An inequality compares two quantities using <, >, ≤ or ≥. Solve a linear inequality like an equation, using inverse operations on both sides, but if you multiply or divide by a negative number the inequality sign reverses. On a number line, an open circle shows that a value is not included and a filled circle shows that it is.
Inequality notation
- x > 3: x is greater than 3
- x ≤ 2: x is less than or equal to 2
- Number line: open circle for < or >, filled circle for ≤ or ≥
- Multiplying or dividing by a negative reverses the sign
Inequality symbols and number lines
x > 3 means x is greater than 3, and x ≤ 2 means x is less than or equal to 2. On a number line, an open circle shows the end value is not included, and a filled circle shows it is included. An arrow or line shows all the values that satisfy the inequality.
Solving inequalities
Solve an inequality in the same way as an equation, using inverse operations on both sides. For 3x − 5 > 13, add 5 to get 3x > 18, then divide by 3 to get x > 6. The answer is a range of values, not a single number.
Multiplying or dividing by a negative
If you multiply or divide both sides by a negative number, reverse the inequality sign. For 5 − 2x ≥ 11, subtract 5 to get −2x ≥ 6, then divide by −2 and reverse the sign: x ≤ −3. You can avoid this by collecting the x terms on the side where the coefficient is positive.
Integer solutions
Often a question asks for the integers that satisfy an inequality. For −3 < x ≤ 2, the integers are −2, −1, 0, 1 and 2: −3 is not included because the sign is <, but 2 is included because of ≤. Watch the ends carefully.
Double inequalities
An inequality such as 1 < 2x + 3 ≤ 11 has three parts. Do the same to all three parts: subtract 3 to get −2 < 2x ≤ 8, then divide by 2 to get −1 < x ≤ 4.
Worked examples
Example 1. n is an integer and n > 7. Write down the smallest possible value of n.
- n must be greater than 7, so 7 itself is not allowed.
- The next integer is 8.
Answer: 8.
Example 2. Solve 3x − 5 > 13.
- Add 5: 3x > 18.
- Divide by 3.
Answer: x > 6.
Example 3. List the integers that satisfy −3 < x ≤ 2.
- −3 is not included (the sign is <).
- 2 is included (the sign is ≤).
Answer: −2, −1, 0, 1, 2.
Example 4. Solve 7x − 3 < 4x + 9 and write down the largest integer that satisfies it.
- Subtract 4x: 3x − 3 < 9.
- Add 3: 3x < 12, so x < 4.
Answer: x < 4; the largest integer is 3.
Example 5. Solve 5 − 2x ≥ 11.
- Subtract 5: −2x ≥ 6.
- Divide by −2 and reverse the sign.
Answer: x ≤ −3.
Example 6. Solve 1 < 2x + 3 ≤ 11.
- Subtract 3 from all three parts: −2 < 2x ≤ 8.
- Divide all three parts by 2.
Answer: −1 < x ≤ 4.
Common mistakes
- Not reversing the sign for a negative. Dividing by −2 turns ≥ into ≤.
- Including an end value that is excluded. For x > 7 the smallest integer is 8, not 7.
- Using the wrong circle on a number line. Open for < or >, filled for ≤ or ≥.
- Writing an equation instead of an inequality. Keep the inequality sign throughout your working.
- Only operating on two parts of a double inequality. Apply each step to all three parts.
How inequalities are examined
Linear inequalities are Foundation content in the 1MA1 and 4MA1 specifications, and Higher papers add quadratic inequalities and regions on graphs. Questions ask you to:
- write down the inequality shown on a number line, or draw one
- solve a linear inequality, sometimes with the unknown on both sides
- list the integer values that satisfy an inequality
- solve a double inequality
Practise with the inequalities worksheet, which includes answers, and the GCSE algebra revision engine.
Solving inequalities uses the same steps as linear equations. At Higher tier, see quadratic inequalities.
Frequently asked questions
What are the steps to solve a linear inequality?
Like an equation: use inverse operations on both sides. Reverse the sign if you multiply or divide by a negative number.
Why does dividing by a negative reverse the inequality?
Because it reverses the order of numbers: 2 < 5, but −2 > −5.
What does a filled circle on a number line mean?
That the end value is included, so the inequality uses ≤ or ≥.