Inequalities
An inequality says one side is bigger or smaller than the other, so the answer is a range of values rather than one number. You solve it almost exactly like an equation, with one extra rule.
Part 1 of 3: Learn it
In short
- < and > are strict (open circle); ≤ and ≥ include the end value (filled circle).
- Solve like an equation, but if you multiply or divide by a negative number, flip the sign.
- "Integers" means whole numbers, including zero and negatives.
Where this is in your specification
Spec points: DfE A22 (quadratic inequalities and regions are Higher tier)
| Board | Topic: Linear equations and inequalities |
|---|---|
| DfE content | A17, A21, A22 |
| AQA 8300 | Algebra |
| Edexcel 1MA1 | Algebra |
| OCR J560 | 6 Algebra |
| Eduqas C300 | HA17, HA21, HA22 |
| Cambridge IGCSE 0580 | E2.5, E2.6 |
| National 5 C847 75 | Algebraic skills: linear equations and inequations |
Symbols and number lines
| Symbol | Means | On a number line |
|---|---|---|
| x > 3 | greater than 3 | open circle at 3, arrow right |
| x ≥ 3 | 3 or more | filled circle at 3, arrow right |
| x < −1 | less than −1 | open circle at −1, arrow left |
| −2 < x ≤ 4 | between −2 and 4, including 4 | open at −2, filled at 4, line between |
Solving
Add, subtract, multiply and divide both sides as you would in an equation. The one change: multiplying or dividing both sides by a negative number reverses the inequality. For example, from −3x < 12, dividing by −3 gives x > −4.
A double inequality such as 5 < 2x + 1 ≤ 13 has three parts. Do the same to all three: 4 < 2x ≤ 12, so 2 < x ≤ 6.
Listing integers
Read the ends carefully. The integers that satisfy 2 < x ≤ 6 are 3, 4, 5 and 6. For −3 ≤ n < 1 they are −3, −2, −1 and 0.
Regions on a graph (Higher)
Draw each boundary line: solid for ≤ or ≥, dashed for < or >. Test a point such as (0, 0) to see which side satisfies the inequality, then shade or label the region the question asks for.
Solve 5 − 2x ≥ 13.
Show the answer
−2x ≥ 8, so x ≤ −4 (the sign flips).
Part 2 of 3: See it worked
Worked examples
Example 1
Solve 3x − 7 > 11 and show the answer on a number line.
- Add 7: 3x > 18
- Divide by 3: x > 6
- Open circle at 6, arrow to the right
Answer: x > 6.
Example 2
Solve −4 ≤ 3n + 2 < 14 and list the integer values of n.
- Subtract 2 from all three parts: −6 ≤ 3n < 12
- Divide by 3: −2 ≤ n < 4
- Integers: −2, −1, 0, 1, 2, 3
Answer: n = −2, −1, 0, 1, 2, 3.
Common mistakes
- Not flipping the sign after dividing by a negative number.
- Writing x = 6 as the answer to an inequality. The answer is a range, so keep the sign.
- Including an end value that the strict sign leaves out when listing integers.
- Forgetting zero and negative numbers in a list of integers.
List the integers n with −1 < n ≤ 3.
Show the answer
0, 1, 2, 3.
Part 3 of 3: Test yourself
Check yourself
Answer each one in your head or on paper first, then open it to check.
Solve 5 − 2x ≥ 13.
−2x ≥ 8, so x ≤ −4 (the sign flips).
List the integers n with −1 < n ≤ 3.
0, 1, 2, 3.
Solve 7 < 2x + 3 ≤ 15.
4 < 2x ≤ 12, so 2 < x ≤ 6.
Jobs that use this
Each link opens the job profile on the National Careers Service (England). In the rest of the UK: My World of Work (Scotland), Careers Wales, nidirect careers (Northern Ireland).
Full lessons and marked practice for this course are coming soon to Brainlag Learn. See courses