Robinson Tuition
Grade 6 · Higher · free GCSE maths worksheet

Regions from inequalities on a graph

Shading regions given by lines.

16 original questions, 59 marks. Print the worksheet, mark it with the answers sheet, and use the solutions to see every step written out by hand.

Worksheet PDFWorked solutions PDFAnswers PDF

The questions

1

On the grid, shade the region that satisfies all three of these inequalities.

x ≥ 1

y > 2

x + y < 6

Label the region R.

−2−11234567−2−11234567Oxy
3 marks
Show answer
Triangle region with vertices (1, 2), (4, 2), (1, 5): solid boundary on x = 1, dashed boundaries on y = 2 and x + y = 6
2

On the grid, shade the region that satisfies all three of these inequalities.

y ≤ 3

y ≥ x - 1

x > 0

Label the region R.

−2−1123456−2−1123456Oxy
3 marks
Show answer
Region bounded by y = 3 (solid, below), y = x - 1 (solid, above) and x = 0 (dashed, right of); vertices (0, -1), (0, 3), (4, 3)
3

When a region is shaded on a graph, the boundary line for each inequality is drawn either solid or dashed.

Write "solid" or "dashed" for the boundary line of each inequality.

  1. y > 1
  2. x ≤ 5
  3. y ≥ x + 2
  4. x + y < 6
2 marks
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a) dashed b) solid c) solid d) dashed
4

The region R satisfies all three of these inequalities.

x ≥ 1

y ≥ 2

x + y ≤ 5

Find all the points in R whose coordinates are both integers.

3 marks
Show answer
(1, 2), (1, 3), (1, 4), (2, 2), (2, 3), (3, 2)
5

The region R satisfies all three of these inequalities.

y > x

x ≥ 0

y ≤ 4

Amy says the point (2, 2) is in R.

Is Amy correct? You must explain your answer.

3 marks
Show answer
No: at (2, 2) the inequality y > x is not satisfied because 2 is not greater than 2; the point is on the boundary line y = x, which is not included
6

A region R is drawn on a coordinate grid.

R is the triangle with vertices (2, 1), (6, 1) and (2, 5), including its boundary.

The three sides of the triangle lie on the lines x = 2, y = 1 and x + y = 7

Find the three inequalities that define R.

−2−11234567−2−11234567OxyR
4 marks
Show answer
a) x >= 2 b) y >= 1 c) x + y <= 7
7

On the grid, shade the region that satisfies all four of these inequalities.

x > 0

y > 1

y < 2x

x + y < 6

Label the region R.

−2−11234567−2−11234567Oxy
4 marks
Show answer
The interior of the triangle with boundary intersections (1/2, 1), (5, 1), (2, 4); all three edges are dashed and excluded. x > 0 holds automatically and x = 0 is not an edge of R.
8

The region R satisfies all three of these inequalities.

x > 0

y ≥ 1

x + y < 5

Write down the coordinates of all the points in R with whole number coordinates.

3 marks
Show answer
(1, 1), (1, 2), (1, 3), (2, 1), (2, 2), (3, 1)
9

The region R satisfies all three of these inequalities.

x + 2y ≤ 10

x ≥ 0

y > 1

Which of these points are in R?

P(4, 3)    Q(6, 2)    S(2, 1)    T(0, 5)

3 marks
Show answer
P, Q and T
10

The region R is bounded by three lines: x = 1, y = 2 and x + y = 8.

R is to the right of x = 1, above y = 2 and below x + y = 8.

The lines x = 1 and x + y = 8 are solid. The line y = 2 is dashed.

Write down the three inequalities that define R.

3 marks
Show answer
x ≥ 1, y > 2, x + y ≤ 8
11

A region R is drawn on a coordinate grid.

R is the triangle with vertices (0, 1), (4, 1) and (4, 5), including its boundary.

Find the three inequalities that define R.

−2−11234567−2−11234567OxyR
4 marks
Show answer
a) y >= 1 b) x <= 4 c) y <= x + 1
12
  1. On the grid, shade the region R that satisfies all three of these inequalities.

    y ≥ 1

    x + y ≤ 8

    y ≤ 3x - 1

  2. Priya says the point (3, 8) is in R. Explain why Priya is wrong.
−2−1123456789−2−1123456789Oxy
5 marks
Show answer
a) Shaded triangle bounded by y = 1, x + y = 8 and y = 3x - 1, all solid; vertices (2/3, 1), (7, 1), (9/4, 23/4) b) At (3, 8), x + y = 11, which is greater than 8, so the inequality x + y <= 8 fails
13

A straight line passes through (0, 1) and (4, 3).

The region R is on or above this line, on or below the line y = 3, and on or to the right of the y-axis.

Write down the three inequalities that define R.

4 marks
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y ≥ x/2 + 1, y ≤ 3, x ≥ 0
14

The region R satisfies all four of these inequalities.

x ≥ 0

y ≥ 0

2x + y ≤ 10

y ≤ x + 2

The point (x, y) is in R and both coordinates are integers.

Find the greatest possible value of x + y

5 marks
Show answer
7
15

A region R is drawn on a coordinate grid.

R is the triangle with vertices (0, 1), (6, 1) and (2, 5), including its boundary.

Find the three inequalities that define R.

−2−11234567−2−11234567OxyR
5 marks
Show answer
a) y >= 1 b) y <= 2x + 1 c) x + y <= 7
16

The region R satisfies all three of these inequalities.

x ≥ 1

y > 2

x + y ≤ 8

  1. Work out how many points in R have whole number coordinates.
  2. Work out the greatest value of 2x + y for points in R with whole number coordinates.
5 marks
Show answer
a) 15 b) 13

Every question here is original and written by Robinson Tuition. The grade follows the Edexcel grade descriptors and suits AQA and OCR too; this is not an exam-board paper.

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