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 Video by Zeeshan Zamurred Equations and Inequalities

Edexcel AS level Maths: 3.4 Linear InequalitiesWatch the full walkthrough 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.

Divide or multiply by a negative \Rightarrow flip the sign

e.g.

- 2 x < 6

becomes

x > - 3

(sign reversed).

1Add 4:

3 x \leq 15

2Divide by 3 (positive, no flip):

x \leq 5

Answer

x \leq 5

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

Double inequalities

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

1Subtract 3 throughout:

- 4 < 2 x \leq 4

2Divide by 2 throughout:

- 2 < x \leq 2

Answer

- 2 < x \leq 2

Writing the solution

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

Formula recap

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

Add/subtract: sign unchanged.

\times or \div by negative \Rightarrow flip

The one rule to remember.

{x : - 2 < x \leq 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.

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