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.

Start practising Read the lesson

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 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
2
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
3
The region $R$ is defined by

$x \geqslant 0, \qquad y \geqslant 0, \qquad y \leqslant 6 - x$

Find the area of $R$ . (3 marks)
Level 3 · 3 marksTry it
4
The region $R$ is defined by $y \geqslant x^{2}$ and $y \leqslant x + 2$ .
(a) Find the coordinates of the points where the boundaries intersect. (3 marks)
(b) Determine whether the point $(0, 1)$ lies in $R$ . (1 mark)
Level 3 · 4 marksTry it
5
The region $R$ is defined by $y \geqslant x^{2} - 1$ and $y \leqslant 3$ .
(a) Find the $x$ -coordinates of the points where the two boundaries meet. (2 marks)
(b) State the range of possible $x$ -values for points of $R$ . (2 marks)
Level 3 · 4 marksTry it

3 more questions on inequalities on graphs and regions

Free with an account — just an email, no card. The whole bank, every mark scheme and worked solution.

Start free