A22AlgebraFoundation & HigherGrades 3–6

Inequalities

Q: Can I write 4 4?

Yes, they mean the same thing. Most people find x > 4 easier to read, so it is worth rewriting at the end.

Q: What is the difference between "integers" and "solutions"?

The solution -2 < x 3 includes every value in that range, such as 0.5. The integer solutions are just the whole numbers.

An inequality describes a range of values rather than a single one: $x > 3$ is true for 3.1, 50 and a million, but not for 3 itself. You solve inequalities almost exactly like equations — with one rule that catches people out every year.

40 min Video by Maths Genie AQA GCSE Maths

The big picture

Adding or subtracting the same number keeps an inequality true: if $5 > 2$ then $5 + 4 > 2 + 4$ . Multiplying or dividing by a positive number does too. But multiplying or dividing by a $negative$ number reverses the order: $5 > 2$ , yet $- 5 < - 2$ .

AQA questions combine three skills: solving, showing the answer on a number line, and listing the integers that satisfy it. Each is a separate mark, so learn the conventions precisely.

Solving InequalitiesCurated by Kepler, not produced by us. Watch it, then read the notes below.

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What you'll learn

Use the symbols $<$ , $>$ , $\leq$ and $\geq$ and show inequalities on a number line
List the integers that satisfy an inequality
Solve linear inequalities, reversing the sign when dividing by a negative
Solve double inequalities

Symbols and number lines

$<$ means "less than" and $>$ means "greater than". $\leq$ and $\geq$ include the end value: "less than or equal to" and "greater than or equal to".

On a number line, an $open$ circle means the end value is not included ( $<$ or $>$ ). A $filled$ circle means it is included ( $\leq$ or $\geq$ ).

Open circle at

- 2

(not included), filled circle at

3

(included). The integers are

- 1, 0, 1, 2, 3

- 2

is not included (strict

<

3

is included (

\leq

Answer

- 1, 0, 1, 2, 3

Your turnSame method, new numbers — try it before moving on.

List the integers $n$ such that $- 4 \leq n < 1$ .

Your answer

Separate values with commas.

Tip — Read $- 2 < n$ from the letter's point of view: " $n$ is greater than $- 2$ ".

Solving linear inequalities

Solve as you would an equation: expand brackets, collect the letter on one side, and use inverse operations on both sides.

Keep the inequality sign in every line instead of writing $=$ . Your answer should look like $x > 4$ or $x \leq - 1.5$ .

1Add 3:

5 x \geq 20

2Divide by 5:

x \geq 4

Answer

x \geq 4

Your turnSame method, new numbers — try it before moving on.

Solve $3 x + 8 < 29$ .

Your answer

Type it like x < 7 or x >= 2.

Your turnSame method, new numbers — try it before moving on.

Solve $2 (3 x - 1) \leq 4 x + 9$ .

Your answer

Type ≤ as <=, e.g. x <= 5.5.

The sign flip

When you multiply or divide both sides by a negative number, reverse the inequality sign.

You can always avoid dividing by a negative by collecting the letter on the side where its coefficient is positive instead.

1Subtract 7:

- 2 x > 6

2Divide by

- 2

and reverse the sign:

x < - 3

Answer

x < - 3

Your turnSame method, new numbers — try it before moving on.

Solve $10 - 3 x \leq 1$ .

Your answer

Type ≥ as >=, e.g. x >= 3.

Why the sign flips

Check with a number. The answer $x < - 3$ claims $x = - 4$ works: $7 - 2 (- 4) = 15 > 13$ ✓. And $x = 0$ should fail: $7 > 13$ ✗ ✓. If you had forgotten to flip, $x = 0$ would have "worked" — and it does not.

Double inequalities

A double inequality such as $- 1 < 2 x + 3 \leq 9$ has three parts. Do the same operation to all three parts at once.

1Subtract 3 from all three parts:

- 4 < 2 x \leq 6

2Divide all parts by 2:

- 2 < x \leq 3

3Integers:

- 1, 0, 1, 2, 3

Answer

- 2 < x \leq 3

; integers

- 1, 0, 1, 2, 3

Your turnSame method, new numbers — try it before moving on.

Solve $5 \leq 3 x - 4 < 17$ .

Your answer

Type it like 3 <= x < 7.

Tip — Write the answer with the smaller number on the left, matching the number line.

Think like an examiner

An answer of $x = 4$ to "solve $5 x - 3 \geq 17$ " usually loses the final mark — the inequality sign is part of the answer.
In "list the integers" questions, check both end values against the symbols: this is where the mark is won or lost.
On a number line, draw the circle exactly over the value and make clear whether it is open or filled.

Rules

a > b \Rightarrow a + c > b + c

Adding or subtracting keeps the sign.

a > b, k > 0 \Rightarrow k a > k b

Positive multiplier keeps the sign.

a > b \Rightarrow - a < - b

Negative multiplier or divisor reverses it.

Watch out for these

- 3 x < 12 \Rightarrow x < - 4

Dividing by

- 3

reverses the sign:

x > - 4

Including

- 2

in the integers for

- 2 < n \leq 3

<

is strict, so

- 2

is excluded.

Using a filled circle for

x > 1

Strict inequalities use an open circle.

Solving

- 1 < 2 x + 3 \leq 9

by subtracting 3 only from the middle.

Apply every operation to all three parts.

Stretch yourself

$n$ is an integer. $3 n - 5 > 7$ and $2 n + 1 \leq 19$ . Find all the possible values of $n$ .

Hint — Solve each inequality separately, then find the integers that satisfy both.

Questions students ask

Can I write

4 < x

instead of

x > 4

Yes, they mean the same thing. Most people find

x > 4

easier to read, so it is worth rewriting at the end.

Do I flip the sign when I subtract a negative number?

No. Only multiplying or dividing by a negative reverses the sign. Adding and subtracting never does.

What is the difference between "integers" and "solutions"?

The solution

- 2 < x \leq 3

includes every value in that range, such as 0.5. The integer solutions are just the whole numbers.

Key takeaways

$<$ , $>$ exclude the end value (open circle); $\leq$ , $\geq$ include it (filled circle).
Solve like an equation, keeping the inequality sign throughout.
Multiplying or dividing by a negative reverses the sign.
For double inequalities, do the same to all three parts.
Check your answer by testing a value inside and outside the range.

Exam-style questions

Practise it the way it is examined

3 original questions · 10 marks, written to match the style and mark allocation of the real papers. Work on paper, then open the mark scheme and tick the marks you earned — method marks count even if the final answer slips.

Question 1Grade 32 marks

Write down all the integers that satisfy $- 3 \leq n < 2$ .[2]

Question 2Grade 54 marks

(a)Solve $6 x - 7 > 2 x + 5$ .[2]
(b)Solve $4 - 5 x \geq 19$ .[2]

Question 3Grade 64 marks

Solve $- 7 < \frac{3 x - 1}{2} \leq 4$ and write down the largest integer solution.[4]

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Inequalities

Symbols and number lines

Solving linear inequalities

The sign flip

Double inequalities

Practise it the way it is examined

Skill drill

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Related topics

Inequalities

Symbols and number lines

Solving linear inequalities

The sign flip

Double inequalities

Practise it the way it is examined

Skill drill

Practise this topic

Related topics