1MA1 · A22AlgebraFoundation & HigherGrades 3–6

Linear Inequalities

Q: Why does the sign flip for negatives?

Multiplying by a negative reverses the order of numbers on the number line: 3 -5.

Q: Do I flip the sign when subtracting a negative?

No. Only multiplying or dividing both sides by a negative reverses the inequality.

Q: What does a dashed boundary line mean?

Points on the line are not included, because the inequality is strict.

An equation like $2x + 3 = 11$ has one answer. An inequality like $2x + 3 < 11$ has infinitely many: every number less than 4. Solving inequalities uses the same steps as solving equations, with a single extra rule that catches almost everyone at least once — and the answers are shown on number lines, as lists of integers, or as shaded regions on a graph.

45 min Equations & Inequalities

The big picture

Adding, subtracting, and multiplying or dividing by a positive number all keep an inequality true. Multiplying or dividing by a $\textit{negative}$ number reverses the order of numbers — $2 < 5$ but $-2 > -5$ — so the inequality sign must flip.

Edexcel asks for inequalities in several forms: solve and show on a number line (Foundation and Higher), list the integers satisfying a double inequality, and at Higher tier identify or shade a region defined by several inequalities on a coordinate grid.

What you'll learn

Solve linear inequalities, reversing the sign when multiplying or dividing by a negative
Represent solutions on a number line and in set notation
Solve double inequalities and list integer solutions
Represent and identify regions defined by inequalities on a graph (Higher)

Solving inequalities

Use inverse operations exactly as for equations, keeping the inequality sign in place of the equals sign.

If you multiply or divide both sides by a negative number, $\textbf{reverse the sign}$ : $<$ becomes $>$ and $\le$ becomes $\ge$ .

You can often avoid this by collecting the unknown on the side where its coefficient is positive.

Example 1Solve

5 - 2x \ge 11

1Subtract 5:

-2x \ge 6

2Divide by

-2

and reverse the sign.

Answer

x \le -3

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

Solve $7 - 3x > 19$ .

Your answer

Type it like x < -4.

Show worked solution

Solution

1Subtract 7: $-3x > 12$ .
2Divide by $-3$ and reverse: $x < -4$ .

Answer: $x < -4$

Example 2Solve

3(x - 4) < 5x + 2

1Expand:

3x - 12 < 5x + 2

2Subtract

3x

and 2:

-14 < 2x

3Divide by 2:

-7 < x

Answer

x > -7

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

Solve $4(x + 2) \ge 2x - 6$ .

Your answer

Type ≥ as >=.

Show worked solution

Solution

1Expand: $4x + 8 \ge 2x - 6$ .
2 $2x \ge -14$ , so $x \ge -7$ .

Answer: $x \ge -7$

Why this works

Checking with a number confirms the flip. For $5 - 2x \ge 11$ , try $x = -4$ (which satisfies $x \le -3$ ): $5 + 8 = 13 \ge 11$ ✓. Try $x = 0$ : $5 \ge 11$ is false, as the solution predicts.

Number lines and set notation

On a number line, an $\textbf{open circle}$ means the endpoint is not included ( $<$ or $>$ ); a $\textbf{filled circle}$ means it is included ( $\le$ or $\ge$ ). An arrow or line shows the rest of the solution.

A solution such as $-2 < x \le 5$ is drawn as a line segment from an open circle at $-2$ to a filled circle at $5$ .

In set notation, $\{x : x > 3\}$ means "the set of all $x$ such that $x > 3$ ".

Tip — Match the circle to the symbol: the sign with a line underneath ( $\le$ , $\ge$ ) gets the filled-in circle.

Double inequalities and integers

For a double inequality like $-4 < 3n + 2 \le 11$ , apply each step to all three parts.

Integer solutions are the whole numbers in the solution set — watch whether the endpoints are included.

Example 1Find the integers

n

such that

-4 < 3n + 2 \le 11

1Subtract 2 throughout:

-6 < 3n \le 9

2Divide by 3:

-2 < n \le 3

-2

is excluded;

3

is included.

Answer

n = -1, 0, 1, 2, 3

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

List the integers $n$ such that $-3 \le 2n - 1 < 6$ .

Your answer

Separate values with commas.

Show worked solution

Solution

1Add 1: $-2 \le 2n < 7$ .
2Divide by 2: $-1 \le n < 3.5$ .

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

Regions on a graph

An inequality in $x$ and $y$ describes a region. Draw the boundary line as if it were an equation: a $\textbf{solid}$ line for $\le$ or $\ge$ (included), a $\textbf{dashed}$ line for $<$ or $>$ (not included).

Test a point not on the line, such as the origin, to decide which side satisfies the inequality.

For several inequalities, the required region is where all are satisfied. Edexcel usually asks you to shade it and label it $R$ .

Example 1How many points with integer coordinates satisfy

x \ge 1

y > x

and

x + y < 7

1For

x = 1

y > 1

and

y < 6

, so

y = 2, 3, 4, 5

— four points.

2For

x = 2

y > 2

and

y < 5

, so

y = 3, 4

— two points.

3For

x = 3

y > 3

and

y < 4

— none. Larger

x

gives none.

Answer

6

points

Tip — When counting integer points, check each boundary: points on a dashed line do not count, points on a solid line do.

Think like an examiner

Show the step where the sign is reversed, and say why.
Use the correct circles on number lines.
List integer solutions separated by commas, checking both endpoints.
Use solid and dashed boundary lines correctly when shading regions.

Inequalities

a < b \Rightarrow -a > -b

Multiplying by a negative reverses the sign.

\le,\ \ge: \text{ filled circle, solid line}

Included boundaries.

<,\ >: \text{ open circle, dashed line}

Excluded boundaries.

Watch out for these

Dividing

-3x > 12

-3

to get

x > -4

Reverse the sign:

x < -4

Using a filled circle for

x < 2

Strict inequalities use an open circle.

Including

-2

in the integers for

-2 < n \le 3

The strict inequality excludes

-2

Writing the answer as

x = 4

for

2x < 8

The solution is a range:

x < 4

Stretch yourself

Find the largest integer $n$ that satisfies both $4n - 7 < 2n + 9$ and $3(n + 1) \ge 5n - 11$ .

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

Reveal solution

SolutionFirst:

4n - 7 < 2n + 9

gives

2n < 16

, so

n < 8

. Second:

3n + 3 \ge 5n - 11

gives

14 \ge 2n

, so

n \le 7

. Both hold when

n \le 7

(since

n \le 7

already implies

n < 8

). The largest integer is

n = 7

. Check:

28 - 7 = 21 < 23

✓ and

24 \ge 24

✓.

\blacksquare

Questions students ask

Why does the sign flip for negatives?

Multiplying by a negative reverses the order of numbers on the number line:

3 < 5

but

-3 > -5

Do I flip the sign when subtracting a negative?

No. Only multiplying or dividing both sides by a negative reverses the inequality.

What does a dashed boundary line mean?

Points on the line are not included, because the inequality is strict.

Key takeaways

Solve inequalities like equations, keeping the inequality sign.
Reverse the sign when multiplying or dividing by a negative.
Open circles for strict inequalities, filled circles for inclusive ones.
Apply each step to all parts of a double inequality.
Regions: solid or dashed boundaries, test a point, shade where all inequalities hold.

Exam-style questions

Practise it the way it is examined

3 original questions · 6 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

Solve $6x - 5 \le 2x + 9$ .[2]
Show mark scheme
- M1 $4x \le 14$
- A1 $x \le 3.5$
Answer: $x \le 3.5$

Question 2Grade 42 marks

Write down all the integers $n$ that satisfy $-7 < 3n + 2 \le 8$ .[2]
Show mark scheme
- M1 $-3 < n \le 2$
- A1 $-2, -1, 0, 1, 2$
Answer: $-2, -1, 0, 1, 2$

Question 3Grade 62 marks

How many points with integer coordinates satisfy all of $x \ge 0$ , $y \ge 1$ and $x + y \le 4$ ?[2]
Show mark scheme
- M1Identifies the region or lists points systematically
- A1 $10$
Answer: $10$

Examiner note

List by $x$ : $x = 0$ gives 4 points, $x = 1$ gives 3, $x = 2$ gives 2 and $x = 3$ gives 1.

Unlimited practice

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

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