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All courses Math I · A-REI.3.1 10 of 59
Solve and graph one-variable absolute-value equations and inequalities; interpret solutions in context.

Solve an absolute-value equation by splitting it into two linear cases

Problem
Solve the absolute-value equation |x - 3| = 5.
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Hint

Rewrite the absolute-value equation as two cases: x - 3 = 5 or x - 3 = -5.

If |A| = 5, then A can be 5 or -5.

Solution walkthrough

01

Start with the absolute-value equation

\[|x~-~3|~=~5\]

An absolute value measures distance, so x - 3 can be 5 or -5.

02

Write the two cases

\[x~-~3~=~5~\quad~\text{or}~\quad~x~-~3~=~-5\]

These are the only two ways for the absolute value to equal 5.

03

Solve both equations

\[x~=~8~\quad~\text{or}~\quad~x~=~-2\]

Add 3 to both sides in each case.

04

State the solution

\[x~=~-2~\text{ or }~x~=~8\]

Both values are 5 units away from 3, so both satisfy the equation.

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Another way

  1. You can think on a number line: numbers 5 units from 3 are 3 - 5 = -2 and 3 + 5 = 8.

!

Common mistake

A common mistake is to solve only x - 3 = 5 and forget the negative case x - 3 = -5.