How to Graph Absolute Value Inequalities?

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

• How to Solve an Absolute Inequality?
• How to Solve Absolute Equations?

A 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.

• \(\color{blue}{4|x| ≥ 12}\)

• \(\color{blue}{|x-2| < 1}\)

• \(\color{blue}{|x+2| ≥ 2}\)

• \(\color{blue}{5|x+2| ≥ 15}\)