Sketch the parabola. For ax² + bx + c 0: between the roots. For > 0: outside the roots.

Inequalities (GCSE)

Q: Number line conventions?

Open circle (), closed circle (≤ or ≥); arrow shows direction.

Linear (both tiers); quadratic and number-line/set notation (Higher).

Linear: same rules as KS3 (flip when dividing by a negative). Quadratic: factorise, sketch the parabola, identify the regions where the inequality holds.

Worked examples

3x − 5 < 7 ⇒ x < 4.

x² − 4 ≤ 0 ⇒ (x+2)(x−2) ≤ 0 ⇒ −2 ≤ x ≤ 2.

Set notation: {x : −2 ≤ x ≤ 2}. The dots are interval brackets in some textbooks: [−2, 2].

Frequently asked questions

Quadratic regions?

Sketch the parabola. For ax² + bx + c < 0 with a > 0: between the roots. For > 0: outside the roots.

Number line conventions?

Open circle (< or >), closed circle (≤ or ≥); arrow shows direction.