Year 11 (2026) Maths Extension 1 (2026) Functions

Inequalities -Quadratic

7 practice questions 1 video lesson Theory + worked examples

Every question with a fully worked solution.

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Theory

Quadratic Inequalities Theory

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          Key concept: A quadratic inequality can be in two forms: the factored form (x−a)(x−b)>0 or the unfactored form ax2+bx+c≤0.
        

To solve an unfactored quadratic inequality the inequality is factorised and the 'critical points' or zeros are determined.

A number line is then used to determine where the values of $x$ are valid for a specific inequality.

Example 1 — Factored Form

Solve

(x + 1) (x - 3) \leq 0

Solution

$x = - 1$ and $x = 3$ are the $x$ intercepts.

We have to draw a graph of $y = (x + 1) (x - 3)$

$(x + 1) (x - 3) \leq 0$ is below or on the $x$ axis for $- 1 \leq x \leq 3$

Example 2 — Unfactored Form

Solve

x^{2} - 6 x + 8 < 0

Solution

\begin{aligned} x^{2} - 6 x + 8 & < 0 \\ x^{2} - 4 x - 2 x + 8 & < 0 \\ x (x - 4) - 2 (x - 4) & < 0 \\ (x - 4) (x - 2) & < 0 \end{aligned}

$x = 4$ and $x = 2$ are the $x$ intercepts.

We have to draw the graph of $y = (x - 4) (x - 2)$

$(x - 4) (x - 2) < 0$ is below the $x$ axis for \(2