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
(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
2
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
3
The diagram shows the lines $y = x$ , $y = 3$ and $x = 2$ , 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
4
A triangular region is bounded by the three inequalities

$y \ge 0, \qquad y \le 2x, \qquad x \le 3$ Work out the coordinates of the three vertices of the region.
Level 4 · 3 marksTry it
5
The diagram shows the vertical line $x = -1$ (solid), the line $y = x - 2$ (dashed), and the horizontal line $y = 4$ (solid). The region $R$ lies to the right of the vertical line, above the dashed line, and below the horizontal line.Write down the three inequalities that define the region $R$ .

(A dashed line is not included in the region; a solid line is.)
Level 4 · 3 marksTry it

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