x+2 ≤ -3\).

Subtract \(2\) from each side of each inequality:

\(x+2-2 ≥ 3-2\) → \(x ≥ 1\)

or

\(x+2-2 ≤ -3-2\) → \(x ≤ -5\)

\(x ≥ 1\) or \(x ≤ -5\).

Graph the numbers that satisfy both conditions:

## Exercises for Graph Absolute Value Inequalities

### Solve and graph the following inequalities.

- \(\color{blue}{4|x| ≥ 12}\)

- \(\color{blue}{|x-2| < 1}\)

- \(\color{blue}{|x+2| ≥ 2}\)

- \(\color{blue}{5|x+2| ≥ 15}\)

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