Rational Inequalities

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:

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.

Critical numbers

Factor numerator and denominator. Critical numbers come from:

• numerator zeros, where the expression may equal zero;
• denominator zeros, where the expression is undefined.

Place both kinds in order on a number line. They divide the domain into intervals of consistent sign.

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

Factoring before the chart

Consider

Factor:

Although $x+2$ cancels algebraically, $x=-2$ remains an excluded hole and must appear on the number line. The simplified sign matches $(x-2)/(x-3)$ for allowed inputs, but the original domain must be preserved.

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

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.

Graphical interpretation

Graph the rational expression and locate where it lies above, on, or below the $x$-axis. Holes and vertical asymptotes split the graph and must be visible in the solution intervals.

Graphing technology can verify a sign chart but may not display a tiny hole clearly, so algebraic restrictions remain essential.

Contextual inequalities

Rational inequalities can describe rates, average costs, concentration, and efficiency thresholds. Intersect the algebraic solution with contextual conditions such as positive time or a limited production range.

Interpret excluded values: they may represent a division by zero, impossible operating condition, or model breakdown.

Why cross multiplication is risky

Cross multiplication preserves an inequality only when the multiplier's sign is known. A variable denominator may be positive on one interval and negative on another, which would change the inequality direction.

Moving to one side and using a sign chart handles all intervals consistently.

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.

Quick self-check

• Is everything combined on one side?
• Are numerator and denominator completely factored?
• Have zeros and undefined values both been marked?
• Is each interval sign justified?
• Are numerator zeros included only when allowed by the symbol?
• Are all denominator zeros excluded and contextual restrictions applied?

Related topics

• Rational Functions
• Rational Expressions
• Rational Equations
• Polynomial Inequalities