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Rational Inequalities Practice Problems

Twelve hand-built rational inequalities with full worked solutions. Each gets one side to zero, combines into a single fraction, factors, tests each interval, and reports the answer — with denominator zeros always excluded.

  1. 1.

    Solve .

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    Critical values (num, include), (den, exclude). Test: , , .

    Answer: .

  2. 2.

    Solve .

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    Critical values (num, include), (den, exclude). Negative between.

    Answer: .

  3. 3.

    Solve .

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    Numerator always positive; the fraction is positive where the denominator is positive.

    Answer: .

  4. 4.

    Solve .

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    Critical values . Negative between.

    Answer: .

  5. 5.

    Solve .

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    Critical values (num, include), (den, exclude). Negative between.

    Answer: .

  6. 6.

    Solve .

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    . Critical values (num), (den). Positive on .

    Answer: .

  7. 7.

    Solve .

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    . Critical values (num, include), (den, exclude). Positive on .

    Answer: .

  8. 8.

    Solve .

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    . Critical values (num, include), (den, exclude). Test: , , .

    Answer: .

  9. 9.

    Solve .

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    . Critical values (den, exclude), (num, include). Test: , , , .

    Answer: .

  10. 10.

    Solve .

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    Critical values (num, from squared factor), (den, exclude). Sign: on , on , on — no flip at .

    Want : , plus (numerator zero).

    Answer: .

  11. 11.

    Solve .

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    . Critical values (num, include), (den, exclude). Test: , , , .

    Answer: .

  12. 12.

    Solve .

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    . Critical values (num, include), (den, exclude). Positive on .

    Answer: .
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MPhil in Mathematics, Founder of Mathovia
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Rational Inequalities Practice Problems

The Rational Inequalities lesson covers the sign-chart method with the two adjustments for fractions; this set is 12 to practice on, each fully solved.

How to use this set

Get one side to zero, combine into a single fraction, factor, list all critical values (numerator and denominator), test each interval, and select the matching intervals — including numerator zeros for / , never denominator zeros. Enter the solution set and press Check Answers. Brackets matter — a denominator zero is always excluded — but the notation style does not: (-inf, -3) U [2, inf) and the inequality form are treated the same.

Common mistakes to watch for

Cross-multiplying or clearing the denominator — its sign is unknown.

Including a denominator zero — never, even for / .

Forgetting the denominator’s zeros as critical values.

Leaving the inequality as “fraction vs. a nonzero number” instead of reducing to “vs. zero”.

Where to go next

The sign-chart mechanics are the same as the Polynomial Inequalities Practice Problems. Combining fractions is the Rational Expressions Practice Problems skill. The answer format is Interval Notation. For the full method, see the Rational Inequalities lesson.

Frequently Asked Questions

Why can't I multiply both sides by the denominator?+

The denominator's sign is unknown — sometimes positive, sometimes negative — and multiplying an inequality by a negative flips it. Instead, move everything to one side, combine into one fraction, and analyze its sign.

What are the critical values?+

Values that make the numerator zero (where the fraction can be zero) and values that make the denominator zero (where it is undefined). Both split the number line.

Are denominator zeros ever in the solution?+

Never — the expression is undefined there. Even for or , use a parenthesis at those points.

How do I handle a non-strict inequality's numerator zeros?+

Include them. For , a numerator zero makes the fraction zero, which satisfies .

Why combine into a single fraction first?+

An inequality like can't be sign-charted as written. Subtract 2 and combine to get a single fraction versus zero.

How do I show that a denominator zero is excluded?+

With a parenthesis rather than a bracket. The checker compares brackets exactly, so (-inf,-3) U [2,inf) and (-inf,-3] U [2,inf) are different answers — which is the whole point on this topic, since a value that makes the denominator zero can never be included.

More practice problems

Polynomial Inequalities Practice Problems Rational Expressions Practice Problems Linear Inequalities Practice Problems