GCSE Maths · Algebra

Linear Inequalities GCSE Questions and Worked Answers

A linear inequality describes a range of allowed values. Read the sign, keep the comparison true while simplifying, then show all the solutions with an inequality or a number line.

Foundation & HigherOne-variable linear inequalities10 original questions

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Hi, I’m Ari. Let’s start with what an inequality allows, then work through signs, number lines and solving one step at a time.

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Start with an allowed amount

What you need to know about linear inequalities

A lift can hold at most 8 people: 6, 7 and 8 are allowed, but 9 is not. Let n stand for the number of people. Writing n ≤ 8 means n is less than or equal to 8. Unlike n = 8, it describes several possible values. An inequality is a comparison that tells us which values are allowed.

A limit can allow many answers

Read the sign before solving

The pointed end faces the smaller value. A line underneath adds ‘or equal to’. In algebra, solutions can include fractions and decimals unless a question asks for integers (whole numbers, including negative whole numbers).

Strictx < 2 or x > 2less than or greater than; 2 excluded

Inclusivex ≤ 2 or x ≥ 2the value 2 is also allowed

On a number lineopen or filled endpointopen excludes; filled includes

An open circle at 2 means 2 itself is excluded. The arrow selects values greater than 2, including decimals such as 2.1.

For example, $x+3>7$ asks: which numbers become greater than 7 when we add 3? Subtracting 3 from both sides shifts both values equally, so their order stays the same:

x+3>7

x>4

$5$ works because $5+3=8>7$ . So does $4.5$ . But $4$ gives equality and is excluded. “Linear” means the unknown occurs only to the first power, not squared or in a denominator.

From problem to method

Typical problems you need to be able to solve

These are the main kinds of question in this topic. Each card first states the problem, then explains how to solve it.

Show allowed values on a number line

What the problem asks: Represent x > 2, or −1 < x ≤ 5.

How to solve it: Mark each boundary. Use an open circle if the endpoint is excluded and a filled circle if included. Shade in the direction of the allowed values.

List integer solutions

What the problem asks: List all integers satisfying −3 < n ≤ 2.

How to solve it: Choose only whole numbers in the interval: −2, −1, 0, 1 and 2. Check the endpoints separately.

Solve a linear inequality

What the problem asks: Find all x satisfying 4x + 3 ≤ 19.

How to solve it: Subtract 3 from both sides and divide both sides by positive 4. These changes preserve the order and give x ≤ 4.

Solve when the coefficient is negative

What the problem asks: Solve −3x < 12.

How to solve it: Divide both sides by −3 and reverse the comparison to get x > −4. Multiplication by a negative reverses order: 2 < 5 becomes −2 > −5.

Form a limit from a cost problem

What the problem asks: A fixed charge plus a cost per item must fit a budget.

How to solve it: Define the item count, write total cost ≤ budget, then solve. If only whole items can be bought, select the greatest allowed integer.

A reliable routine

Solve a one-variable linear inequality

Use this method when the task asks for every value of one unknown that makes a linear comparison true. Adding or subtracting the same amount preserves order. Multiplying or dividing by a positive number preserves it; a negative number reverses it.

Expand brackets if present.
Collect the unknown terms on one side and numbers on the other by doing the same operation to both sides.
Divide by the coefficient of the unknown; reverse the sign if that coefficient is negative.
Write the full range of solutions, not just one allowed number.
Substitute a value inside the range into the original inequality to check.

Check: The sign changes on multiplying or dividing both sides by a negative number, not merely because a negative number appears. This page covers one variable; quadratic inequalities and shaded 2D regions are separate Higher topics.

Fully worked

Linear inequalities GCSE worked examples

Each solution explains the clue, why the method fits and how to check the result.

Example 1

Read and draw an interval

2 marks

Question

Represent $-1<x\le5$ on a number line.

The left sign excludes $-1$ , so place an open circle there. The right sign includes $5$ , so place a filled circle there. Join them to show all values between, not only the integers.

The open circle excludes −1. The filled circle includes 5. The line between them includes every real value between the endpoints.

Example 2

Solve with a positive coefficient

2 marks

Question

Solve $5x-7<18$ .

Add $7$ to both sides.

5x<25

Divide by positive $5$ , preserving the comparison.

\boxed{x<5}

Check: $x=4$ gives $13<18$ . The boundary $x=5$ gives equality, so it is excluded.

Example 3

Reverse the comparison

3 marks

Question

Solve $7-3x\le19$ .

Subtract $7$ from both sides.

-3x\le12

Divide by $-3$ . The negative divisor reverses the order.

\boxed{x\ge-4}

Check: $x=0$ gives $7\le19$ ; $x=-5$ gives $22\le19$ , which is false.

Example 4

Handle brackets and both sides

3 marks

Question

Solve $3(x+2)>x+14$ .

Multiply both terms inside the bracket by $3$ .

3x+6>x+14

Subtract $x$ from both sides.

2x+6>14

Subtract $6$ , then divide by positive $2$ .

2x>8

\boxed{x>4}

Tip: check the original brackets, not just the final line.

Example 5

Solve a double inequality

3 marks

Question

Solve $-5<2x+1\le9$ .

Both comparisons must hold. Subtract $1$ from all three parts.

-6<2x\le8

Divide all three parts by positive $2$ .

\boxed{-3<x\le4}

The left endpoint is still excluded; the right endpoint is still included.

Example 6

Find a maximum whole-number purchase

4 marks

Question

A printing service charges £7 setup plus £3 per poster. You have £30. Find the maximum number of posters you can buy.

Let $n$ be the number of posters. Spending must be at most £30.

7+3n\le30

3n\le23

n\le\frac{23}{3}

Only whole posters can be bought, so the maximum is $\boxed{7}$ .

Check: seven cost £28; eight cost £31. Do not round $7.66\ldots$ up.

10 original questions · total 25 marks

Linear inequalities GCSE exam-style questions

Try each question before opening its fully worked answer. The difficulty rises through the set.

Before you startAllow about 30 minutes · Show each operation. Preserve strict or inclusive endpoints and state all solutions. List only integers when requested. · answers start collapsed

Translate a limit

1 mark

Write ‘x is at least 6’ using an inequality sign.

Show worked answer

At least means 6 is included and larger values are allowed: $\boxed{x\ge6}$ .

List all allowed integers

2 marks

List the integers satisfying $-4<n\le1$ .

Show worked answer

Exclude $-4$ , include $1$ , and list every whole number between them:

\boxed{-3,-2,-1,0,1}

Read an open endpoint

2 marks

A number line has an open circle at 2 and an arrow to the right. Write the inequality.

An open circle at 2 means 2 itself is excluded. The arrow selects values greater than 2, including decimals such as 2.1.

Show worked answer

The open circle excludes $2$ . Right means greater than:

\boxed{x>2}

Undo addition and multiplication

2 marks

Solve $4x+5\le25$ .

Show worked answer

Subtract $5$ :

4x\le20

Divide by positive $4$ :

\boxed{x\le5}

Divide by a negative

2 marks

Solve $-2x>10$ .

Show worked answer

Divide both sides by $-2$ and reverse the sign:

\boxed{x<-5}

For example, $x=-6$ gives $12>10$ .

Collect terms from both sides

3 marks

Solve $7x-4\ge3x+12$ .

Show worked answer

Subtract $3x$ from both sides.

4x-4\ge12

4x\ge16

\boxed{x\ge4}

Expand before solving

3 marks

Solve $2(3x-1)<4x+8$ .

Show worked answer

6x-2<4x+8

Subtract $4x$ , then add $2$ .

2x-2<8

2x<10

\boxed{x<5}

Work with three parts

3 marks

Solve $2\le3x-4<14$ .

Show worked answer

Add $4$ to every part.

6\le3x<18

Divide every part by $3$ .

\boxed{2\le x<6}

Solve a fractional expression

Harder3 marks

Solve $\dfrac{x-3}{4}>2$ .

Show worked answer

The whole numerator is divided by $4$ . Multiply both sides by positive $4$ .

x-3>8

Add $3$ .

\boxed{x>11}

Respect a delivery budget

4 marks

Delivery costs £5 plus £4 for each box. The total must be no more than £42. Find the maximum whole number of boxes.

Show worked answer

Let $b$ be the number of boxes.

5+4b\le42

4b\le37

b\le9.25

The greatest allowed integer is $\boxed{9}$ : nine boxes cost £41, while ten cost £45.

Examiner-style feedback

Common linear inequalities mistakes

Giving only the boundary

x = 5 is not a complete answer to x < 5. The question asks for a whole range.

Reversing the sign after subtraction

Subtracting a negative number does not itself reverse order. Only multiplication or division by a negative does.

Using a filled circle for a strict sign

A strict < or > excludes its endpoint, so use an open circle.

Rounding an item count up

A budget maximum must stay inside the allowed range. Check the next whole item would exceed the budget.

30-second recap

Choose, carry out, check

Translate the wording before you reach for a rule.

Read what the sign includes.
Keep both sides comparable at every step.
Reverse order when dividing by a negative.
Give the whole allowed range.

Quick answers

Linear inequalities FAQ

Are linear inequalities Foundation or Higher?

Solving one-variable linear inequalities and showing solutions on a number line are assessed across both tiers. Quadratic inequalities and two-variable graphical regions are Higher content and outside this page.

Do inequalities allow decimal solutions?

Yes, unless the problem restricts the variable to integers or a whole-number count.

Why does a negative divisor reverse the sign?

Multiplying by a negative reflects values across zero. For example, 2 is below 5, but −2 is above −5.

Build connected skills

What to revise next

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Fractions

Handle fractions before combining them with algebra.

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More practice

GCSE Maths topics

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Content standards

Curriculum and rights review

Reviewed 7 September 2026 against GCSE A22. This guide covers one-variable linear inequalities. Questions and diagrams are original; suggested marks are Pass an Exam estimates.

Official specification references

AQA GCSE algebra: A17 and A22