Algebra 2 · Unit 2 · Topic 2.5
Quadratic Inequalities and Sign Intervals
A quadratic inequality is solved by identifying real zeros and testing the sign on the intervals they separate.
Definition and explanation
A quadratic inequality is solved by identifying real zeros and testing the sign on the intervals they separate. Include boundary roots for ≤ or ≥ and exclude them for strict inequalities.
Worked example
Solve x² − x − 12 ≤ 0. Factor (x − 4)(x + 3). The product is negative between −3 and 4 and zero at the endpoints, so the solution is [−3, 4]. Testing x = 0 gives −12 ≤ 0.
Common mistake
Solving the associated equation finds boundaries, but the inequality generally includes an interval of values.
Key takeaway
A quadratic inequality is solved by identifying real zeros and testing the sign on the intervals they separate.
Explain each step of the worked example, then check the units, signs, and conditions that apply.
Further reading
Topic coverage was checked against the openly available OpenStax Intermediate Algebra 2e, Algebra and Trigonometry 2e, and Introductory Statistics 2e textbooks. The explanations and examples on this site are independently written; textbook prose, exercises, and artwork are not reproduced.