Rational Inequality
A rational inequality is an inequality that contains a rational expression — a fraction where the numerator and/or denominator includes a variable. Solving one means finding all values of the variable that make the inequality true.
A rational inequality is an inequality in which at least one side involves a rational expression, that is, a ratio of two polynomials. To solve a rational inequality, you move all terms to one side, find the critical values (where the expression equals zero or is undefined), and then use a sign chart or test points to determine which intervals satisfy the inequality. Solutions are expressed as unions of intervals, always excluding values that make any denominator zero.
Worked Example
Problem: Solve the rational inequality:
Step 1: Find the critical values: Set the numerator and denominator each equal to zero. The numerator gives x = 2, and the denominator gives x = −3. These split the number line into intervals.
Step 2: Note any excluded values: Since x = −3 makes the denominator zero, it must be excluded from the solution regardless of the inequality symbol.
Step 3: Build a sign chart: Test one value from each interval: (−∞, −3), (−3, 2), and (2, ∞). Pick x = −5, x = 0, and x = 4.
Step 4: Determine which intervals satisfy the inequality: The expression is positive on (−∞, −3) and (2, ∞), and negative on (−3, 2). Since the inequality is ≥ 0, we want where the expression is positive or zero. The expression equals zero at x = 2, so include that endpoint.
Step 5: Write the solution: Combine the valid intervals, excluding x = −3 and including x = 2.
Answer: The solution set is .
Visualization
Why It Matters
Rational inequalities appear whenever you need to know when a rate, ratio, or proportion exceeds or falls below a threshold. In science and economics, models often involve fractions with variables — for instance, determining when a concentration stays below a safe level, or when average cost per unit drops below a target price. Mastering the sign-chart method also builds skills you'll reuse in calculus when analyzing the behavior of functions.
Common Mistakes
Mistake: Cross-multiplying both sides as if it were an equation
Correction: When you multiply both sides of an inequality by an expression containing a variable, you don't know whether that expression is positive or negative, so you don't know whether to flip the inequality sign. Instead, move everything to one side and use a sign chart.
Mistake: Including values that make the denominator zero in the solution
Correction: A rational expression is undefined when its denominator is zero. Even with a ≤ or ≥ sign, those values must always be excluded from the solution set — use a parenthesis, never a bracket, at those points.