3.4PureCore

Linear Inequalities

Inequalities work much like equations — you can add, subtract, multiply and divide — with one crucial twist: multiplying or dividing by a negative number flips the inequality sign. Solutions are ranges of values, not single numbers.

25 min Equations and Inequalities

Edexcel AS level Maths: 3.4 Linear InequalitiesCurated by Kepler, not produced by us. Watch it before the notes below.

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What you'll be able to do

Solve linear inequalities and state the solution set
Know when to reverse the inequality sign
Solve “double” inequalities
Represent solutions on a number line and in set notation

Solving like an equation — with one rule

Treat the inequality like an equation: do the same to both sides to isolate $x$ . The only difference is the negative-number rule.

\text{Divide or multiply by a negative} \Rightarrow \text{flip the sign}

e.g.

-2x < 6

becomes

x > -3

(sign reversed).

Example 1Solve

3x - 4 \le 11

1Add 4:

3x \le 15

2Divide by 3 (positive, no flip):

x \le 5

Answer

x \le 5

Example 2Solve

5 - 2x < 9

1Subtract 5:

-2x < 4

2Divide by

-2

and flip:

x > -2

Answer

x > -2

Tip — Only a negative multiply/divide flips the sign. Adding or subtracting never does.

Double inequalities

A double inequality like $-1 < 2x + 3 \le 7$ traps $x$ between two values. Do the same operation to all three parts at once.

Example 1Solve

-1 < 2x + 3 \le 7

1Subtract 3 throughout:

-4 < 2x \le 4

2Divide by 2 throughout:

-2 < x \le 2

Answer

-2 < x \le 2

Writing the solution

A solution can be shown on a number line (open circle for $<$ or $>$ , filled for $\le$ or $\ge$ ) or in set notation such as $\{x : x \le 5\}$ .

Formula recap

a < b \Rightarrow a + c < b + c

Add/subtract: sign unchanged.

\times \text{ or } \div \text{ by negative} \Rightarrow \text{flip}

The one rule to remember.

\{x : -2 < x \le 2\}

Set notation for a range.

Common mistakes to avoid

Forgetting to flip the sign when dividing by a negative.

−2x < 6 gives x > −3, not x < −3.

Treating an inequality solution as a single value.

The solution is a range of values, e.g. x ≤ 5.

Key takeaways

Solve like an equation, isolating x.
Flip the inequality only when multiplying or dividing by a negative.
For double inequalities, operate on all three parts together.
State the answer as a range, on a number line or in set notation.

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.

Solve $x + 5 > 9$ .
- $x > 4$
- $x < 4$
- $x > 14$
- $x > -4$
Show answer and explanation
Answer: $x > 4$
Subtract 5 from both sides.
Solve $2x \le 10$ .
- $x \le 5$
- $x \ge 5$
- $x \le 20$
- $x < 5$
Show answer and explanation
Answer: $x \le 5$
Divide by 2 (positive), no flip.
Solve $-3x > 12$ .
- $x < -4$
- $x > -4$
- $x < 4$
- $x > 4$
Show answer and explanation
Answer: $x < -4$
Divide by −3 and flip the sign.
Solve $4 - x \ge 1$ .
- $x \le 3$
- $x \ge 3$
- $x \le -3$
- $x \ge -3$
Show answer and explanation
Answer: $x \le 3$
$-x \ge -3$ , divide by −1 and flip: $x \le 3$ .
Solve $-2 \le x + 1 < 4$ .
- $-3 \le x < 3$
- $-1 \le x < 5$
- $-3 < x \le 3$
- $-2 \le x < 4$
Show answer and explanation
Answer: $-3 \le x < 3$
Subtract 1 from all three parts.
On a number line, a filled (solid) circle represents:
- $\le$ or $\ge$
- $<$ or $>$
- No solution
- Only equalities
Show answer and explanation
Answer: $\le$ or $\ge$
A solid circle includes the endpoint (≤ or ≥).
Which operation never changes the inequality direction?
- Subtracting a number
- Dividing by a negative
- Multiplying by a negative
- Negating both sides
Show answer and explanation
Answer: Subtracting a number
Adding/subtracting keeps the sign; only negative multiply/divide flips it.

Next step

Keep practising

Start with the quiz above to check the key ideas. You can also choose another topic from the practice library.

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Related lessons

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Put the methods into practice: see how A-Level maths duels work. An account is required to play; the overview and sample are public.