Inequalities are a fundamental part of the GCSE Maths curriculum across all UK exam boards (AQA, Edexcel, OCR). They describe a range of values that satisfy a condition, rather than a single answer. Mastering them is crucial for topics like solving equations, coordinate geometry, and real-world problem-solving. This guide will walk you through solving linear inequalities and representing the solutions clearly on number lines and in regions.
What is an Inequality?
An inequality shows the relationship between two expressions that are not equal. Instead of an equals sign (=), we use symbols:
< less than
> greater than
≤ less than or equal to
- ≥ greater than or equal to
For example, (x > 3) means "x is greater than 3". This could be 3.1, 5, or 100, but not 3 itself.
Step 1: Solving Linear Inequalities
Solving an inequality is very similar to solving an equation. The key rule is:
> You can add, subtract, multiply, or ÷ both sides of an inequality by the same number, just like an equation.
However, there is one critical exception:
> If you multiply or ÷ both sides by a negative number, you must reverse (flip) the inequality sign.
Let's work through an example.
Example 1: Solve (5x - 2 ≤ 18).
- Treat it like an equation. Add 2 to both sides to isolate the term with (x).
5x - 2 + 2 ≤ 18 + 2
5x ≤ 20
- Divide both sides by 5. Since 5 is positive, the sign stays the same.
5x5 ≤ 205
x ≤ 4
So the solution is all numbers less than or equal to 4.
Example 2: Solve ( -3x > 12 ).
Here, we need to ÷ by -3.
- Divide both sides by -3.
-3x-3 < 12-3
Crucially, because we divided by a negative, we flip the > to a <.
- Simplify.
x < -4
flowchart TD
A["Start: Solve Inequality"] --> B["Multiply or divide
by a negative number?"]
B -- No --> C["Proceed as normal.
Inequality sign stays the same."]
B -- Yes --> D["Flip the inequality sign!, e.g. < becomes >, ≤ becomes ≥"]
C --> E["Solution Found"]
D --> E
Step 2: Representing Solutions on a Number Line
Once you have your solution (like (x ≤ 4)), you need to show it visually. This is a common exam question.
Key conventions for number lines:
Closed circle (●) for ≤ (less than or equal to) or ≥ (greater than or equal to). This shows the number is included.
Open circle (○) for < (less than) or > (greater than). This shows the number is not included.
- Arrow pointing in the direction of all numbers that satisfy the inequality.
Let's show (x ≤ 4) and (x < -4) from our examples.
flowchart LR
subgraph “Inequality: x ≤ 4”
direction LR
A1["“ ”"] --- B1["“ ”"] --- C1["“ ”"] --- D1["“ ”"] --- E1["“● 4”"] --- F1["“ ”"]
style E1 fill:#f9f,stroke:#333
linkStyle 0,1,2,3 stroke:red,stroke-width:4px
linkStyle 4 stroke:none
end
subgraph “Inequality: x < -4”
direction LR
A2["“ ”"] --- B2["“○ -4”"] --- C2["“ ”"] --- D2["“ ”"] --- E2["“ ”"] --- F2["“ ”"]
style B2 fill:#f9f,stroke:#333
linkStyle 5 stroke:none
linkStyle 6,7,8,9 stroke:red,stroke-width:4px
end
For (x ≤ 4), we place a closed circle on 4 and draw an arrow pointing left, towards the smaller numbers.
- For (x < -4), we place an open circle on -4 and draw an arrow pointing left.
Step 3: Shading Regions on Graphs
For inequalities with two variables (like (y > 2x + 1)), the solution is a set of coordinate points that form a region on a graph.
How to shade the region for (y > 2x + 1):
Treat the inequality as an equation ((y = 2x + 1)) and plot the line. This line is the boundary of the region.
Choose the line style:
Use a solid line for ≤ or ≥ (points on the line are included).
- Use a dashed line for < or > (points on the line are not included).
Since we have (>), we draw a dashed line for (y = 2x + 1).
Choose which side to shade. Pick a test point that is not on the line. The origin ((0,0)) is often easiest.
- Substitute ((0,0)) into the inequality: (0 > 2(0) +
- 1) → (0 > 1). This is false.
- Since ((0,0)) makes the inequality false, we shade the region on the opposite side of the line from ((0,0)).
Common Pitfalls and Exam Tips
Flipping the Sign: Always double-check if you multiplied or divided by a negative. This is the number one mistake!
Number Line Circles: Mixing up open and closed circles loses marks. Remember: Closed for "or equal to".
Reading Inequality Word Problems: Phrases like "at least" mean (≥), "at most" mean (≤), "more than" means (>), "less than" means (<).
- Double Inequalities: For something like (-3 < x ≤ 2), solve each part separately. On a number line, this is a line segment between -3 (open circle) and 2 (closed circle).
Practice Question (GCSE Style)
> 1. Solve (7 - 2x ≥ 1).
> 2. Show your solution to part (1) on the number line below.
Walkthrough:
Subtract 7 from both sides: (-2x ≥ -6).
Divide both sides by -2. Remember to flip the sign! (x ≤ 3).
- On the number line, draw a closed circle at 3 and an arrow pointing left.
Summary
Tackling inequalities successfully requires careful arithmetic and attention to visual conventions. Remember the golden rule about flipping the sign with negatives, and practice drawing clear number lines and shaded regions. This topic is a regular feature on GCSE papers, so mastering it will definitely help you secure those valuable marks.
- Next Steps: Try past paper questions on inequalities from your exam board's website. Look for questions that combine inequalities with other topics, like solving quadratic inequalities or finding integer solutions that satisfy two inequalities at once.