2.1.1—Modulus equations

|f(x)|=a with a≥0 means f(x)=a or f(x)=−a. If a<0 there is no real solution. For inequalities, the sign of the comparison determines whether the solution lies inside or outside an interval.

Solve each branch, intersect with any domain, and test boundary points. Nested moduli require a second case split rather than one global square.

|2x−3|=5 gives 2x−3=5 or −5, so x=4 or x=−1.

Squaring can introduce solutions when the right side is not known non-negative; always substitute candidates into the original modulus equation.