Absolute Value Equations

Absolute value measures distance from zero, and two different numbers can sit the same distance away. So these equations usually have two answers.

|x| = 7 means x = 7 or x = −7.

When there is more inside, split into two equations:
|2x − 3| = 11 becomes 2x − 3 = 11 and 2x − 3 = −11, giving x = 7 and x = −4.

Isolate the absolute value bars before splitting. In 3|x + 1| − 2 = 10, first get |x + 1| = 4, then split.

If the isolated absolute value equals a negative number, stop: no solution. Distance is never negative.

Using the piecewise definition, |u| = a with a > 0 is equivalent to the disjunction u = a or u = −a. For a = 0 the two branches coincide, giving one solution; for a < 0 the solution set is empty because |u| ≥ 0 always.

That case analysis is why isolating the bars is mandatory: the split is only valid once the equation reads exactly |u| = a. Splitting 3|x+1| − 2 = 10 prematurely produces wrong branches.

When absolute values appear on both sides, |u| = |v| gives u = v or u = −v, and both must be checked in the original equation. More generally, whenever the right-hand side contains the variable — as in |x − 2| = 3x — the sign of that side is not guaranteed, so every candidate root requires verification. Extraneous solutions are common and are not detectable from the algebra alone.

Geometrically, |x − c| = r says x is exactly r units from c, which makes the two solutions c ± r readable without any algebra at all.