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GCSE Maths · Algebra
Graphical inequalities GCSE Questions and Worked Answers
A graphical inequality describes a region of allowed coordinate pairs. Draw its boundary, decide whether the boundary is included, and keep the side whose points make the inequality true.
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Start with the meaning
What you need to know about graphical inequalities
Suppose a stall can display at most four items. Let x count mugs and y count plates. One mug and two plates fit because their total is three; three mugs and two plates do not fit because their total is five. The instruction x + y ≤ 4 means ‘x plus y is less than or equal to four’. On a coordinate grid each point (x, y) is one pair of values. The graph collects all allowed pairs into a region.
See the idea first
A region contains many solutions
Counts cannot be negative, so x ≥ 0 and y ≥ 0 keep us in the first quadrant. The line x + y = 4 is the limit. It belongs to the solution because exactly four items are allowed. For actual mugs and plates, only whole-number points within the region represent possible counts; the shaded region also shows real-number solutions.
From problem to method
Typical problems you need to be able to solve
These are the main kinds of question in this topic. Each card first states the problem, then explains how to solve it.
Shade one side of a line
What the problem asks: Shade y > x + 1.
How to solve it: First draw y = x + 1 dashed, because equality is excluded. The point (0, 2) works since 2 > 1; shade the side containing it.
Combine several conditions
What the problem asks: Shade x ≥ 0, y ≥ 0 and x + y ≤ 4.
How to solve it: Keep only points satisfying all three. The overlap is the closed triangle with corners (0, 0), (4, 0), (0, 4).
Read inequalities from a region
What the problem asks: A region lies below the solid line y = 3 and right of the dashed line x = 1. Write its conditions.
How to solve it: Below including the edge gives y ≤ 3; right excluding the edge gives x > 1. Both conditions must hold.
A reliable routine
Draw a region for linear inequalities
This method applies to straight-line boundaries. Each line separates the plane into two sides; testing a point not on the line tells you which side satisfies the condition.
- Replace the inequality sign by = to draw the boundary line.
- Use a solid line for ≤ or ≥ and a dashed line for < or >.
- Test a convenient point not on the boundary in the original inequality.
- Repeat and retain the overlap; shade or label it as the question instructs.
Check: A point on the boundary cannot tell you which side to shade. If the boundary goes through (0, 0), choose a different test point. The diagram's triangular region is an example, not a universal shape.
Fully worked
Graphical inequalities GCSE worked examples
Each solution explains the clue, why the method fits and how to check the result.
Example 1
Vertical boundary
Question
Describe the region x < 2.
Draw vertical x = 2 dashed. (0, 0) satisfies 0 < 2, so shade left. The line itself is excluded.
Example 2
Horizontal boundary
Question
Describe y ≥ −1.
Draw horizontal y = −1 solid. (0, 0) works, so shade above, including the line.
Example 3
Sloping boundary
Question
Shade y ≤ 2x + 1.
Boundary points include (0, 1) and (1, 3). Join with a solid line. Test (0, 0):
This is true, so retain the side containing the origin, below the line.
Example 4
Overlapping conditions
Question
Draw x ≥ 0, y ≥ 0, x + y ≤ 4.
The axes are included. The third boundary joins (0, 4) to (4, 0) and is solid. Test (0, 0): 0 ≤ 4. Keep the triangle in the first quadrant, including all edges.
Example 5
Whole-number solutions
Question
List integer pairs with x > 0, y > 0 and x + y ≤ 4.
For x = 1, y can be 1, 2 or 3. For x = 2, y can be 1 or 2. For x = 3, y can be 1. The six pairs are (1,1), (1,2), (1,3), (2,1), (2,2), (3,1).
Example 6
Rearrange carefully
Question
Describe 2x − y < 3 as a region.
Subtract 2x, then divide by −1. A negative multiplier reverses order: 1 < 2 but −1 > −2. So the inequality sign reverses.
Draw y = 2x − 3 dashed and shade above.
10 original questions · total 23 marks
Graphical inequalities GCSE exam-style questions
Try each question before opening its fully worked answer. The difficulty rises through the set.
Before you startAllow about 28 minutes · Show your working. Give exact answers unless a question asks you to round, and include units where needed. · answers start collapsed
Test a point
Does (2, 3) satisfy x + y < 5?
Show worked answer
No: 5 is equal to 5, not less.
Boundary style
Is the boundary for y ≥ x − 2 solid or dashed?
Show worked answer
Solid: ≥ includes points where the two sides are equal.
Shade left
Describe the region x ≤ −1.
Show worked answer
Use the solid vertical line x = −1; shade left including the line.
A strip
Describe −2 < y ≤ 3.
Show worked answer
Shade the horizontal strip above dashed y = −2 and below solid y = 3.
A diagonal
Give two points on the boundary x + y = 6 and the side for x + y > 6.
Show worked answer
(0, 6) and (6, 0) lie on it. Draw it dashed. (0, 7) works, so shade the side away from the origin.
Both conditions
Does (3, 2) satisfy y ≥ 1 and x + y ≤ 6?
Show worked answer
Yes: both conditions hold.
Read a region
A region lies above solid y = 2x and left of dashed x = 4. Write the inequalities.
Show worked answer
Solid includes equality; dashed excludes it.
Count integer pairs
How many integer pairs satisfy 0 ≤ x ≤ 2 and 1 ≤ y ≤ 3?
Show worked answer
Three choices for x and three for y are independent.
Choose a test point
Why is (0, 0) unsuitable for choosing a side of y > 2x?
Show worked answer
It lies on the boundary y = 2x. Instead test (0, 1): 1 > 0, so keep the side above.
Excluded corner
Does (0, 4) belong to x ≥ 0, y ≥ 0, x + y < 4?
Show worked answer
It satisfies the first two but fails the strict third condition:
Therefore it is excluded.
Examiner-style feedback
Common graphical inequalities mistakes
The solution to simultaneous conditions is their overlap.
A dashed boundary excludes equality.
Real-number regions include fractions; item counts require integer coordinates.
30-second recap
Choose, carry out, check
Translate the wording before you reach for a rule.
- Draw the equality boundary.
- Choose solid or dashed.
- Test an off-line point.
- Keep the common region.
Quick answers
Graphical inequalities FAQ
Are graphical inequalities Higher tier?
Two-variable graphical inequality regions are Higher content in the GCSE subject content (A22), shared by AQA, Edexcel and OCR; this guide is labelled Higher.
Should I shade unwanted regions?
Follow the question's instruction. If you shade excluded parts, clearly label the remaining required region.
Content standards
Curriculum and rights review
A22 Higher: two-variable linear inequalities and graphical regions; no A-level optimisation. Questions, suggested marks, solutions and diagrams are original practice material, not official past-paper questions.
Official specification references