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Math Practice

Polynomial Inequalities Practice Problems

Twelve hand-built polynomial inequalities with full worked solutions. Each moves everything to one side, factors, finds the critical values, tests each interval, and reports the solution in interval notation.

  1. 1.

    Solve \(x^{2} - x - 6 > 0\).

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    \((x - 3)(x + 2) > 0\); critical values \(-2, 3\). Positive outside the roots.

    Answer: \((-\infty, -2) \cup (3, \infty)\).

  2. 2.

    Solve \(x^{2} + 2x - 8 \le 0\).

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    \((x + 4)(x - 2) \le 0\); critical values \(-4, 2\). Negative between; include endpoints.

    Answer: \([-4, 2]\).

  3. 3.

    Solve \(x^{2} - 16 \ge 0\).

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    \((x - 4)(x + 4) \ge 0\). Positive outside; include endpoints.

    Answer: \((-\infty, -4] \cup [4, \infty)\).

  4. 4.

    Solve \(x^{2} + 5x < 0\).

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    \(x(x + 5) < 0\); critical values \(-5, 0\). Negative between.

    Answer: \((-5, 0)\).

  5. 5.

    Solve \(x^{2} - 3x - 10 \ge 0\).

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    \((x - 5)(x + 2) \ge 0\); critical values \(-2, 5\).

    Answer: \((-\infty, -2] \cup [5, \infty)\).

  6. 6.

    Solve \(2x^{2} - 3x - 2 < 0\).

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    \((2x + 1)(x - 2) < 0\); critical values \(-\tfrac{1}{2}, 2\). Negative between.

    Answer: \(\left(-\dfrac{1}{2}, 2\right)\).

  7. 7.

    Solve \(x^{2} \le 5x - 6\).

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    \(x^{2} - 5x + 6 \le 0 \Rightarrow (x - 2)(x - 3) \le 0\); critical values \(2, 3\). Negative between.

    Answer: \([2, 3]\).

  8. 8.

    Solve \(x^{2} - 4x + 4 > 0\).

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    \((x - 2)^{2} > 0\): positive everywhere except \(x = 2\).

    Answer: \((-\infty, 2) \cup (2, \infty)\).

  9. 9.

    Solve \(x^{3} - 9x > 0\).

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    \(x(x - 3)(x + 3) > 0\); critical values \(-3, 0, 3\). Test: \(-\), \(+\), \(-\), \(+\).

    Answer: \((-3, 0) \cup (3, \infty)\).

  10. 10.

    Solve \((x - 2)^{2}(x + 4) \ge 0\).

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    Critical values \(-4, 2\). Sign: \(-\) on \((-\infty, -4)\), \(+\) on \((-4, 2)\), \(+\) on \((2, \infty)\) — no flip at \(2\).

    Answer: \([-4, \infty)\) (includes \(x = 2\) as equality).

  11. 11.

    Solve \(x^{3} - x^{2} - 6x \le 0\).

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    \(x(x - 3)(x + 2) \le 0\); critical values \(-2, 0, 3\). Test: \(-\), \(+\), \(-\), \(+\).

    Answer: \((-\infty, -2] \cup [0, 3]\).

  12. 12.

    Solve \((x + 1)(x - 3)(x - 6) > 0\).

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    Critical values \(-1, 3, 6\). Test: \(-\), \(+\), \(-\), \(+\).

    Answer: \((-1, 3) \cup (6, \infty)\).
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Polynomial Inequalities Practice Problems

The Polynomial Inequalities lesson lays out the sign-chart method step by step; this set is 12 to practice on, each fully solved.

How to use this set

These problems are hand-written, not auto-generated. Move all terms to one side, factor, list the critical values, test one point per interval, and select the intervals whose sign matches — adding the critical values for / . Then compare with the solution.

Common mistakes to watch for

Dividing by a variable factor. is not ; move everything over and factor.

Not setting one side to zero first.

Forgetting the endpoints for / , or including them for / .

Assuming the sign always alternates — it doesn’t at a squared factor.

Where to go next

Finding the critical values uses Factoring Quadratics. The next step, with denominators, is the Rational Inequalities Practice Problems. The answer format is Interval Notation. For the full method, see the Polynomial Inequalities lesson.

Frequently Asked Questions

Why can't I solve a polynomial inequality by isolating x?+

Isolating would require dividing by variable factors whose sign is unknown, and dividing an inequality by a negative flips it. The sign-chart method avoids this by analyzing the sign of one factored expression versus zero.

What are critical values?+

The roots of the factored polynomial — the only places the polynomial can change sign. They split the number line into test intervals.

How do I test an interval?+

Pick any number strictly inside it, substitute into the factored form, and record whether the result is positive or negative. One test point settles the whole interval.

Are the critical values included?+

For or , no (parentheses). For or , yes (brackets) — the polynomial equals zero there, which satisfies a non-strict inequality.

What happens at a repeated factor like (x − 2)²?+

The polynomial touches zero at but does not change sign, because a squared factor is never negative.

Why is this a curated set?+

The answers are intervals or unions, not single numbers. Each problem has a full worked solution with the sign table.

More practice problems

Quadratic Equations Practice Problems Rational Inequalities Practice Problems Linear Inequalities Practice Problems