Rational inequalities

Rational inequalities

Solve rational inequalities by combining algebraic simplification with sign reasoning. This precalculus practice focuses on comparisons with a nonzero right side, endpoint decisions and restrictions that survive cancellation. It connects earlier polynomial sign charts with the additional care required when a denominator can change sign or vanish.

Questions cover strict and non-strict inequalities, even factors, cancelled inputs, comparisons between two fractions and a positive-domain application. Single-select and multiselect tasks ask for complete solution intervals and for explanations of why a tempting algebraic shortcut fails.

Review factoring, interval notation and rational domains before starting. Practice progresses from a direct comparison to several critical values and logical diagnosis. No graphing calculator is required: testing signs in intervals and checking endpoints gives exact conclusions.

First form one fraction compared with zero. Preserve every original denominator exclusion even if a factor cancels. Then distinguish zeros, which may be included depending on the inequality, from undefined inputs, which are never solutions. Detailed explanations make the interval signs explicit and connect each accepted endpoint to the original expression. Use the final context problem to translate an algebraic solution back to a physically allowed input range.