Rational Inequalities

A rational inequality compares a fraction with a variable in its denominator to zero (or to another value): , . It is solved with the same sign chart as a Polynomial Inequality, with two adjustments.

First, you cannot clear the denominator by multiplying — its sign is unknown, and multiplying an inequality by a negative flips it. So you get everything onto one side and combine into a single fraction first. Second, the values that make the denominator zero are always excluded, because the expression is undefined there — even when the inequality is or .

The Rational Inequality Formula

Numerator zeros can be included (for ); denominator zeros never can.

The Method

1. Move every term to one side so the other side is .
2. Combine into a single fraction over a common denominator.
3. Factor the numerator and the denominator.
4. Find all critical values — set the numerator to zero, and set the denominator to zero.
5. Split the number line at every critical value.
6. Test one point per interval in the factored fraction; record or .
7. Select the matching intervals. For / , include numerator zeros (brackets); never include denominator zeros.
8. Write the answer in interval notation.

Worked Example A: Already One Fraction

Solve .

Steps 1–3 — already a single fraction, already factored.

Step 4 — critical values: numerator zero at ; denominator zero at (excluded).

Steps 5–6 — test the three intervals:

Step 7 — want : the positive intervals; include (numerator zero), exclude .

Answer: .

Worked Example B: Combine First

Solve .

Step 1 — subtract :

Step 2 — common denominator:

Step 4 — critical values: numerator zero at ; denominator zero at (excluded).

Step 7 — want : the negative intervals; strict, so no endpoints, and excluded anyway.

Answer: .

Worked Example C: Quadratic Numerator

Solve .

Factor: .

Critical values: (numerator, includable); (denominator, excluded).

Want : the negative intervals, plus numerator zeros and ; exclude .

Answer: .

Worked Example D: A Common Trap

Solve .

Do not cross-multiply. Subtract and combine:

Critical values: (numerator), (denominator, excluded).

Want : the middle interval, strict.

Answer: . Cross-multiplying would have given , i.e. — wrong, because it silently assumed .

Why Cross-Multiplying Is Illegal Here

With an equation like , cross-multiplying is fine: multiply both sides by and the equals sign is unaffected. With an inequality, multiplying both sides by an expression flips the symbol whenever that expression is negative — and the sign of a variable expression like is not fixed. Since you cannot know in advance whether you are multiplying by a positive or a negative, there is no single valid version of the multiplication. Combining into one fraction and reading its sign directly sidesteps the whole problem.

The trap in Worked Example D shows this concretely: cross-multiplying to silently assumes , and gets the wrong answer for the half of the number line where .

The Two Kinds of Critical Value

Every critical value comes from either the numerator or the denominator, and they behave differently at the endpoints:

• Numerator zeros make the whole fraction equal to . For a non-strict inequality ( or ), they satisfy it, so they are included — bracket.
• Denominator zeros make the fraction undefined. They can never satisfy anything, so they are always excluded — parenthesis, no matter which inequality symbol is used.

Both kinds still split the number line into test intervals, because the fraction can change sign at either.

Worked Example E: A Quadratic in Both Numerator and Denominator

Solve .

Step 1 — factor top and bottom:

Step 2 — critical values: makes both top and bottom zero (a hole, not an asymptote); is a numerator zero (includable); is a denominator zero (excluded).

Step 3 — test each interval in the factored form:

Step 4 — we want : the negative interval, plus the numerator zero ; exclude (the fraction is undefined there — it is a hole) and .

Answer: . The value is excluded even though it “looks like” a numerator zero, because it also makes the denominator zero.

Common Mistakes to Avoid

• Cross-multiplying or clearing the denominator. The denominator’s sign is unknown; combine into one fraction instead.
• Including a denominator zero. Never — the expression is undefined there, even for / . Always a parenthesis.
• Forgetting the denominator’s zeros as critical values. The sign can change there too, so they must split the number line.
• Leaving the inequality as “fraction vs. a nonzero number.” Always reduce to “single fraction vs. ” first.
• Excluding a numerator zero on a non-strict inequality. For / , a numerator zero makes the fraction , which qualifies — include it.
• Assuming the sign alternates every time. Test each interval; a repeated factor does not flip the sign.

Where This Shows Up Later

• Graphing rational functions. The sign chart tells you where the graph is above or below the -axis and how it behaves on each side of a vertical asymptote (a denominator zero).
• Domains. A rational expression’s domain excludes exactly its denominator zeros — the same points excluded here.
• Limits and asymptotes in calculus. Behavior near a vertical asymptote is a sign question about the intervals on either side.
• Optimization with a rational model. Constraints like “average cost below \$5” become rational inequalities.

Practice Problems

Work each problem yourself before opening the answer. Give solutions in interval notation.

Problem 1. Solve .

Problem 2. Solve .

Problem 3. Solve .

Problem 4. Solve .

Problem 5. Solve .

Problem 6. Solve .

Problem 7. Solve .

Problem 8. Solve .

Problem 9. Solve .

Problem 10. Solve .

Problem 11. Solve .

Problem 12. Solve .

Problem 13. Solve .

Problem 14. Solve .

Quick Reference

The sign-chart mechanics are the same as Polynomial Inequalities; combining fractions is the Rational Expressions skill; the answer format is Interval Notation. More Algebra lessons are available too.