Solving Rational Inequalities- Practice Problems with Solutions

What Are Rational Inequalities?

A rational inequality is an inequality that contains a rational expression—basically a fraction where the numerator and denominator are polynomials. You're asked to find the values of the variable that make the inequality true.

The standard forms look like this:

Where R(x) is some rational expression like (x+2)/(x-3) or (x²-4)/(x+1).

The Critical Rule Most People Forget

Never multiply both sides by the denominator unless you know its sign. This is where students consistently make mistakes.

When you multiply by a negative number, the inequality sign flips. When you multiply by a positive number, it stays the same. Since the denominator changes sign at its zeros, guessing is not an option.

How to Solve Rational Inequalities

Step 1: Move Everything to One Side

Get zero on one side of the inequality. Your expression should look like:

R(x) < 0 or R(x) > 0

Step 2: Find the Critical Points

Critical points come from two sources:

Step 3: Plot on a Number Line

Mark these critical points on a number line. Use open circles for points that make the denominator zero (never included in the solution). Use closed circles for points that make the numerator zero (included when the inequality is ≥ or ≤).

Step 4: Test the Intervals

Pick a test value from each interval created by your critical points. Plug it into the simplified expression. Check if the result is positive or negative.

Step 5: Write the Solution

Based on your inequality sign, select the intervals that satisfy the condition. Combine with the appropriate critical points.

Practice Problems with Solutions

Problem 1: Basic Case

Solve: (x-2)/(x+3) > 0

Step 1: Already in correct form.

Step 2: Critical points are x = 2 (numerator) and x = -3 (denominator).

Step 3: Number line with points at -3 and 2.

Step 4: Test intervals:

Solution: x < -3 or x > 2

Write it in interval notation: (-∞, -3) ∪ (2, ∞)

Problem 2: Including the Zero

Solve: (x+1)/(x-4) ≤ 0

Critical points: x = -1 and x = 4

Test intervals:

Solution: -1 ≤ x < 4

Interval notation: [-1, 4)

The x = -1 is included because the inequality is ≤ and the numerator equals zero makes the whole expression zero. The x = 4 is excluded because the denominator is zero.

Problem 3: Factored Numerator

Solve: (x² - 9)/(x + 2) < 0

Factor: ((x-3)(x+3))/(x+2) < 0

Critical points: x = -3, x = 3, x = -2

Test intervals:

Solution: x < -3 or -2 < x < 3

Interval notation: (-∞, -3) ∪ (-2, 3)

Problem 4: Quadratic in Denominator

Solve: (x + 2)/(x² - 4) ≥ 0

Factor denominator: (x + 2)/((x-2)(x+2)) ≥ 0

Simplify: Cancel the common factor (x + 2), but remember x ≠ -2 from the original denominator.

Critical points: x = 2 (from denominator), x = -2 (from original denominator)

After simplification to 1/(x-2):

Test:

Solution: x > 2

Interval notation: (2, ∞)

The x = -2 is excluded because it made the denominator zero in the original expression. No values satisfy the ≥ 0 condition at the critical points.

Common Mistakes to Avoid

Quick Reference Table

Inequality When Numerator = 0 When Denominator = 0
R(x) > 0 Not included (value is 0) Never included
R(x) < 0 Not included (value is 0) Never included
R(x) ≥ 0 Included (value is 0) Never included
R(x) ≤ 0 Included (value is 0) Never included

Getting Started: Your Checklist

Before you start solving any rational inequality, run through this:

If you answered yes to all of these, your solution is solid.

When to Use a Sign Chart

A sign chart is the fastest method for rational inequalities with multiple factors. Instead of testing every interval with numbers, you track the sign of each factor.

For (x+2)(x-3)/(x+1) > 0:

This method gets fast with practice. It's the technique you'll use on timed tests.