Quadratic inequality
A standard method from Inequalities and linear programming: the steps and worked examples.
Steps
- Move everything to one side and factorise (or find roots with the formula).
- Sketch the parabola through the roots (a > 0 opens up).
- "< 0" means below the x-axis: between the roots.
- "> 0" means above: outside the roots.
y = (x + 2)(x − 3)
\ /
---•-------•--- x² − x − 6 < 0 → −2 < x < 3 (below)
−2 \_/ 3 x² − x − 6 > 0 → x < −2 or x > 3 (above)Worked examples
Example 1: Quadratic inequality (NF): solve x² − x − 6 < 0.
(x + 2)(x − 3) < 0, so −2 < x < 3.
Example 2: Triangle inequality: two sides of a triangle are 4 cm and 9 cm. Find the range of the third side x.
4 + 9 > x → x < 13
4 + x > 9 → x > 5
9 + x > 4 → always true
5 < x < 13For the whole topic, read Inequalities and linear programming: key points, common mistakes and how the DSE tests it.