Absolute Value Equations

The Key Insight

If $|x| = 5$, then $x$ could be 5 or -5.

Why? Both are 5 units away from zero.

Absolute value equations usually have two solutions.

The Pattern

For $|A| = B$ where $B \geq 0$:

Split into two cases:

• $A = B$
• $A = -B$

Solve each case separately.

Example

When There’s No Solution

$|x| = -3$ has no solution.

Absolute value is always non-negative. It can never equal a negative number.

If $|A| = B$ and $B < 0$, there is no solution.