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 $\textbf{negative}$ 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.

Open on YouTube

What you'll learn

Use the symbols $<$ , $>$ , $\le$ and $\ge$ 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". $\le$ and $\ge$ include the end value: "less than or equal to" and "greater than or equal to".

On a number line, an $\textbf{open}$ circle means the end value is not included ( $<$ or $>$ ). A $\textbf{filled}$ circle means it is included ( $\le$ or $\ge$ ).

Open circle at

-2

(not included), filled circle at

3

(included). The integers are

-1, 0, 1, 2, 3

Example 1List the integers

n

such that

-2 < n \le 3

-2

is not included (strict

<

3

is included (

\le

Answer

-1, 0, 1, 2, 3

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

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

Your answer

Separate values with commas.

Show worked solution

Solution

1 $-4$ is included ( $\le$ ); $1$ is not ( $<$ ).
2Integers: $-4, -3, -2, -1, 0$ .

Answer: $-4, -3, -2, -1, 0$

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 $x \le -1.5$ .

Example 1Solve

5x - 3 \ge 17

1Add 3:

5x \ge 20

2Divide by 5:

x \ge 4

Answer

x \ge 4

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

Solve $3x + 8 < 29$ .

Your answer

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

Show worked solution

Solution

1Subtract 8: $3x < 21$ .
2Divide by 3: $x < 7$ .

Answer: $x < 7$

Example 2Solve

4(x + 2) > 2x + 3

1Expand:

4x + 8 > 2x + 3

2Subtract

2x

2x + 8 > 3

3Subtract 8:

2x > -5

, so

x > -2.5

Answer

x > -2.5

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

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

Your answer

Type ≤ as <=, e.g. x <= 5.5.

Show worked solution

Solution

1Expand: $6x - 2 \le 4x + 9$ .
2Subtract $4x$ and add 2: $2x \le 11$ .
3Divide by 2: $x \le 5.5$ .

Answer: $x \le 5.5$

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.

Example 1Solve

7 - 2x > 13

1Subtract 7:

-2x > 6

2Divide by

-2

and reverse the sign:

x < -3

Answer

x < -3

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

Solve $10 - 3x \le 1$ .

Your answer

Type ≥ as >=, e.g. x >= 3.

Show hint

After subtracting 10 you divide by $-3$ — what happens to the sign?

Show worked solution

Solution

1Subtract 10: $-3x \le -9$ .
2Divide by $-3$ and reverse: $x \ge 3$ .

Answer: $x \ge 3$

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 $-1 < 2x + 3 \le 9$ has three parts. Do the same operation to all three parts at once.

Example 1Solve

-1 < 2x + 3 \le 9

and list the integer solutions.

1Subtract 3 from all three parts:

-4 < 2x \le 6

2Divide all parts by 2:

-2 < x \le 3

3Integers:

-1, 0, 1, 2, 3

Answer

-2 < x \le 3

; integers

-1, 0, 1, 2, 3

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

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

Your answer

Type it like 3 <= x < 7.

Show worked solution

Solution

1Add 4 to all parts: $9 \le 3x < 21$ .
2Divide all parts by 3: $3 \le x < 7$ .

Answer: $3 \le x < 7$

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 $5x - 3 \ge 17$ " 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 ka > kb

Positive multiplier keeps the sign.

a > b \Rightarrow -a < -b

Negative multiplier or divisor reverses it.

Watch out for these

-3x < 12 \Rightarrow x < -4

Dividing by

-3

reverses the sign:

x > -4

Including

-2

in the integers for

-2 < n \le 3

<

is strict, so

-2

is excluded.

Using a filled circle for

x > 1

Strict inequalities use an open circle.

Solving

-1 < 2x + 3 \le 9

by subtracting 3 only from the middle.

Apply every operation to all three parts.

Stretch yourself

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

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

Reveal solution

Solution

3n - 5 > 7

gives

3n > 12

, so

n > 4

2n + 1 \le 19

gives

2n \le 18

, so

n \le 9

. Together

4 < n \le 9

, so

n = 5, 6, 7, 8, 9

\blacksquare

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 \le 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); $\le$ , $\ge$ 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 $-3 \le n < 2$ .[2]
Show mark scheme
- B1At least four correct values with no more than one error
- B1 $-3, -2, -1, 0, 1$
Answer: $-3, -2, -1, 0, 1$

Question 2Grade 54 marks

(a)Solve $6x - 7 > 2x + 5$ .[2]
Show mark scheme (a)
- M1Collects terms correctly: $4x > 12$
- A1 $x > 3$
Answer: $x > 3$
(b)Solve $4 - 5x \ge 19$ .[2]
Show mark scheme (b)
- M1 $-5x \ge 15$ or $4 - 19 \ge 5x$
- A1 $x \le -3$
Answer: $x \le -3$

Examiner note

In (b), $x \ge -3$ is the most common wrong answer — dividing by $-5$ reverses the sign.

Question 3Grade 64 marks

Solve $-7 < \dfrac{3x - 1}{2} \le 4$ and write down the largest integer solution.[4]
Show mark scheme
- M1Multiplies all parts by 2: $-14 < 3x - 1 \le 8$
- M1 $-13 < 3x \le 9$
- A1 $-\frac{13}{3} < x \le 3$
- B1Largest integer is 3 (follow through)
Answer: $-\frac{13}{3} < x \le 3$ ; largest integer $3$

Unlimited practice

Skill drill

A new question every time, marked instantly, with a full worked solution. Aim for a streak of five, then move up a level.

0 / 0 first-time correct Streak 0Best at this level: 0

Generating a question…

Test yourself

Answer a few questions, then review anything you missed.

Prefer paper? Open the written self-check

Answer each question before opening its answer. This self-check does not record a quiz score.

Which value satisfies $x > 5$ ?
- $5.1$
- $5$
- $4.9$
- $-6$
Show answer and explanation
Answer: $5.1$
$5$ itself is not included.
Which integers satisfy $-1 \le n < 3$ ?
- $-1, 0, 1, 2$
- $0, 1, 2$
- $-1, 0, 1, 2, 3$
- $0, 1, 2, 3$
Show answer and explanation
Answer: $-1, 0, 1, 2$
$-1$ is included; $3$ is not.
Solve $x + 4 > 9$ .
- $x > 5$
- $x > 13$
- $x < 5$
- $x = 5$
Show answer and explanation
Answer: $x > 5$
Subtract 4 from both sides.
Solve $4x - 1 \le 11$ .
- $x \le 3$
- $x \ge 3$
- $x \le 2.5$
- $x \le 12$
Show answer and explanation
Answer: $x \le 3$
$4x \le 12$ .
Solve $-2x > 8$ .
- $x < -4$
- $x > -4$
- $x > 4$
- $x < 4$
Show answer and explanation
Answer: $x < -4$
Divide by $-2$ and reverse the sign.
On a number line, $x \ge 2$ is shown with…
- a filled circle at 2 and an arrow right
- an open circle at 2 and an arrow right
- a filled circle at 2 and an arrow left
- an open circle at 2 and an arrow left
Show answer and explanation
Answer: a filled circle at 2 and an arrow right
$\ge$ includes 2; greater means to the right.
Solve $3 < 2x - 5 < 11$ .
- $4 < x < 8$
- $-1 < x < 3$
- $8 < x < 16$
- $4 < x < 3$
Show answer and explanation
Answer: $4 < x < 8$
Add 5 to all parts: $8 < 2x < 16$ .
How many integers satisfy $-5 < 3n + 1 \le 13$ ?
- $6$
- $5$
- $7$
- $4$
Show answer and explanation
Answer: $6$
$-2 < n \le 4$ : $-1, 0, 1, 2, 3, 4$ .

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