Inequalities appear on every major GCSE Maths paper and span a range of difficulty levels. At foundation, you're solving linear inequalities and reading number lines. At higher, you're identifying integer solutions and shading graphical regions defined by multiple inequalities. AQA (8300), Edexcel (1MA1), and OCR (J560) all test this topic — it typically appears on both the non-calculator and calculator papers, worth 2–5 marks per question. Get the rules straight and you can pick up marks here quickly.
What You Need to Know First
Make sure you're confident with the following before working through the methods below:
- Solving linear equations — rearranging equations by adding, subtracting, multiplying, dividing
- The four inequality symbols — < (less than), > (greater than), ≤ (less than or equal to), ≥ (greater than or equal to)
- Plotting straight-line graphs — needed for graphical inequalities; use y = mx + c confidently
- Negative number arithmetic — particularly multiplying and dividing negatives
- Factorising quadratics — required for quadratic inequalities at higher tier
Solving Linear Inequalities
Solving an inequality works almost identically to solving an equation. The key difference: if you multiply or divide both sides by a negative number, you must flip the inequality sign.
Worked Example 1 — Basic Linear Inequality
Solve 3x − 5 > 7.
Step 1: Add 5 to both sides.
3x > 12
Step 2: Divide both sides by 3.
x > 4
That's it. The solution is x > 4, meaning any value of x greater than 4 satisfies the inequality.
Worked Example 2 — Dividing by a Negative
Solve 2 − 4x ≥ 14.
Step 1: Subtract 2 from both sides.
−4x ≥ 12
Step 2: Divide both sides by −4. Flip the sign.
x ≤ −3
This is the most commonly tested "trap" in inequality questions. Students who forget to flip the sign get x ≥ −3, which is the wrong direction entirely.
Representing on a Number Line
Once you've solved an inequality, you may need to show it on a number line:
- Open circle (○) — use for < or > (the value is not included)
- Closed/filled circle (●) — use for ≤ or ≥ (the value is included)
- Draw an arrow pointing in the correct direction from the circle
For x > 4: open circle at 4, arrow pointing right. For x ≤ −3: closed circle at −3, arrow pointing left.
Finding Integer Solutions
If the question asks for integer solutions, list every whole number that satisfies the inequality. Always work out both boundaries first.
Example: List the integers satisfying −2 < x ≤ 5.
The integers are: −1, 0, 1, 2, 3, 4, 5.
Note: −2 is excluded (strict inequality), but 5 is included (≤).
Graphical Inequalities and Regions
For higher tier, you'll need to identify and shade the region on a graph that satisfies one or more inequalities. Each inequality represents a half-plane — one side of a straight line.
The Method
- Treat the inequality as an equation and draw the boundary line.
- Use a solid line for ≤ or ≥ (the boundary is included).
- Use a dashed line for < or > (the boundary is not included).
- Test a point — usually (0, 0) — to see which side of the line satisfies the inequality.
- Shade the region that satisfies the inequality (or leave the satisfying region unshaded, if the question asks you to shade the region that does NOT satisfy it — read the question carefully).
Worked Example 3 — Single Graphical Inequality
Show the region satisfying y ≥ 2x − 1 on a graph.
Step 1: Draw the line y = 2x − 1.
When x = 0: y = −1. Plot (0, −1). When x = 3: y = 5. Plot (3, 5). Draw a solid line through these points (≥ means the boundary is included).
Step 2: Test the point (0, 0).
y ≥ 2x − 1 → 0 ≥ 2(0) − 1 → 0 ≥ −1 ✓
True — so (0, 0) is in the required region.
Step 3: Shade the region above the line (since (0, 0) satisfied the inequality and it lies above the line).
Worked Example 4 — Three Simultaneous Inequalities
Find the region satisfying all three: x ≥ 0, y ≥ 1, and x + y ≤ 5.
Line 1: x = 0 — the y-axis. Solid line. Shade to the right (x ≥ 0).
Line 2: y = 1 — a horizontal line. Solid line. Test (0, 0): 0 ≥ 1 is false, so shade above this line.
Line 3: x + y = 5. Plot (0, 5) and (5, 0). Solid line. Test (0, 0): 0 + 0 ≤ 5 ✓. Shade below and left of this line.
The required region is where all three shadings overlap — a triangle bounded by the y-axis on the left, y = 1 at the bottom, and x + y = 5 on the upper right.
On AQA and Edexcel papers, the required region is often labelled R. You may be asked to label it yourself or to identify which label corresponds to the region that satisfies all given inequalities.
Common Mistakes Students Make
Forgetting to flip the sign when dividing by a negative. This is the single most frequent error. Write a reminder in your working: "÷ by negative → flip sign." Check every step where you divide or multiply.
Using the wrong type of circle on a number line. Open circles for strict inequalities, closed for "or equal to." If you draw a closed circle for x > 4, you've included 4 — which is wrong.
Drawing a solid line for a strict inequality on a graph. If the inequality is < or >, the boundary line must be dashed. A solid line implies the boundary is included, which loses the mark.
Testing the wrong point when shading. Most students test (0, 0), which works unless the boundary line passes through the origin. If it does, test a different point, like (0, 1) or (1, 0).
Only satisfying some of the inequalities for the region. When there are multiple inequalities, the required region must satisfy all of them simultaneously. Shade each one separately on the same diagram and find the overlap.
Exam Tips for Inequalities
When a question asks for integer solutions, always find both boundaries before listing. Solve both ends of a double inequality (e.g. −1 < 2x + 3 ≤ 9) separately, then list the integers between them.
For graphical regions, a table of values is your safest method for drawing boundary lines accurately. Rushing and estimating costs marks — plot at least two clear points per line.
AQA awards the method mark (M1) for drawing the correct boundary line, even if you shade the wrong side. So always draw the line — you keep the method mark and only lose the accuracy mark for incorrect shading.
If the region has vertices, the exam may ask you to write down the coordinates of a vertex. Read intersecting lines carefully or find them algebraically if needed.
On the calculator paper, use your calculator to check solutions to inequalities involving fractions or decimals. Substituting a boundary value back into the original inequality takes ten seconds and confirms your answer.
Practise Now
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