GCSE Maths revision · Higher: number and algebra

Quadratic inequalities

Higher only

The idea

To solve $x^2 - x - 6 < 0$ , find the roots ( $x = -2$ and $x = 3$ ) and sketch the U-shaped curve. It is below the axis between the roots, so $-2 < x < 3$ .

Solve the equals version to find the roots.
Sketch the graph.
For a U-shaped curve, $< 0$ means below the axis (between the roots) and $> 0$ means above (the two outside parts).

Watch out The answer is a range of values, not just the two roots.

See it worked

Worked exampleNon-calculator3 marks

Solve each inequality.

(a) Solve $x^2 - 9 < 0$ .

(b) Solve $x^2 \ge 16$ .

(a) $x^2 - 9 = (x + 3)(x - 3)$ is zero at $x = \pm 3$ .
The graph of $y = x^2 - 9$ is below the $x$ -axis between the roots, so $-3 < x < 3$ .
(b) $x^2 = 16$ at $x = \pm 4$ . $x^2$ is at least 16 outside the roots: $x \le -4$ or $x \ge 4$ .

Answer (a) $-3 < x < 3$ ; (b) $x \le -4$ or $x \ge 4$

Your turn

Question 1Non-calculator1 mark

Which is the solution of $x^{2}$ $-$ 7x + 10 < 0?

Question 2Non-calculator4 marks

$(x+2)(x-5)<0$ .

(a) Solve the inequality.

(b) Find the number of integer solutions.

Question 3Calculator allowed3 marks

x and y are positive integers. They satisfy x + 2y $\le$ 13 and 3x + y $\le$ 17. Find the greatest possible value of 4x + 5y.

In past papers

41 official questions in 37 papers since 2017. The latest:

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