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All courses Algebra I · A-REI.3.1 10 of 76
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 \(|x~-~3|~=~5\).
Your answer
Show answer choicesHide answer choices Work through the mathematics first, then compare your reasoning.Select the answer that matches your work.
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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

Decode absolute value as distance

\[|x-3|=\text{distance}~\text{from}~x~\text{to}~3\]

The equation asks for numbers whose distance from 3 is exactly 5. A positive distance has positions on both sides of the center.

02

Write both signed cases

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

An expression has absolute value 5 precisely when it equals either positive 5 or negative 5. These are the only cases.

03

Solve the positive case

\[x-3=5~->~x=8\]

Adding 3 to both sides places the right-hand solution 5 units above 3.

04

Solve the negative case

\[x-3=-5~->~x=-2\]

Adding 3 to both sides places the left-hand solution 5 units below 3.

05

Check both distances and answer

\[|-2-3|=|-5|=5;~|8-3|=|5|=5;~x~=~-2~\text{or}~x~=~8\]

Both values satisfy the original absolute-value equation, so the complete solution exactly matches choice A.

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Another way

  1. On a number line, move 5 units left and right from center 3: 3 - 5 = -2 and 3 + 5 = 8.

!

Common mistake

Do not solve only x - 3 = 5. A positive absolute-value target requires both the positive and negative signed cases.

Solution walkthrough video