How to Solve Absolute Value Equations?

The absolute value of a number is its distance from zero. The distance is always positive. For example \(2\) and \(-2\) have the same absolute value \(2\). The absolute value of \(a\) is written as \(|a|\). If \(a\) is positive, \(|a|\) is equal to \(a\). If \(a\) is negative, then the absolute value is its opposite: \(|a|=-a\)

Related Topics

• How to Solve an Absolute Value Inequality
• How to Graph Absolute Value Inequalities

A step-by-step guide to solving absolute value equations

To solve an absolute value equation, follow four steps:

• Step 1: Isolate the absolute value expression.
• Step 2: set its contents equal to both the positive and negative value of the number on the other side of the equation.
• Step 3: Solve both equations.
• Step 4: Check the solutions.

Solving Absolute value Equations – Example 1:

Solve \(|x|-6=4\).

Solution:

Add \(6\) to both sides of equation: \(|x|-6+6=4+6\) Then \(|x|=10\).

Set the contents of the absolute value portion equal to \(+10\) and \(-10\).

\(x=10\) or \(x=-10\)

Now, check the solutions:

\(|10|-6=10-6=4\) and \(|-10|-6=10-6=4\)

The answers are \(10\) and \(-10\).

Solving Absolute value Equations – Example 2:

Solve \(|x-3|=8\).

Solution:

\(x-3=8\) or \(x-3=-8\)

Solve the equation \(x-3=8:\). add \(3\) to both sides: \(x-3+3=8+3\) then \(x=11\).

Solve the equation \(x-3=-8:\). add \(3\) to both sides: \(x-3+3=-8+3\) then \(x=-5\).

Now, check the solutions:

\(x=11\) → \(|11-3|=|8|=8\)

\(x=-5\) → \(|-5-3|=|-8|=8\)

The answers are \(11\) and \(-8\).

Exercises for Absolute Value Equations

Solve each absolute value equation.

• \(\color{blue}{|x|-2=5}\)
• \(\color{blue}{|x+3|=9}\)
• \(\color{blue}{|x-1|=3}\)
• \(\color{blue}{2|x+4|=12}\)

• \(\color{blue}{x=7}\)
• \(\color{blue}{x=6, x=-12}\)
• \(\color{blue}{x=4, x=-2}\)
• \(\color{blue}{x=2, x=-10}\)