Quadratic inequalities revisited
Objective. Solve compound quadratic inequalities using sign analysis.
- 1. Learn
- 2. Worked example
- 3. Practice
- 4. Feedback
- 5. Continue
Learn: the key idea
Quadratic inequalities revisited ask for the range of values of x satisfying an inequality involving an x² term, found by combining roots with the shape of the parabola. First rearrange so one side is zero, factorise or use the formula to find the critical values, then sketch the curve to see where it lies above or below the x-axis. For x² − 5x + 6 > 0, the roots are 2 and 3, and since the parabola opens upward, the inequality holds outside the roots: x < 2 or x > 3.
Worked example
Solve the quadratic inequality x² − x − 12 ≤ 0.
- 1Rearrange to standard form: x² − x − 12 ≤ 0 (already zero on one side)
- 2Factorise: (x − 4)(x + 3) ≤ 0
- 3Critical values are x = 4 and x = −3; the upward parabola is ≤ 0 between them
- 4State the solution as the interval between the roots inclusive
Answer: −3 ≤ x ≤ 4
Concept mastery
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Practice
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Start practisingCommon mistake. Students often write the solution as a single combined inequality like 2 < x < 3 when the parabola actually points upward and is positive outside the roots. Always sketch the graph to check which region satisfies the inequality before writing the final answer.
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