A-level · Algebra & functions

Inequalities on graphs and regionsExam-style questions

Reading an inequality as a region of the plane. The boundary as the matching equation; which side of a curve or line to shade, and testing a point; dashed boundaries for strict inequalities and solid ones for inclusive; a region as the overlap of several inequalities; finding its corners by solving the boundaries together; and writing down the inequalities for a shaded region.

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18 questions · A-level level · mark scheme and worked solution on each · no sign-up to try

DifficultyAll Level 2 Level 3 Level 4 Level 5

1
The inequality $y > 3x - 2$ defines a region of the plane.
(a) Determine whether the point $(2, 5)$ lies in the region. (2 marks)
(b) Determine whether the point $(3, 5)$ lies in the region. (1 mark)
Level 2 · 3 marksTry it
2
A student is asked to shade the region satisfying $y > 2x + 1$ .
(a) State whether the boundary line should be drawn solid or dashed, and why. (2 marks)
(b) Describe how to decide which side of the line to shade. (1 mark)
Level 2 · 3 marksTry it
3
The diagram shows the curve $y = 9 - x^2$ and the line $y = x + 3$ . The region $R$ , shaded in the diagram, is bounded by the curve, the line and the $x$ -axis.Write down three inequalities that define the region $R$ .

(3 marks)
Level 2 · 3 marksTry it
4
The region $R$ is bounded below by the curve $y = x^{2}$ and above by the line $y = 2x + 3$ .
(a) Write down two inequalities that define $R$ . (2 marks)
(b) Find the $x$ -coordinates of the points where the boundaries intersect. (2 marks)
Level 3 · 4 marksTry it
5
The region $R$ is defined by the inequalities

$y < 4 - x, \qquad y > 1, \qquad x > 0$

Find all the points in $R$ whose coordinates are both integers. (4 marks)
Level 3 · 4 marksTry it

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