GCSE · Inequalities

Inequalities as regions on a graphExam-style questions

Why an inequality in x and y claims a HALF-PLANE, why testing one point settles which side, and how several inequalities carve out a region together.

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11 questions · GCSE (Higher) level · mark scheme and worked solution on each · no sign-up to try

DifficultyAll Level 3 Level 4 Level 5

1
The diagram shows the line $x = 2$ and the line $y = 1$ .Diagram NOT accurately drawn

The region $R$ is to the right of the line $x = 2$ and below the line $y = 1$ .

Points on the line $x = 2$ are included in $R$ ; points on the line $y = 1$ are not.

Write down the two inequalities that define $R$ .
Level 3 · 2 marksTry it
2
A region is defined by the three inequalities

$x \ge 0, \qquad y < x + 1, \qquad x + y \le 6$

Here are three points: $\;A(2, 1) \qquad B(4, 3) \qquad C(1, 4)$

Which of the points lies in the region? You must show how you decide.
Level 3 · 3 marksTry it
3
A region $R$ is defined by the two inequalities

$x < 3 \qquad \text{and} \qquad y \ge 2$ (a) Is the point $(1, 4)$ in the region $R$ ? Give a reason.

(b) Is the point $(3, 2)$ in the region $R$ ? Give a reason.
Level 3 · 3 marksTry it
4
(a) On the grid, shade the region $R$ that satisfies all of these inequalities.

$y \ge -1 \qquad 2x + y \le 7 \qquad y \le 2x + 3 \qquad x \ge -3$

Label the region $R$ .(4 marks)

Only three of the four inequalities are needed to describe the region $R$ .
(b) Write down the inequality that is not needed.

(1 mark)
Level 4 · 5 marksTry it
5
The diagram shows three straight lines and a region $R$ bounded by them.Diagram NOT accurately drawn

All three boundary lines are included in $R$ .

Write down the three inequalities that define $R$ .
Level 4 · 3 marksTry it

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