Shade the region that satisfies several inequalities, choose dashed or solid lines, and find integer points inside. It is Higher only, aimed at grade 6. Free questions with answers are on this page. Worked examples show each line before you try the questions.
A linear inequality such as y>2x−1 is a region. Draw the boundary y=2x−1: dashed if < or >, solid if ≤ or ≥. Then test a point not on the line (the origin, if it is free) to see which side to shade. Several inequalities at once: the solution is the overlap. Label it R.
Quadratic inequalities on a graph are 'where is the curve above (or below) the axis / the line'. On an x-number-line they become two intervals or a between. Integer points inside R can be listed if asked. Do not shade the wrong side because the test point was on the line — pick a clearly free point.
Solid boundary for ≤ or ≥; dashed for < or >. Test a point (the origin if it is not on the line) to decide which side to shade. Integer points must sit strictly inside open inequalities.
Objectives
Say whether a point is above, below, or on a line.
Test a point in y < x.
Read the gradient and intercept of a boundary.
Key points
Draw the boundary line for each inequality by treating it as an equation.
Use a solid line for ≤ or ≥ and a dashed line for < or >.
Test a point to decide which side of the line to shade.
Find the region satisfied by all the inequalities at once.
Worked examples
Example 1
Is the point (1, 5) above, below or on the line y=2x+1?
2 marks
Substitute x = 1 into the line equation: y = 2(1) + 1 = 3.
Compare the point y-coordinate (5) with the line y-value (3): 5 > 3.
State that the point is above the line.
Answer: above
Common slips
Slip: Compare y with 0 instead of the line. We must evaluate the line height at the given x-coordinate.
Slip: Kim said below. 5 is greater than 3, so the point lies above the line.
Slip: State on the line. 5 is not equal to 3, so it is strictly above.
Example 2
Does the point (1, 4) satisfy the inequality y<x?
2 marks
Substitute x = 1 and y = 4 into the inequality: 4 < 1.
Evaluate the statement: 4 < 1 is false.
Conclude that the point does not satisfy the inequality: no.
Answer: no
Common slips
Slip: Swap x and y. The x-coordinate is 1 and the y-coordinate is 4.
Slip: Len said yes. 4 is greater than 1, not less than 1.
Slip: Conclude yes. The inequality is not satisfied.
Example 3
Find the y-intercept of the boundary y=3x-4.
3 marks
Recall that in y = mx + c, the constant term c is the y-intercept.
Read the constant term with its sign: c = -4.
The y-intercept is -4.
Answer: -4
Common slips
Slip: Mia wrote 3. 3 is the gradient, which is the coefficient of x.
Slip: Drop the negative sign to write 4. The minus sign belongs to the constant term.
Slip: Write y = 3x - 4 as the intercept. The intercept is the constant on its own, -4, not the whole equation.
Questions and answers
GCSE Maths inequalities on graphs questions and answers. Try each one, then open the answer. Easy questions come first.
Question 1 Easy2 marks
Is the point (0, 4) above, below or on the line y=2x+1?
Show answer
Answer: above
Line at x=0 is y=1.
4 > 1, so the point is above.
M1 — Line at x=0 is y=1.
A1 — 4 > 1, so the point is above.
Question 2 Easy2 marks
Is the point (3, 2) above, below or on the line y=x+4?
Show answer
Answer: below
Line at x=3 is y=7.
2 < 7, so the point is below.
M1 — Line at x=3 is y=7.
A1 — 2 < 7, so the point is below.
Question 3 Medium2 marks
Does the point (4, 1) satisfy the inequality y<x?
Show answer
Answer: yes
Test 1 < 4: true.
Answer yes.
M1 — Test 1 < 4: true.
A1 — Answer yes.
Question 4 Medium2 marks
Does the point (2, 2) satisfy the inequality y<x?
Show answer
Answer: no
Test 2 < 2: false.
Answer no.
M1 — Test 2 < 2: false.
A1 — Answer no.
Question 5 Hard2 marks
Find the y-intercept of the boundary y=2x+1.
Show answer
Answer: 1
Read constant term c = 1.
Intercept is 1.
M1 — Read constant term c = 1.
A1 — Intercept is 1.
Question 6 Hard2 marks
Find the y-intercept of the boundary y=4x-3.
Show answer
Answer: -3
Read constant term c = -3.
Intercept is -3.
M1 — Read constant term c = -3.
A1 — Intercept is -3.
Question 7 Exam-style4 marks
Is the point (3, 6) inside the region where y > 2x - 1 and y < x + 4? Answer yes or no.
Show answer
Answer: yes
At x = 3 the line y = 2x - 1 has y = 5, and the line y = x + 4 has y = 7.
Test y > 2x - 1: 6 > 5 is true.
Test y < x + 4: 6 < 7 is true.
Both are true, so the answer is yes.
M1 — Substitutes x = 3 into both lines: 5 and 7
M1 — Tests 6 > 5
M1 — Tests 6 < 7
A1 — Yes, because both inequalities are true
Common mistakes
Most lost marks on this topic are one slip.
Calling a point above the line when its y is smaller
Compare the point’s y with the line’s y
Saying equal values satisfy a strict less-than
y < x needs a smaller y
Reading the gradient as the intercept
The intercept is the constant term
Spot the slip
The point (1, 5) and the line y=2x+1. Where is the point?
above
below
on the line
Show the slip
above On the line, x=1 gives y=3. The point has y=5, which is higher.
Exam tips
Be ready to do this without a calculator. Paper 1 is the non-calculator paper.
If the question says “show that”, the result is given. The marks are for the steps that reach it.
The method is the same on AQA-style and Edexcel-style papers.
Label the region R exactly as the question asks, so the marker knows which part is your answer.
Write your test point and the result next to each line ('(0,0): 0≥−2 true') — this shows method if the shading is unclear.
Printable worksheet
The same questions are in a PDF you can print. Answers start on a later page.