Absolute Value Equations

The split: two cases, two equations

If $|x - 5| = 9$, then the expression $x - 5$ is nine units from zero — so it's either $9$ or $-9$. Write both cases: $x - 5 = 9$ gives $x = 14$, and $x - 5 = -9$ gives $x = -4$. Both answers are correct, and a full answer lists both.

The negative case is where mistakes happen. You make the whole right side negative — the entire value, not just part of it. From $|2x + 1| = 11$, the second case is $2x + 1 = -11$, and you solve it exactly like any two-step equation.

Isolate the absolute value first

If anything is added, subtracted, or multiplied outside the absolute value bars, deal with it before splitting. In $|x + 6| - 2 = 10$, add $2$ to both sides first so the equation reads $|x + 6| = 12$. Only then split into cases.

Splitting too early is the classic error. $|x + 6| - 2 = 10$ does not split into $x + 6 - 2 = 10$ and its negative twin — the bars only protect $x + 6$, so the absolute value must stand alone before the cases make sense.

The no-solution trap

An absolute value is a distance, so it can never be negative. If the equation says $|x - 3| = -4$, stop — no value of $x$ makes a distance equal $-4$. The answer is no solution, and there's nothing to solve.

Watch for this after isolating, too. $|2x - 1| + 6 = 2$ looks harmless, but subtracting $6$ gives $|2x - 1| = -4$, and that's the same dead end. One special middle case: if the absolute value equals exactly $0$, there's only one solution, because $0$ and $-0$ are the same number.

Worked examples

Try one yourself

Common questions