To represent an inequality on a graph, draw the boundary line (solid for ≤ or ≥, dashed for < or >) and shade the region that satisfies the inequality. When several inequalities are combined, the solution region is where all conditions are met simultaneously — usually shown as the unshaded area in UK GCSE exams.
What is the difference between a solid and a dashed boundary line?
The boundary line is the graph of the equation obtained by replacing the inequality symbol with an equals sign.
- Solid line (─): used for
≤or≥. The boundary itself is included in the solution. Points on the line satisfy the inequality. - Dashed line (- - -): used for
<or>. The boundary is excluded from the solution. Points exactly on the line do not satisfy the inequality.
Memory aid: ≤ and ≥ are "or equal to" — the line is part of the solution, so draw it solid. < and > exclude equality — draw it dashed.
How do you decide which side of the line to shade?
After drawing the boundary line, test a point that is clearly not on the line:
- Choose a convenient test point — usually (0, 0) unless the line passes through the origin.
- Substitute the test point's coordinates into the inequality (not the equation).
- If the inequality is satisfied (true), shade the side containing the test point.
- If the inequality is not satisfied (false), shade the other side.
Example: Shade the region satisfying y > 2x + 1.
- Draw the line
y = 2x + 1(dashed, because strictly greater than). - Test (0, 0): Is
0 > 2(0) + 1? Is0 > 1? No — (0, 0) does not satisfy the inequality. - Shade the other side from (0, 0) — the region above the line.
Step-by-step worked example: single inequality
Show the region satisfying 3x + 2y ≤ 12.
- Draw the boundary: rearrange to
y = (12 − 3x)/2 = 6 − 1.5x. Plot the line through (0, 6) and (4, 0). Draw it solid (≤ includes the boundary). - Test (0, 0):
3(0) + 2(0) = 0 ≤ 12. True — (0, 0) satisfies the inequality. - Shade the region containing (0, 0) — below and to the left of the line.
How do you find the solution region for multiple inequalities?
For two or more simultaneous inequalities, the solution region satisfies all of them at once. Each inequality defines a half-plane; the intersection of all these half-planes is the solution region.
Worked example: Find and indicate the region satisfying all three inequalities:
y ≥ 0x ≥ 1x + y ≤ 5
Step 1: Draw each boundary:
y = 0: the x-axis (solid line, shading above for y ≥ 0)x = 1: vertical line at x = 1 (solid line, shading to the right for x ≥ 1)x + y = 5: line through (5, 0) and (0, 5) (solid line, shading below for x + y ≤ 5)
Step 2: Identify the region that satisfies all three:
| Inequality | Region to shade |
|---|---|
| y ≥ 0 | Above or on the x-axis |
| x ≥ 1 | To the right of or on x = 1 |
| x + y ≤ 5 | Below or on the line x + y = 5 |
Step 3: The solution region is the triangle with vertices at (1, 0), (5, 0), and (1, 4). Points inside and on the edges of this triangle satisfy all three inequalities.
In UK GCSE exams, the convention is to shade the region you do not want (leaving the solution region unshaded and labelling it R).
How do you find integer coordinate points in the solution region?
Once the solution region is identified, list the coordinate pairs (x, y) where x and y are integers and the point lies inside (or on the boundary of, if lines are solid) the region.
For the example above, integer points include: (1, 0), (1, 1), (1, 2), (1, 3), (1, 4), (2, 0), (2, 1), (2, 2), (2, 3), (3, 0), (3, 1), (3, 2), (4, 0), (4, 1), (5, 0).
What are the most common mistakes?
- Using a solid line for a strict inequality. A dashed line is required for
<and>. - Shading the wrong region. Always test a specific point — do not guess based on the visual impression of "above" or "below".
- Forgetting to satisfy all inequalities. In multi-inequality questions, the solution region must pass every test, not just the most recently drawn boundary.
- Not labelling the region. Most mark schemes require you to label the solution region clearly (e.g. "R") or show which region is intended.
Frequently asked questions
What does "shade the region not required" mean?
In UK GCSE exams, the standard instruction is to leave the solution region unshaded and shade everything else. This prevents overlapping shading from making the diagram unreadable. Always label the unshaded solution region with the letter R (or as instructed).
How do you handle a vertical boundary line (e.g. x < 3)?
Draw a dashed vertical line at x = 3. Test a point: (0, 0) gives x = 0 < 3 — true. The solution region is to the left of x = 3 (the side containing the test point). Shade right to indicate the region that does not satisfy x < 3.
Can the solution region be infinite?
Yes — two intersecting inequalities such as y > x and y < 2x + 3 define an infinite wedge-shaped region that extends indefinitely. Only when three or more inequalities enclose a finite area does the solution region become bounded (a polygon).
Is this topic on GCSE foundation or higher tier?
Graphical inequalities (representing a single inequality on a graph) may appear on both tiers. Finding a solution region for two or more simultaneous inequalities is typically a higher-tier topic. Check your specification — on AQA GCSE Mathematics, graphical inequalities appear on higher tier (grade 5+).
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