3.1.1—Modulus equations

For |f(x)|=g(x), first require g(x)≥0, then solve f(x)=g(x) or f(x)=−g(x). For inequalities, split according to whether the comparison side is non-negative.

Nested or variable-right-side moduli may need several intervals. Keep endpoints and test each candidate in the unsquared original equation.

|x−2|=x gives x≥0 and branches x−2=x (impossible) or 2−x=x, giving x=1.

Squaring both sides without checking the sign of the right side can add invalid solutions.