Solve: \(x + 4 < 10\)
Subtract 4 from both sides.
Subtract 4 from both sides: \(x < 10 - 4\), so \(x < 6\).
Solving Inequalities
Inequalities show a range of possible values rather than a single solution. In GCSE Maths, you need to solve them accurately and represent solutions clearly on number lines. They can also be shown graphically using inequalities on graphs.
Overview
Key Definitions
Inequality
A statement showing one value is greater or smaller than another.
Variable
A letter representing an unknown value.
Solve
Find the range of values that makes the inequality true.
Range
A group of values that all satisfy the inequality.
Number Line
A visual way to show which values are included.
Reverse the Sign
When multiplying or dividing by a negative number, the inequality sign changes direction.
Key Rules
Solve like an equation
Add, subtract, multiply or divide both sides as needed.
Show the answer as a range
For example, \(x > 4\) means many possible values.
Use open or closed circles correctly
\(>\) and \(<\) are open, \(\geq\) and \(\leq\) are closed.
Reverse the sign for negatives
If you multiply or divide by a negative, flip the inequality.
Quick Pattern Check
One-step
\(x + 2 < 9\)
Two-step
\(3x - 1 \geq 8\)
Negative coefficient
\(-2x > 6\)
Compound inequality
\(2 < x \leq 7\)
How to Solve
What is an inequality?
An inequality shows a range of values instead of one exact answer. You solve it in a similar way to an equation.
\( x + 4 > 10 \)
Subtract 4 from both sides: \( x > 6 \).
This means all values greater than 6 are solutions.
Step-by-step method
- Solve like a normal equation.
- Keep the inequality sign the same.
- Reverse the sign if multiplying or dividing by a negative.
- Write the solution clearly.
One-step inequalities
\( x - 3 \leq 5 \)
Add 3 to both sides.
Answer: \( x \leq 8 \).
Two-step inequalities
\( 2x + 1 < 9 \)
Subtract 1: \(2x < 8\).
Divide by 2: \(x < 4\).
Important rule: reversing the sign
If you multiply or divide by a negative number, the inequality sign must reverse.
\( -3x > 12 \)
Divide by \(-3\).
Reverse the sign.
Answer: \( x < -4 \).
Exam tip: This is the most common mistake.
Showing solutions on a number line
Solutions are often shown graphically using a number line.
Open circle
\(x > 3\) or \(x < 3\) (not included)
Closed circle
\(x \geq 3\) or \(x \leq 3\) (included)
Compound inequalities
\( 2 < x \leq 7 \)
x is greater than 2 but less than or equal to 7.
Use an open circle at 2 and a closed circle at 7.
Exam thinking
Always check if you divided or multiplied by a negative.
Write the inequality sign correctly.
Exam tip: Show each step clearly for full marks.
Example Questions
Edexcel
Exam-style questions inspired by Edexcel GCSE Mathematics.
Edexcel
Solve \( x + 5 < 12 \).
Edexcel
Solve \( 3x \geq 15 \).
Edexcel
Solve \( 2x - 7 > 9 \).
Edexcel
Solve \( -4x \leq 8 \).
Edexcel
Represent the solution to \( x > -2 \) on a number line.
AQA
Exam-style questions based on the AQA GCSE Mathematics specification, focusing on reasoning and multi-step inequalities.
AQA
Solve \( 5x - 3 \leq 2x + 12 \).
AQA
Solve \( 3(2x - 1) > 4x + 5 \).
AQA
Solve \( \frac{x - 4}{3} \geq 2 \).
AQA
Find all integer values of \( x \) that satisfy \( -2 < x \leq 4 \).
AQA
A student solves \( -2x > 6 \) and writes \( x > -3 \). Explain the mistake and give the correct solution.
OCR
Exam-style questions aligned with OCR GCSE Mathematics, emphasising reasoning, graphical interpretation, and real-life applications.
OCR
Solve \( -3 \leq 2x + 1 < 7 \).
OCR
Solve \( 0.5x - 1 < 4 \).
OCR
The cost of entry to a theme park is £12 plus £3 per ride. Write and solve an inequality to find the maximum number of rides a person can take if they have £30.
OCR
Find the range of integer values that satisfy \( 4 < 2x \leq 12 \).
OCR
The perimeter of a rectangle is less than 50 cm. If the length is \( x \) cm and the width is 8 cm, form and solve an inequality to find the possible values of \( x \).
Exam Checklist
Step 1
Solve the inequality like an equation.
Step 2
Watch carefully for any multiplication or division by a negative.
Step 3
Write the answer as a range, not a single number.
Step 4
Use the correct circle and shading on the number line if needed.
Most common exam mistakes
Forgotten sign flip
Not reversing the inequality after dividing by a negative.
Wrong symbol direction
Mixing up less than and greater than.
Wrong number line endpoint
Using open instead of closed, or closed instead of open.
Single-answer thinking
Forgetting that inequalities usually describe many possible values.
Common Mistakes
These are common mistakes students make when solving inequalities in GCSE Maths.
Forgetting to reverse the inequality sign
Incorrect
A student keeps the same sign after multiplying or dividing by a negative number.
Correct
When you multiply or divide both sides by a negative, you must reverse the inequality sign.
Using a closed circle incorrectly
Incorrect
A student includes a value that should not be part of the solution.
Correct
Use a closed (filled) circle only when the value is included, such as \(\leq\) or \(\geq\).
Using an open circle incorrectly
Incorrect
A student excludes a value that should be included.
Correct
Use an open circle when the value is not included, such as \(<\) or \(>\).
Treating the solution as a single value
Incorrect
A student gives only one answer.
Correct
Inequalities represent a range of values, not a single solution. Write the full interval or show it on a number line.
Mixing up the direction of the sign
Incorrect
A student writes the inequality symbol in the wrong direction.
Correct
Check your final answer carefully to ensure the inequality sign correctly represents the solution.
Try It Yourself
Practise solving linear inequalities and representing them on number lines.
Foundation
Foundation Practice
Solve simple inequalities and understand inequality signs.
Solve: \(x - 3 > 8\)
Add 3 to both sides.
Add 3 to both sides: \(x > 11\).
Solve: \(2x < 14\)
Divide both sides by 2.
Divide both sides by 2: \(x < 7\).
Solve: \(5x \geq 20\)
Divide both sides by 5.
Divide both sides by 5: \(x \geq 4\).
Solve: \(x + 7 \leq 15\)
Undo the +7.
Subtract 7 from both sides: \(x \leq 8\).
Solve: \(3x + 2 < 17\)
Subtract 2 first, then divide by 3.
Subtract 2: \(3x < 15\). Divide by 3: \(x < 5\).
Which inequality means x is greater than 3?
The open side of the sign faces the bigger value.
\(x > 3\) means x is greater than 3.
Which inequality includes the value 5?
\(\geq\) means greater than or equal to.
\(x \geq 5\) includes 5 because the sign means greater than or equal to.
A student solves \(x + 6 < 10\) and gets \(x < 16\). What mistake did they make?
To undo +6, use the opposite operation.
They should subtract 6 from both sides: \(x < 4\), not \(x < 16\).
Solve: \(4x - 1 \leq 11\)
Add 1 first, then divide by 4.
Add 1: \(4x \leq 12\). Divide by 4: \(x \leq 3\).
Higher
Higher Practice
Solve inequalities with brackets, negatives, fractions and unknowns on both sides.
Solve: \(3x + 5 > x + 13\)
Collect the x terms on one side.
Subtract \(x\): \(2x + 5 > 13\). Subtract 5: \(2x > 8\). Divide by 2: \(x > 4\).
Solve: \(5x - 4 \leq 2x + 8\)
Move \(2x\) to the left first.
Subtract \(2x\): \(3x - 4 \leq 8\). Add 4: \(3x \leq 12\). Divide by 3: \(x \leq 4\).
Solve: \(2(x + 3) < 18\)
Divide by 2 first or expand first.
Divide by 2: \(x + 3 < 9\). Subtract 3: \(x < 6\).
Solve: \(4(x - 2) \geq 20\)
Divide both sides by 4 first.
Divide by 4: \(x - 2 \geq 5\). Add 2: \(x \geq 7\).
Solve: \(-2x < 10\)
When you divide by a negative number, reverse the inequality sign.
Divide both sides by \(-2\). The sign reverses, so \(x > -5\).
Solve: \(-3x \geq 12\)
Dividing by a negative reverses the sign.
Divide by \(-3\). The inequality reverses: \(x \leq -4\).
Solve: \(7 - 2x > 15\)
After subtracting 7, you will divide by \(-2\).
Subtract 7: \(-2x > 8\). Divide by \(-2\) and reverse the sign: \(x < -4\).
Solve: \(\frac{x}{3} + 2 < 7\)
Subtract 2, then multiply by 3.
Subtract 2: \(\frac{x}{3} < 5\). Multiply by 3: \(x < 15\).
A student solves \(-4x < 20\) and writes \(x < -5\). What mistake did they make?
The sign changes when dividing by a negative.
Dividing by \(-4\) reverses the inequality. The correct answer is \(x > -5\).
Solve: \(2x + 9 > 5x - 6\)
Move the x terms to one side. Be careful if the x coefficient becomes negative.
Subtract \(5x\): \(-3x + 9 > -6\). Subtract 9: \(-3x > -15\). Divide by \(-3\) and reverse the sign: \(x < 5\).
Games
Practise this topic with interactive games.
Games coming soon.
Frequently Asked Questions
What is an inequality?
A statement showing a range of values.
What happens when multiplying by a negative?
You reverse the inequality sign.
How are answers shown?
On a number line or as an interval.