Polynomial and Rational Inequalities
A free College Algebra lesson from the “Linear Equations and Inequalities” unit, with a worked example and practice problems including step-by-step solutions.
To solve f(x) > 0 (or <, >=, <=), find the values where f(x) = 0 (and where f(x) is undefined, for rational expressions). Those critical values split the number line into intervals. Test a point in each interval: if f is positive there, the whole interval satisfies > 0. For rational inequalities, denominator zeros are always EXCLUDED from the solution (never include).
What you'll learn
- Solve a polynomial inequality by finding zeros and using a sign chart on the resulting intervals
- Solve a rational inequality the same way, marking values that make the denominator zero as excluded
- Express solutions in interval notation
Worked example
Problem. Solve (x - 1)(x - 3) > 0. Enter the upper boundary of the leftmost solution interval.
- Zeros at x = 1 and x = 3.
- Test intervals: x < 1 (positive), 1 < x < 3 (negative), x > 3 (positive).
- Solution: x < 1 OR x > 3. The leftmost interval ends at x = 1.
Answer: 1
Lesson preview: this page shows 5 practice items. The interactive activity contains 28 practice items and 15 quiz items (typed response and multiple choice). Counts include repeated prompts and variants.
Practice problems
1. Solve x^2 - 9 > 0. Enter the smaller boundary of the solution.
Show solution
- x^2 - 9 = (x - 3)(x + 3). Zeros at -3 and 3.
- Solution x < -3 or x > 3. Smaller boundary is -3.
Answer: -3
2. Same inequality. Enter the larger boundary.
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- Larger boundary is 3.
Answer: 3
3. Solve x^2 < 16. Enter the upper boundary of the solution.
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- x^2 - 16 < 0 -> (x - 4)(x + 4) < 0.
- Solution -4 < x < 4. Upper boundary 4.
Answer: 4
4. Solve (x - 1)(x - 3) > 0. Enter the LOWER boundary of the rightmost interval.
Show solution
- Zeros at 1 and 3. Solution x < 1 or x > 3.
- Rightmost interval starts at x = 3.
Answer: 3
5. Solve x^2 - 5x + 6 <= 0. Enter the smaller boundary.
Show solution
- Factor: (x - 2)(x - 3) <= 0.
- Solution 2 <= x <= 3. Smaller boundary 2.
Answer: 2
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