Write down the three inequalities that define the shaded region R.
Worked Solution & Answer
Dashed boundaries use strict or non-strict per region: x ≥ 1, y ≥ 2, x + y ≤ 9 B1
Answer: x ≥ 1, y ≥ 2, x + y ≤ 9
Inequalities on graphs — Notes & Worked Examples Show / Hide
Explanation:
To represent inequalities on a graph, first draw each boundary line by replacing the inequality sign with =. Use a solid line for inclusive inequalities (≤ or ≥) and a dashed line for strict inequalities (< or >).
To decide which side of a line satisfies the inequality, substitute a simple test point not on the line—such as (0, 0) or (1, 1)—into the inequality and check if it is true.
Example 1: Write down the three inequalities that define the region R (containing (2, 0)) bounded by solid lines x = 1 and y = −2, and dashed line x + y = 4.
Solution:
Right of solid x = 1 → x ≥ 1; above solid y = −2 → y ≥ −2
Below dashed x + y = 4 (test (2,0): 2 + 0 < 4) → x + y < 4
Answer: x ≥ 1, y ≥ −2, and x + y < 4
Example 2: Find all integer coordinates (x, y) that satisfy x > 1, y > 2, and 2x + y ≤ 7.
Solution:
Since x and y are integers: x ≥ 2 and y ≥ 3
Try x = 2: 2(2) + y ≤ 7 → y ≤ 3, so y = 3 works → (2, 3)
Try x = 3: 2(3) + y ≤ 7 → y ≤ 1 (no integers with y > 2)
Answer: (2, 3)
On the grid, shade the region R that satisfies all the following inequalities.
x ≤ 2 y ≥ -3 y ≤ 2x + 1 3x + 2y ≤ 6
Label the region R.
x ≤ 2 y ≥ -3 y ≤ 2x + 1 3x + 2y ≤ 6
Label the region R.
On the grid, shade the region R that satisfies all the following inequalities.
x ≤ 2 y ≥ -3 y ≤ 2x + 1 3x + 2y ≤ 6
Label the region R.
x ≤ 2 y ≥ -3 y ≤ 2x + 1 3x + 2y ≤ 6
Label the region R.
Worked Solution & Answer
Plot boundary lines x = 2, y = –3, y = 2x + 1 (through (–2,–3), (0,1), (2,5)), and 3x + 2y = 6 (through (0,3), (2,0)) M2
Shade interior region R. A1
Shade interior region R. A1
Answer: Quadrilateral region R bounded by x = 2, y = –3, y = 2x + 1 and 3x + 2y = 6 with vertices (–2, –3), (2, –3), (2, 0) and (6/7, 19/7)
On the grid, shade the region that satisfies all these inequalities.
y > 1 x + y < 5 y > 2x
Label the region **R**.
y > 1 x + y < 5 y > 2x
Label the region **R**.
Worked Solution & Answer
Draw lines y = 1, x + y = 5 and y = 2x M2
shade the region above y = 1, below x + y = 5 and to the left of y = 2x. A1
shade the region above y = 1, below x + y = 5 and to the left of y = 2x. A1
Answer: Region R shaded above y = 1, below x + y = 5 and above/left of y = 2x
On the grid show, by shading, the region that satisfies all of these inequalities.
2y + 4 < x x < 3 y < 6 - 3x
Label the region **R**.
2y + 4 < x x < 3 y < 6 - 3x
Label the region **R**.
Worked Solution & Answer
Draw boundary lines x = 3, y = 0.5x – 2, and y = 6 – 3x M2
Shade the region satisfying all three inequalities. A1
Shade the region satisfying all three inequalities. A1
Answer: Region R shaded below 2y + 4 = x, below y = 6 – 3x, and left of x = 3
The region R, shown shaded in the diagram, is bounded by three straight lines.
Write down the three inequalities that define R.
Write down the three inequalities that define R.
Worked Solution & Answer
Horizontal line is y = –3 (region above: y ≥ –3) M1
line through (0, 1) and (1, 0) is x + y = 1 (region below: x + y ≤ 1) M1
given line y = 2x + 2 (region below: y ≤ 2x + 2) A1
line through (0, 1) and (1, 0) is x + y = 1 (region below: x + y ≤ 1) M1
given line y = 2x + 2 (region below: y ≤ 2x + 2) A1
Answer: y ≥ -3, x + y ≤ 1, y ≤ 2x + 2
On the grid, identify the region represented by
x ≤ 5 y ≤ 4 x + y > 6
Label the region R.
x ≤ 5 y ≤ 4 x + y > 6
Label the region R.
Worked Solution & Answer
Draw vertical line x = 5 (solid), horizontal line y = 4 (solid), and line x + y = 6 (dashed) M2
The region R is the triangle bounded by these lines with vertices at (2, 4), (5, 4), and (5, 1). A1
The region R is the triangle bounded by these lines with vertices at (2, 4), (5, 4), and (5, 1). A1
Answer: Region R shaded between x = 5, y = 4, and dashed line x + y = 6
On the grid, identify the region represented by
x > 3 \quadand y > 1 \quadand x + y ≤ 7
Label the region R.
x > 3 \quadand y > 1 \quadand x + y ≤ 7
Label the region R.
Worked Solution & Answer
1. Draw dashed vertical line x = 3 (strict inequality). M1
2. Draw dashed horizontal line y = 1 (strict inequality).
3. Draw solid line x + y = 7 through (0, 7) and (7, 0). M1
4. Shade the triangular region between x > 3, y > 1, and below/on x + y ≤ 7 with vertices at (3, 1), (6, 1), and (3, 4).
5. Label this region R. A1
2. Draw dashed horizontal line y = 1 (strict inequality).
3. Draw solid line x + y = 7 through (0, 7) and (7, 0). M1
4. Shade the triangular region between x > 3, y > 1, and below/on x + y ≤ 7 with vertices at (3, 1), (6, 1), and (3, 4).
5. Label this region R. A1
Answer: Region R shaded on diagram
The shaded region shown on the grid is bounded by four straight lines.
Find the four inequalities that define the shaded region.
Worked Solution & Answer
Vertical line x = –3 (right of line: x ⩾ –3) M1
horizontal line y = 6 (below line: y ⩽ 6) M1
line through (–2, 0) and (0, 6) is y = 3x + 6 (above: y ⩾ 3x + 6) M1
line through (0, 1) and (2, 0) is y = –1/2 x + 1 (above: y ⩾ –1/2 x + 1) A1
horizontal line y = 6 (below line: y ⩽ 6) M1
line through (–2, 0) and (0, 6) is y = 3x + 6 (above: y ⩾ 3x + 6) M1
line through (0, 1) and (2, 0) is y = –1/2 x + 1 (above: y ⩾ –1/2 x + 1) A1
Answer: x ≥ -3, y ≤ 6, y ≥ 3x + 6, y ≥ -12x + 1
(a) Shade the region R satisfying x ≥ 0, x ≤ 2, y ≤ x + 3, 2x + 3y ≥ 6 (4)
(a) On the grid show, by shading, the region that satisfies all these inequalities.
x ≥ 0 x ≤ 2 y ≤ x + 3 2x + 3y ≥ 6
Label the region **R**. [4 marks]
x ≥ 0 x ≤ 2 y ≤ x + 3 2x + 3y ≥ 6
Label the region **R**. [4 marks]
Worked Solution & Answer
(a)
Draw lines x = 0, x = 2, y = x + 3, 2x + 3y = 6 and shade enclosed trapezium M3 A1
Answer (a): Trapezium region R bounded by x = 0, x = 2, y = x + 3 and 2x + 3y = 6
x + y < 5 y > 1 x > 2 y < 3x - 2
(a) On the grid, show by shading, the region that satisfies all of these inequalities.
Label the region **R**. [4 marks]
Label the region **R**. [4 marks]
(b) Which inequality can be removed so that the region **R** will not change? [1 mark]
Worked Solution & Answer
(a)
Draw dashed boundary lines x + y = 5, y = 1, x = 2, y = 3x – 2 and shade the bounded triangular region M3 A1
Answer (a): Region R shaded
(b)
The region is already fully enclosed by x + y < 5, y > 1, and x > 2
the line y = 3x – 2 does not cut into or bound this region, so y < 3x – 2 can be removed B1
the line y = 3x – 2 does not cut into or bound this region, so y < 3x – 2 can be removed B1
Answer (b): y < 3x - 2