How to Graph Absolute Value Inequalities?

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

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.

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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.

  • \(\color{blue}{4|x| ≥ 12}\)
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  • \(\color{blue}{|x-2| < 1}\)
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  • \(\color{blue}{|x+2| ≥ 2}\)
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  • \(\color{blue}{5|x+2| ≥ 15}\)
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  • \(\color{blue}{4|x| ≥ 12}\)
  • \(\color{blue}{|x-2| < 1}\)
  • \(\color{blue}{|x+2| ≥ 2}\)
  • \(\color{blue}{5|x+2| ≥ 15}\)
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