Comparing Two Moduli
Both sides non-negative: compare their squares
For , both sides are non-negative, so squaring preserves their order. Move everything to one side and factorise the difference of squares.
Worked example
Solve |2x − 1| ≥ |3 − x|.
The factors vanish at and . These boundary values divide the number line into three intervals. Test one value in each: the product is positive at −3, negative at 0, positive at 2.
Between consecutive roots the linear factors keep their signs, so the product does too. Choose the intervals with the required sign, not just the boundary values. Include the boundaries only if equality is allowed.
Write the complete set of values
Solve .
Show worked answer
The roots are and . The product is positive outside them. Equality is not allowed.
“And” would ask one number to lie in both separate regions at once. Here either region is allowed, so use “or”.
Solve . Check whether 0 belongs to your answer.
Show worked answer
Zero is excluded: the original inequality would read .