Solve linear inequalities like equations, show the solution on a number line and list the integers that satisfy it. It is on Foundation and Higher, aimed at grade 4. Free questions with answers are on this page. Worked examples show each line before you try the questions.
An inequality is an equation with a direction: <,≤,>,≥. Solve it with the same inverses, but flip the inequality if you multiply or divide by a negative. −2x>6 becomes x<−3, not x>−3. Represent the answer on a number line: empty circle for < or >, filled circle for ≤ or ≥.
Integer solutions of 3<2x≤10 need solving both ends, then listing the integers that fit. x>1.5 and x≤5 together give x=2,3,4,5. On a graph, y>2x+1 is the region above the dashed line y=2x+1; dashed for strict inequalities, solid for ≤ or ≥.
Solve like an equation, but if you multiply or divide by a negative, flip the inequality sign. Integer lists must respect < vs ≤. On a number line, an open circle is < or >; a filled circle is ≤ or ≥.
Objectives
< means less than. > means greater than.
In 2x + 1 < 9, subtract 1 and then divide by 2.
If you multiply or divide both sides by a negative number, reverse the sign.
Key points
Read and write the symbols <, >, ≤ and ≥ correctly.
Solve a one-sided inequality by doing the same thing to both sides.
Solve a double inequality such as −2≤2x+4<10 by working on all three parts at once.
Draw the solution on a number line with the right kind of circle and arrow.
Worked examples
Example 1
Solve 2x + 3 < 7.
2 marks
Subtract 3 on both sides: 2x < 4.
Divide both sides by 2: x < 4 ÷ 2.
So x < 2.
Answer: x<2
Common slips
Slip: Subtract 5 on both sides: 2x < 4. That changes a number from the question, so it answers a different question.
Slip: Divide both sides by 5: x < 4 ÷ 2. That changes a number in the working, so the step no longer follows from the one before.
Slip: So x < 6. That does not match the working above it.
Example 2
Solve 3x − 4 > 11.
2 marks
Add 4 on both sides: 3x > 15.
Divide both sides by 3: x > 15 ÷ 3.
So x > 5.
Answer: x>5
Common slips
Slip: Add 6 on both sides: 3x > 15. That changes a number from the question, so it answers a different question.
Slip: Divide both sides by 6: x > 15 ÷ 3. That changes a number in the working, so the step no longer follows from the one before.
Slip: So x > 9. That does not match the working above it.
Example 3
Solve −4x ≤ 12.
3 marks
Divide both sides by −4: 12 ÷ −4 = −3.
Dividing by a negative reverses the sign, so ≤ becomes ≥.
So x ≥ −3.
Answer: x≥-3
Common slips
Slip: Divide both sides by −6: 12 ÷ −4 = −3. That changes a number from the question, so it answers a different question.
Slip: Skip this step: Dividing by a negative reverses the sign, so ≤ becomes ≥. Every step in the working is needed. Skipping one loses the method.
Slip: So x ≥ −7. That does not match the working above it.
Questions and answers
GCSE Maths inequalities questions and answers. Try each one, then open the answer. Easy questions come first.
Question 1 Easy2 marks
Solve x + 3 < 10.
Show answer
Answer: x<7
Subtract the constant from both sides.
The value is x<7.
M1 — Subtract the constant from both sides.
A1 — Answer x<7
Question 2 Easy2 marks
Solve 2x + 1 > 9.
Show answer
Answer: x>4
Subtract 1, then divide by 2.
The value is x>4.
M1 — Subtract 1, then divide by 2.
A1 — Answer x>4
Question 3 Medium2 marks
Solve 5x − 1 < 3x + 7.
Show answer
Answer: x<4
Collect the x terms on the left.
Then 2x < 8.
M1 — Collect the x terms on the left.
A1 — Answer x<4
Question 4 Medium2 marks
Solve 10 − x > 4.
Show answer
Answer: x<6
Subtract 10 from both sides.
Dividing by −1 reverses the inequality.
M1 — Subtract 10 from both sides.
A1 — Answer x<6
Question 5 Hard2 marks
Solve −2 < x − 1 ≤ 4. Give the solution as two inequalities joined by "and", with x in both.
Show answer
Answer: -1<x and x≤5
Add 1 to every part.
Keep both inequality signs.
M1 — Add 1 to every part.
A1 — Answer -1<x and x≤5
Question 6 Hard2 marks
A number x is multiplied by 3, then 4 is added, and the result is at least 13. Solve for x.
Show answer
Answer: x≥3
3x + 4 is at least the total.
Subtract 4 and divide by 3.
M1 — 3x + 4 is at least the total.
A1 — Answer x≥3
Question 7 Exam-style3 marks
Solve −2 < x − 2 ≤ 4. Give the solution as two inequalities joined by "and", with x in both.
Show answer
Answer: 0<x and x≤6
Add 2 to every part.
Keep both inequality signs.
M1 — Add 2 to every part.
M1 — Keep both inequality signs.
A1 — Answer 0<x and x≤6
Common mistakes
Most lost marks on this topic are one slip.
Reversing the sign when dividing by a positive number.
The sign reverses only for a negative multiplier.
Filling the circle for a strict inequality.
A strict inequality has an open circle.
Including the end value in a list when the sign is strict.
x < 4 does not include 4.
Spot the slip
The open circle on a number line means
the number is not included
the number is included
the inequality has no solution
Show the slip
the number is not included An open circle matches < or >.
Exam tips
Be ready to do this without a calculator. Paper 1 is the non-calculator paper.
If the question says “show that”, the result is given. The marks are for the steps that reach it.
The method is the same on AQA-style and Edexcel-style papers.
Copy the inequality symbol on every line of working; a line with = in it can lose the accuracy mark.
For 'list the integers', check each end: filled circle means include it, open circle means leave it out.
Printable worksheet
The same questions are in a PDF you can print. Answers start on a later page.