Edexcel IAL · Maths Pure 1
Inequalities
Inequalities describe a range of values that satisfy a condition — not just one answer, but every value in a whole region. Master how to solve, represent, and sketch them.
3 Main Topics
Worked Examples
Hidden Answers
Works Offline
Chapter Overview
Linear Inequalities — Solve like equations but answers are ranges. Represent on number lines, in set notation, or interval notation.
Quadratic Inequalities — Always sketch the parabola. Identify regions above or below the x-axis to find the solution range.
Inequalities on Graphs — Use two variables (x and y) and shade regions. Dotted = strict (< or >), solid = weak (≤ or ≥).
Golden Rule — If you multiply or divide by a negative number, flip the inequality sign.
Notation — Set notation: {x : x > 3}. Interval notation: (3, ∞). Number line: filled = included, hollow = excluded.
Quadratic Tip — Always rearrange to get a positive x² term before solving. This keeps the U-shape predictable.
Linear Inequalities
A linear inequality is just like a linear equation (no x², no x³, no √x) — except instead of one exact answer, the solution is a whole range of values. Think of it this way: if the equation 2x = 6 has the single answer x = 3, then the inequality 2x < 6 has the answer "every x smaller than 3".
- > — greater than (e.g. 5 > 3 ✓)
- < — less than (e.g. −8 < 7 ✓)
- ≥ — greater than or equal to (includes the boundary)
- ≤ — less than or equal to (includes the boundary)
Three Ways to Write the Same Inequality
The inequality −4 ≤ x < 9 (x is at least −4 but strictly less than 9) can be written in three different ways. All three mean the same thing — exams use all of them, so you need to recognise each one instantly.
| Method |
Example: −4 ≤ x < 9 |
Key Rules |
| Number Line |
Filled circle at −4, hollow circle at 9, line connecting them |
Filled ● = included (≤ or ≥) | Hollow ○ = excluded (< or >) |
| Set Notation |
{x : x ≥ −4} ∩ {x : x < 9} |
Use ∩ for AND (overlap), ∪ for OR (union) |
| Interval Notation |
[−4, 9) |
Square bracket [ or ] = included | Round ( or ) = excluded. ∞ always uses ( or ) |
🔵 Memory Trick
Square brackets [ ] look like they're "grabbing" the number — that's how they include it. Round brackets ( ) curve away from the number — that's how they exclude it.
How to Solve Linear Inequalities
The process is almost identical to solving a linear equation. Collect like terms, isolate x, and you're done. There is one critical exception:
⚠️ The Sign-Flip Rule
If you multiply or divide both sides by a negative number, you MUST flip the inequality sign: < becomes >, ≥ becomes ≤, etc. The safest approach: rearrange first so the x-coefficient is positive, then divide — that way you never need to flip.
Example: Solve 8 − 3x ≥ 5x − 4
1
Add 3x to both sides (avoids dividing by a negative later): 8 ≥ 8x − 4
2
Add 4 to both sides: 12 ≥ 8x
3
Divide both sides by +8 (positive — no flip needed!): x ≤ 3/2
Question
19 − 2x ≥ 5
9 < x + 5