Rational Inequalities

Rational inequalities use zeros, undefined values, and sign charts to determine where a quotient is positive or negative.

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Excluded inputs appear as breaks or vertical asymptotes. Two branches approach dashed horizontal and vertical asymptotes without touching them. The separation emphasizes that denominator restrictions belong to both the algebra and the graph.

A rational expression can change sign at a numerator zero or across a denominator zero, so both types belong on the sign chart.

Write one rational expression

Move all terms to one side and combine them into a single rational expression compared with zero:

\frac{P(x)}{Q(x)}>0,quad \frac{P(x)}{Q(x)}\le0,

or a related form. Unlike a rational equation, multiplying blindly by a variable denominator can reverse the inequality when that denominator is negative. A sign chart avoids this uncertainty.

Inclusion rules

A numerator zero may be included when the inequality uses $\le$ or $\ge$, provided the denominator is nonzero there. A denominator zero is never included because the expression is undefined.

This is why two critical numbers can use different endpoint symbols even in the same answer.

Worked example

Multiplicity and sign changes

At a factor of even multiplicity, the sign does not change. At odd multiplicity, it does. This applies to numerator and denominator factors, though denominator zeros remain excluded.

A test value in each interval is still the safest check when multiple factors are present.

Solving comparisons of functions

To solve

f(x)\ge g(x),

form $f(x)-g(x)\ge0$ and combine into one quotient. Its zeros are intersection inputs; its sign identifies where $f$ lies above or on $g$.

Keep all restrictions from both original functions.

Common mistakes

Multiplying by an unknown-sign denominator without cases. Use a sign chart.

Including a denominator zero with a bracket. Undefined values are always excluded.

Forgetting cancelled restrictions. A hole remains outside the domain.

Testing only numerator signs. The denominator affects the quotient sign.

Solving the related equation only. Critical numbers are boundaries, not the full solution.