Linear Inequalities
“An inequality's answer is a range, not a point — divide by a negative and the direction flips.”
The formula
ax + b > 0How to read it: find the range of x for which the linear expression is greater (or less) than 0
- a, b
- — given numbers (a ≠ 0)
- >
- — the inequality sign — flips direction when you multiply/divide by a negative
- x
- — the range of values meeting the condition
The hook
An equation hunts for 'exactly this value'; an inequality hunts for 'the whole range bigger or smaller than it'.
In plain words
A statement that x is greater or less than something. The answer isn't one number but a 'range'.
The intuition
The answer isn't a single point on the number line but a ray shooting off one way. If x>2, everything to the right of 2 is a solution.
How it's built
You transpose and tidy up exactly like an equation — with one exception. Multiply or divide both sides by a 'negative' number and the inequality sign flips.
Example
2x−4>0 → 2x>4 → x>2. On the number line, the entire region to the right of 2 is the answer.
Common mistake
The classic trap: forgetting to flip the sign when dividing by a negative. −2x>4 becomes x<−2 after dividing by −2 (the > turns into <).
Where it's used
Setting the 'interval' of values that satisfy a condition — 'earn at least this much', 'how many can I buy', 'the safe range'.
Where it came from
The signs < and > were first used by the Englishman Harriot in the 1600s, surviving thanks to the intuition that 'the smaller side is the pointy end'.
Prerequisites
Quick check
What is the solution of −2x + 6 < 0?
- x > 3✓
- x < 3
- x > −3
- x < −3
Practice
What is the solution of x + 4 < 9?
Answer: 0
- Transpose 4: x < 9 − 4
- x < 5
Takeaway: The sign stays the same — you're only moving a positive.
What is the solution of −3x ≥ 9?
Answer: 1
- Divide both sides by −3 → the sign flips
- x ≤ −3
Takeaway: Dividing by a negative flips ≥ into ≤.
Solve 5x − 2 < 3x + 6.
Answer: undefined
- Gather x's: 5x − 3x < 6 + 2
- 2x < 8
- x < 4
Takeaway: Dividing by positive 2 keeps the direction — a negative would have flipped it.