Rational Inequalities — Cheat sheet

Key method

Move all terms to one side, form one fraction, factor numerator and denominator, and test intervals separated by every critical point. Include eligible numerator zeros for non-strict inequalities; never include denominator zeros.

Example

Solve $(x-1)/(x+2)\ge0$. Exclude $-2$, and split at $-2,1$. Testing $-3,0,2$ gives positive, negative, positive. Thus

At $1$ the expression is zero; at $-2$ it is undefined.

Avoid this mistake

Cross-multiplying by $x+2$ without knowing its sign can reverse the inequality in part of the domain and produce the wrong answer.