To solve a linear inequality, treat it like an equation: isolate the unknown by performing the same operation on both sides. The only difference is one extra rule — whenever you multiply or divide both sides by a negative number, the inequality sign must flip direction. Ignore this and the answer will be wrong every time.
What is the difference between an inequality and an equation?
An equation states that two expressions are equal (=) and typically has one solution. An inequality states that one expression is greater than (>), less than (<), greater than or equal to (≥), or less than or equal to (≤) another — and the solution is usually a range of values, not a single number.
| Symbol | Meaning | Example |
|---|---|---|
| > | strictly greater than | x > 3 means x can be 4, 5, 3.1, … but NOT 3 |
| < | strictly less than | x < −1 means x can be −2, −5, −1.7, … but NOT −1 |
| ≥ | greater than or equal to | x ≥ 0 includes 0 itself |
| ≤ | less than or equal to | x ≤ 10 includes 10 itself |
How do you solve a two-step linear inequality?
Use the same balance steps as for a linear equation.
Worked example: solve 3x − 4 > 11.
- Add 4 to both sides: 3x > 15.
- Divide both sides by 3: x > 5.
Worked example: solve 2 − 5x ≤ 17.
- Subtract 2 from both sides: −5x ≤ 15.
- Divide both sides by −5 — FLIP THE SIGN because you are dividing by a negative: x ≥ −3.
The sign flips because dividing by a negative reverses the ordering on the number line.
When exactly must the sign flip?
The sign must flip when (and only when) you multiply or divide both sides by a negative number. Adding or subtracting — even a negative — does not cause a flip.
| Operation | Sign flips? | Example |
|---|---|---|
| Add or subtract any number | No | x + 5 > 3 → x > −2 |
| Multiply or divide by positive | No | 4x < 20 → x < 5 |
| Multiply or divide by negative | Yes | −2x > 8 → x < −4 |
A common way to avoid the flip: rearrange so the x-term is positive before you divide. In −5x ≤ 15, add 5x to both sides and subtract 15: 0 ≤ 15 + 5x → 5x ≥ −15 → x ≥ −3. Same answer, no flip needed.
How do you show the solution on a number line?
Draw a horizontal number line. Use an open circle (○) for strict inequalities (> or <) to show that the endpoint is not included, and a closed circle (●) for ≥ or ≤ to show that the endpoint is included. Shade the arrow in the direction of the solution.
- x > 5: open circle at 5, arrow pointing right.
- x ≥ −3: closed circle at −3, arrow pointing right.
- x < 2: open circle at 2, arrow pointing left.
How do you list integer solutions from an inequality?
Some questions ask "list the integer values of x that satisfy the inequality".
Example: find the integers satisfying −2 ≤ x < 4.
The integers from −2 (included) up to 4 (excluded) are: −2, −1, 0, 1, 2, 3.
Write every whole number in the range. Count them carefully — students frequently miss −2 (because ≥ includes it) or include 4 (because < excludes it).
How do you write the solution in set notation?
GCSE Higher questions sometimes expect the answer in set notation. The standard forms are:
- {x : x > 5} — "the set of all x such that x is greater than 5"
- {x ∈ ℤ : −2 ≤ x < 4} — "the set of all integers x such that x is at least −2 and less than 4"
Use ℤ (integers) when the question limits the domain to whole numbers.
Frequently asked questions
Why does multiplying by a negative flip the inequality?
Think about a simple true statement: 2 < 6. Multiply both sides by −1: −2 > −6. The inequality has flipped because the number line reverses direction under a negative multiplier — what was to the left is now to the right. The rule is not arbitrary; it follows directly from how the number line works.
Can a linear inequality have no solution?
Yes. For example, solving 2x + 1 > 2x + 5 gives 1 > 5 after subtracting 2x from both sides — a false statement. The inequality has no solution because no value of x makes it true.
Can a linear inequality be satisfied by all values of x?
Yes. Solving 3x − 2 < 3x + 1 gives −2 < 1 after subtracting 3x — a statement that is always true regardless of x. The solution is all real numbers.
How do I solve an inequality with the unknown on both sides?
Collect all x-terms on one side, just as with equations. For example, 5x − 3 > 2x + 9: subtract 2x to get 3x − 3 > 9; add 3 to get 3x > 12; divide by 3 to get x > 4. Check: x = 5 gives 25 − 3 = 22 and 10 + 9 = 19; 22 > 19 ✓.
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