Linear Inequalities

Solving linear inequalities, the flip rule for negatives, double inequalities, number lines and listing integer values.

In the exam for Cambridge IGCSE Maths 0580, Edexcel IGCSE Maths 4MA1, Cambridge O Level Maths 4024, Edexcel Further Pure Maths 4PM1 · 90 real past-paper questions below

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What an Inequality Means

An inequality gives a range of answers, not just one. $x > 2$ means any number bigger than 2: 2.5, 7, 100.

Sign	Means	Circle
$<$	less than	open
$>$	more than	open
$\leq$	less than or equal to	filled
$\geq$	more than or equal to	filled

Rule

On a number line, an open circle means the number is not included. A filled circle means it is.

Key idea

Reading the line: look at the circle first, then which way the arrow goes.

Key idea

The signs $<$ and $>$ were first printed in 1631, in a book by the English mathematician Thomas Harriot. He never saw it: the book came out about ten years after he died.

Solve It Like an Equation

Most questions say “Solve the inequality”. Do the same steps as an equation, and keep the sign all the way.

Worked example

1
$6 x - 5 < 2 x + 13$ — the question
2
$4 x - 5 < 13$ — take 2x from both sides
3
$4 x < 18$ — add 5 to both sides
4
$x < 4.5$ — divide both sides by 4: any number below 4.5

Common mistake

Writing $x = 4.5$ at the end. The answer is a range, so it keeps the sign: $x < 4.5$ .

The Flip Rule

One thing is different from an equation. If you multiply or divide by a negative number, the sign turns round.

Example

Why: $2 < 3$ , but multiply both by −1 and $- 2 > - 3$ . The order swaps.

Worked example

1
$5 - 3 x \geq 20$ — The question. The x term has a minus in front.
2
$- 3 x \geq 15$ — Take 5 from both sides: 20 − 5 = 15.
3
$x \leq - 5$ — Divide both sides by −3, so 15 ÷ (−3) = −5. You divided by a negative, so the sign flips.

Don't trust the flip? Keep the x term positive instead. Same question:

Worked example

1
$5 - 3 x \geq 20$ — The same question.
2
$5 \geq 20 + 3 x$ — Add 3x to both sides, so the x term is positive.
3
$- 15 \geq 3 x$ — Take 20 from both sides: 5 − 20 = −15.
4
$- 5 \geq x$ — Divide both sides by 3. 3 is positive, so the sign stays.
5
$x \leq - 5$ — Read it backwards, from x's side. −5 ≥ x says the same thing as x ≤ −5.

Key idea

Both ways give $x \leq - 5$ . Check with x = −6: 5 + 18 = 23, and 23 ≥ 20 is true.

Two Inequalities at Once

A double inequality, like $- 5 < 3 x + 1 \leq 10$ , has three parts. Do each step to all three.

Worked example

1
$- 5 < 3 x + 1 \leq 10$ — the question
2
$- 6 < 3 x \leq 9$ — take 1 from all three parts
3
$- 2 < x \leq 3$ — divide all three parts by 3

Rule

The answer sits between two numbers. Each end keeps its own sign.

Read the Number Line

Exams often draw a number line and ask which inequality it shows. Read the circle first, then the arrow or the joining line.

Worked example

1
$open circle at - 2$ — The circle is empty, so −2 itself is not included.
2
$arrow to the right$ — The arrow points towards bigger numbers.
3
$x > - 2$ — Bigger than −2, but not equal to it. Use the letter the question uses.

Now a line with a circle at each end, labelled y:

Worked example

1
$filled circle at - 1.5$ — The circle is filled, so −1.5 is included: $y \geq - 1.5$ .
2
$open circle at 4$ — The circle is empty, so 4 is not included: $y < 4$ .
3
$- 1.5 \leq y < 4$ — A line joins the circles, so both are true at once. Write one statement, smaller number on the left.

Common mistake

Writing only one end, like $y \geq - 1.5$ . That loses marks. A joined line needs both ends in one statement.

Draw It on the Number Line

Many questions say “Solve, then show your answer on the number line”. Solve first, then draw.

Worked example

1
$5 t - 4 < 2 t + 5$ — The question. There are t terms on both sides.
2
$3 t - 4 < 5$ — Take 2t from both sides, so t is only on the left.
3
$3 t < 9$ — Add 4 to both sides.
4
$t < 3$ — Divide both sides by 3. 3 is positive, so the sign stays.
5
$open circle at 3$ — Draw an empty circle at 3, because 3 itself is not included.
6
$arrow to the left$ — Draw an arrow from the circle to the left, because t is less than 3.

For a double inequality, draw a circle at each end and join them.

Worked example

1
$- 1 \leq x < 4$ — The answer to draw. It has two ends.
2
$filled circle at - 1$ — ≤ includes −1, so fill the circle in.
3
$open circle at 4$ — < does not include 4, so leave it empty.
4
$line from - 1 to 4$ — Join the circles with a straight line. No arrow: x stops at both ends.

Common mistake

Filling the circle for < or >. Only ≤ and ≥ get a filled circle.

Listing the Whole Numbers

“n is an integer” means n is a whole number. It can be 0 or negative. When an end is not a whole number, picture the number line to decide.

Worked example

1
$- 3.4 < n \leq 2$ — The question: n is an integer. The left end is not a whole number.
2
$- 4 < - 3.4 < - 3$ — −3.4 sits between −4 and −3. −4 is to its left, so too small. −3 is to its right, so it is in.
3
$n \leq 2$ — ≤ means 2 itself is allowed.
4
$n = - 3, - 2, - 1, 0, 1, 2$ — Count up in ones from −3 to 2. Six values. Don't skip 0.

Example

“The smallest integer with $d > - 3 \frac{1}{4}$ ”: −3¼ lies between −4 and −3. −4 is to its left, so too small. The answer is −3.

Watch for an extra word. “Positive” means start at 1, not 0.

Worked example

1
$2 n + 9 > 5 n - 2$ — The question asks for the positive integers n.
2
$9 > 3 n - 2$ — Take 2n from both sides, so the n term stays positive.
3
$11 > 3 n$ — Add 2 to both sides.
4
$3 \frac{2}{3} > n$ — Divide both sides by 3: 11 ÷ 3 = 3⅔.
5
$n < 3 \frac{2}{3}$ — Read it backwards, from n's side.
6
$n = 1, 2, 3$ — Positive, so start at 1. Stop at 3, because 4 is too big.

Common mistake

Choosing −4 for a negative end like −3.4. On the negative side, −4 is smaller than −3.4, so it does not fit.

Harder Inequalities (Extended/Higher)

Two harder types come up on the Extended and Higher papers. Both use moves you already know.

1. A fraction: clear it first

Worked example

1
$5 - 2 x \geq \frac{x + 1}{3}$ — The question. The right side is divided by 3.
2
$3 (5 - 2 x) \geq x + 1$ — Multiply both sides by 3. The whole left side is multiplied, not just the 5.
3
$15 - 6 x \geq x + 1$ — Expand: 3 × 5 = 15 and 3 × (−2x) = −6x.
4
$15 \geq 7 x + 1$ — Add 6x to both sides, so the x term is positive and no flip is needed.
5
$14 \geq 7 x$ — Take 1 from both sides.
6
$2 \geq x$ — Divide both sides by 7.
7
$x \leq 2$ — Read it backwards, from x's side.

2. A minus x in the middle

Worked example

1
$- 1 < 7 - 2 x \leq 11$ — The question. The middle has −2x.
2
$- 8 < - 2 x \leq 4$ — Take 7 from all three parts: −1 − 7 = −8 and 11 − 7 = 4.
3
$4 > x \geq - 2$ — Divide all three by −2: −8 ÷ (−2) = 4 and 4 ÷ (−2) = −2. Dividing by a negative, so both signs flip.
4
$- 2 \leq x < 4$ — Rewrite it with the smaller number on the left. Each end keeps its own sign.

Common mistake

Flipping only one of the two signs. Dividing by a negative flips both.

Which Method? Read the Question

Look at the sign and at what the question asks for.

You see	You do
A number line	Circle first: open or filled
“Solve”	Same steps as an equation, keep the sign
Dividing by a negative	Flip the sign
Three parts	Do each step to all three
“Integer”	List the whole numbers, check each end
An $x^{2}$	See Quadratic Inequalities

Key idea

Solving is usually 2 marks, and 3 for a double inequality. Leave the answer as an inequality, not an equals sign.

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