absolute-value equation

Absolute value measures distance from zero, ignoring direction — so |x| = 5 asks “which numbers sit exactly 5 steps from zero?” There are two such numbers, 5 and −5, which is why absolute-value equations so often have two answers.

To solve |expression| = k, first make sure the absolute value is alone on one side. If k is positive, split into two ordinary equations: expression = k and expression = −k. Solve each, and you have the full solution set.

Watch the edge cases honestly. If k = 0 there is exactly one solution, since only zero has zero distance from zero. If k is negative there is no solution at all, because a distance can never be negative — no value of the variable can make it so.