Edexcel IGCSE

Inequalities

An inequality says that one quantity is smaller or larger than another, so its answer is a whole range of numbers rather than a single one. It is section 2.8 of the Edexcel International GCSE Mathematics A specification, and almost every question on it reduces to the same move: solve it exactly as you would solve the matching equation, then ask one extra question — did I multiply or divide by a negative number? If you did, the sign turns round. If you did not, it stays exactly as it was.

Edexcel IGCSE 4MA1Specification 2.844 approved practice questions

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Before you start

What you need to be able to do already

If any of these is shaky, fix it first — every worked example below leans on all three.

Multiply and divide with negative numbers, and keep the sign right
Work out $- 6 \div - 2$ , and then $- 2 \times - 4$ . Both come out positive, $3$ and $8$ , because two negatives multiplied or divided give a positive. If either took thought, fix directed number first: the single biggest error on this page is a sign that should have turned round, and you can neither see it coming nor check for it afterwards while the arithmetic underneath is still effortful.
Undo operations one at a time to get a letter on its own
Rearrange $y = 4 x + 3$ to make $x$ the subject. It is $x = \frac{y - 3}{4}$ : subtract $3$ from both sides, then divide both sides by $4$ . Solving an inequality is that same machinery with one extra rule bolted onto the dividing step, so if the rearranging is still effortful the extra rule will not stick.
Draw the line $y = mx + c$, and say whether a point lies above it or below itGraphs
Where does $y = 2 x + 1$ cross the $y$ -axis, and is the point $(1, 5)$ above the line or below it? It crosses at $(0, 1)$ , and $(1, 5)$ is above, because at $x = 1$ the line has only reached $y = 3$ . Three of the seven objectives in section 2.8 are about regions on a grid, and you cannot shade the correct side of a line you cannot draw.

The idea

What is inequalities?

An inequality is a statement that two quantities are not equal, together with which of them is the larger: $x > 3$ says $x$ is bigger than $3$ , and $x \leq 5$ says $x$ is $5$ or anything below it. Because it fixes a direction rather than a value, the answer is a range and not a number — $x \leq 5$ is satisfied by $5$ , by $4.9$ , by $0$ and by $- 1000$ alike. You solve one with exactly the moves you would use on an equation, with a single extra rule attached to one of those moves, and you report the answer as a range: as an inequality, as circles and an arrow on a number line, as the list of integers inside it, or as a shaded region on a grid.

The notation, and how to read it

Every worked solution below is written in these five symbols.

SymbolReads asMeans
$<$ is less thanThe quantity on the left is the smaller one. With negatives this is where it bites: $- 15 < - 9$ , because $- 15$ sits further left on the number line even though its digits look bigger.
$>$ is greater thanThe quantity on the left is the larger one. The open end always faces the bigger number, so $15 > 12$ and $12 < 15$ are the same statement written from the two ends.
$\leq$ is less than or equal toEverything below the endpoint, and the endpoint itself. $x \leq 5$ includes $5$ , so the circle at $5$ on a number line is filled in and a boundary line on a grid is drawn solid.
$\geq$ is greater than or equal toEverything above the endpoint, and the endpoint itself. $y \geq - 1$ includes $- 1$ . Read the bar underneath as the words ‘or equal to’ — it is the whole difference between this and $>$ .
$(3, 7]$ the interval from 3 to 7Interval notation for $3 < x \leq 7$ . A round bracket excludes its endpoint (open circle); a square bracket includes it (closed circle). The smaller number is always written first, so the brackets are what carry the meaning.

The one idea underneath all of it

An inequality is a statement about order — which of two quantities sits further right on the number line — so every step you take has to preserve that order. Adding or subtracting the same number slides both sides along equally, and multiplying or dividing by a positive number stretches both sides equally; in all of those the order survives and the sign is left alone. Multiplying or dividing by a negative number is the one operation that does not preserve it: it reflects the number line, sending what was on the right over to the left, so the order reverses and the sign must reverse with it. That single idea is the entire difference between solving an inequality and solving an equation, and it is where most of the marks on this topic are lost.

The smallest possible example

Solve $- 2 x < 6$ .

$x > - 3$

Dividing both sides by $- 2$ gives $- 3$ for the number, and turns $<$ into $>$ . Here is why, with no algebra at all: $3 > 1$ is true, but multiply both sides by $- 2$ and you get $- 6$ and $- 2$ , and $- 6 < - 2$ . Multiplying by a negative reflects the number line, so whichever side was bigger has become the smaller one. Check the answer both ways: $x = 0$ is inside $x > - 3$ , and $- 2 \times 0 = 0$ , which is less than $6$ ; $x = - 4$ is outside it, and $- 2 \times - 4 = 8$ , which is not. Every other question on this page is this, with more steps stacked in front of it.

Key terms

The words a question will use

Exam wording is precise. These five carry the meaning a question depends on.

TermWhat it meansNotation
Solution set
Every value that makes the inequality true, which is a range rather than a number. This is the difference between the two kinds of question: $3 x - 7 = 8$ has the single answer $x = 5$ , while $3 x - 7 \leq 8$ is satisfied by $5$ and by everything below it. An answer written as $x = 5$ , or as a bare $5$ , has thrown the range away — our bank records that exact substitution as a misconception on this subtopic. $x \leq 5$
Strict inequality
$<$ or $>$ , where the endpoint itself is not allowed. On a number line that is an open, hollow circle; on a grid the boundary is a dashed line. ‘Greater than $3$ but not including $3$ ’ is the exam's way of telling you which one it wants. $x > 3$
Inclusive inequality
$\leq$ or $\geq$ , where the endpoint is allowed. On a number line that is a closed, filled circle; on a grid the boundary is a solid line. Swapping the two circle conventions round is one of the misconceptions catalogued for this subtopic, and it costs a mark on almost every number-line question it touches. $x \geq - 1$
Compound inequality
Two conditions on the same letter at once, read outward from the middle: $- 2 < x < 7$ says $x > - 2$ and $x < 7$ together. When integers are asked for, count inward from each end and let the symbols decide whether the ends themselves are in — $- 8 < n < - 3$ allows exactly four, $- 7, - 6, - 5$ and $- 4$ , because both ends are strict; with $\leq$ at both ends it would have been six. $- 2 < x < 7$
Region
The set of points on a grid satisfying every one of two or more inequalities at the same time — the overlap of the shaded areas, not all of them put together. Edexcel asks you to shade it and label it $R$ , and the fastest check is to take one point from inside your shading and test it in each inequality in turn. $y \leq 2 x + 5$

The method

How to work out a inequalities

Five steps, in this order. The worked examples below point back to these numbers.

Expand any brackets and collect the letter onto one side.Do this first, exactly as you would for an equation. Moving terms across by adding or subtracting never changes the direction of the sign, so there is nothing to watch for yet. If you would rather not divide by a negative later, this is the moment to choose: $5 - 2 x > 1$ can become $5 > 1 + 2 x$ instead, and it will solve to the same answer.
Add or subtract to leave the letter term on its own. $3 x - 7 \leq 8$ becomes $3 x \leq 15$ . The sign stays exactly as it was, because adding $7$ slides both sides the same distance along the number line and so cannot change which is bigger. On a two-mark question this line is the method mark — write it down before you divide, and it is already earned.
Divide or multiply both sides by the number in front of the letter, and reverse the sign if that number is negative. $3 x \leq 15$ gives $x \leq 5$ with the sign untouched, because $3$ is positive. But $- 2 x < 6$ gives $x > - 3$ : dividing by $- 2$ reflects the number line, so the two sides swap order and $<$ has to become $>$ . It is the only error our question bank records against two different objectives in this section, once for linear inequalities and again for quadratic ones, and the sign is not decoration — the right number with the wrong sign describes the opposite half of the number line.
Write the answer as an inequality in the letter, and mark it on the number line.The answer is $x \leq 5$ — not $5$ , and not $x = 5$ . On the number line use an open circle for $<$ or $>$ and a filled circle for $\leq$ or $\geq$ , then draw the arrow the way the solution runs: to the right for greater than, to the left for less than. The circle and the arrow are usually a mark of their own.
Test one number from inside your answer and one from outside it, in the original inequality.For $x > - 3$ , try $x = 0$ : $- 2 \times 0 = 0$ , and $0 < 6$ is true, as it should be. Then try $x = - 4$ : $- 2 \times - 4 = 8$ , and $8 < 6$ is false, as it should be. If the inside number fails and the outside one works, the sign was turned the wrong way — and this catch takes about ten seconds.

Use it when

The question has one letter, no squared term, and asks you to solve, to list the values that work, or to show the answer on a number line. Anything of the shape $a x + b < c$ is these five steps, including the cases where the letter carries a negative coefficient — $5 - 2 x \geq 1$ is solved the same way, and step 3 is simply where you pay attention.

Do not use it when

The letter is squared. $x^{2} - 5 x < 14$ cannot be handled with the linear routine: rearrange it to $x^{2} - 5 x - 14 < 0$ , factorise to $(x + 2) (x - 7) < 0$ , read off the critical values $x = - 2$ and $x = 7$ , then test a number between them — $x = 0$ gives $- 14$ , which is negative, so the solution is $- 2 < x < 7$ . Dividing by $x$ is not generally valid because its sign is unknown and $x = 0$ may be lost; only do so when the sign and zero case have been established separately. Region questions on a grid are different again: draw each boundary line first and shade the required side.

Worked examples

6 IGCSE Maths practice questions on inequalities, fully worked

One per objective, so the range is covered rather than the same question repeated. Every step is written out, and the answer is shown — this is study material, not a test.

Example 1

Solve the double inequality: 3 < 2x + 5 ≤ 11. Give your answer as an inequality in x, e.g. x ≥ 4. [2 marks]

Understand and use the symbols >, <, >= and <=

Worked solution

Step 1: Split the double inequality into two parts: 3 < 2x + 5 and 2x + 5 ≤ 11.
Step 2: For the first part, subtract 5 from both sides: 3 − 5 < 2x, so −2 < 2x.
Step 3: Divide both sides by 2: −1 < x.
Step 4: For the second part, subtract 5: 2x ≤ 11 − 5, so 2x ≤ 6.
Step 5: Divide both sides by 2: x ≤ 3.
Step 6: Combine the two results: −1 < x and x ≤ 3, which is written as −1 < x ≤ 3.

Answer−1 < x ≤ 3

Example 2

A number line shows the set of numbers greater than 3 and less than or equal to 7. Which interval notation correctly describes this set? [1 mark]

Solving linear inequalities

Worked solution

Step 1: 'Greater than 3' means 3 is not included – we use an open bracket '('.
Step 2: 'Less than or equal to 7' means 7 is included – we use a closed bracket ']'.
Step 3: Put the brackets together: start with '(' for 3 (not included) and end with ']' for 7 (included).
Step 4: The correct notation is (3, 7].

Answer(3, 7]

Example 3

The temperature in a freezer must be kept below -5°C. Represent this inequality on the number line below. Give your answer as an inequality in x, e.g. x ≥ 4. [1 mark]

Representing inequalities on a number line

Worked solution

Step 1: The problem says the temperature must be kept below -5°C.
Step 2: 'Below' means 'less than', so we use the symbol <.
Step 3: Let x represent the temperature.
Step 4: So the inequality is x < -5.

Answerx < -5

Show 3 more worked examples

Example 4

A taxi firm charges a flat fee of £3 and then £2 per mile. Jack wants the total cost to be less than £15. Which inequality represents this situation? $total cost < 15$ ? [1 mark]

Represent simple linear inequalities on rectangular Cartesian graphs

Worked solution

Step 1: The flat fee of £3 is the constant term, added to the cost per mile times the number of miles.
Step 2: Jack wants the total to be less than £15, so the inequality uses a < sign.
Step 3: The correct expression for total cost is $3 + 2$ x, where x is the number of miles.
Step 4: Since total must be less than 15, the inequality is $3 + 2$ x < 15.

Answer

3 + 2 x < 15

Example 5

Work out the region that satisfies both inequalities: x > 3 and y ≤ 2x − 1 [2 marks]

Identify regions on rectangular Cartesian graphs defined by simple linear inequalities

Worked solution

Step 1: Start with the inequality x > 3. This means all points to the right of the vertical line x = 3 (not including points on the line, since it's > not ≥).
Step 2: Next consider y ≤ 2x − 1. This is a line with gradient 2 and y-intercept −1. The region is on or below the line (since y is less than or equal to).
Step 3: Shade the overlapping area of both conditions: points that are to the right of x = 3 and also below the line.
Step 4: The overlapping area where all points satisfy both rules is Region A.

AnswerRegion A

Example 6

Solve the quadratic inequality $x^{2} - 5 x < 14$ and represent the solution set on a number line. [4 marks]

Solve quadratic inequalities in one unknown and represent the solution set on a number line

Worked solution

Step 1: Rearrange the inequality to bring all terms to one side: x² - 5x - 14 < 0.
Step 2: Factorise the quadratic: (x - 7)(x + 2) < 0.
Step 3: Identify the critical points where the expression equals zero: x = 7 and x = -2.
Step 4: Test a value in each interval. For x=0, $(0 - 7) (0 + 2) = - 14$ < 0, so the inequality holds between -2 and 7.
Step 5: Write the solution as an inequality: -2 < x < 7.

Answer-2 < x < 7

Common mistakes

Where marks actually get lost

Learnly tags 10 distinct misconceptions on this subtopic. These are errors that show up in real student work — not generic revision advice.

Examiner traps

Wrong inequality direction
What goes wrong: Students often reverse the inequality sign when representing conditions like 'below -5°C', writing x > -5 instead of x < -5.
What to do instead: Remember: 'below' means less than. If temperature is below -5, it is colder, so x is less than -5. Use x < -5.
Confusing open and closed circles
What goes wrong: Students may use a closed circle (including the endpoint) when the inequality is strict, or an open circle when it includes equality.
What to do instead: Use an open circle for < or > (strict), and a closed circle for ≤ or ≥ (includes the number). For 'below -5°C', use open circle at -5.
Misplacing negative numbers
What goes wrong: Students incorrectly place numbers on the number line, e.g., putting -5 to the right of 0 or shading the wrong side.
What to do instead: On a number line, numbers increase to the right. -5 is left of 0. For x < -5, shade all numbers to the left of -5.
Using wrong variable or no variable
What goes wrong: Students might write the inequality without a variable (e.g., just '-5') or use a different variable than x.
What to do instead: Always write the inequality with the variable given. If the question uses x, write something like x < -5, not just -5.
Forgetting to flip inequality sign
What goes wrong: When multiplying or dividing by a negative number, students often forget to reverse the inequality sign, leading to an incorrect solution set.
What to do instead: Remember: if you multiply or divide by a negative number, flip the inequality sign. For example, -2x < 6 becomes x > -3.
Drawing arrow in wrong direction
What goes wrong: Students sometimes shade the number line in the opposite direction of the inequality, e.g., shading left for x > 2.
What to do instead: Shade to the right for 'greater than' and to the left for 'less than'. Check with a test point.

See 4 more examiner traps

Incorrect solution format
What goes wrong: Students may give the solution as a single number instead of an inequality, or write it incorrectly like 'x = 5' instead of 'x ≤ 5'.
What to do instead: Your answer must be an inequality like x < 3 or x ≥ -1, not just a number. Include the variable.
Incorrect sign when factoring quadratic
What goes wrong: Students may factor the quadratic incorrectly, especially when the constant term is negative, leading to wrong critical values.
What to do instead: Check your factors: for x^ $2 - 4$ x - 5, the factors are (x - 5)(x + 1). The numbers multiply to -5 and add to -4.
Wrong test point selection
What goes wrong: Students may choose test points that are not in the correct intervals or misinterpret the sign of the quadratic expression.
What to do instead: Pick a test point from each interval (e.g., -2, 0, 6) and plug into the factored form to see if the result is positive or negative.
Incorrect number line representation
What goes wrong: Students may use open circles when closed circles are needed, or shade the wrong region on the number line.
What to do instead: Use closed circles (●) for ≤ or ≥ and open circles (○) for < or >. Shade the region where the inequality is true.

Exam technique

How the marks are actually awarded

Method marks are given for working a marker can follow. This is what to put on the page.

Where the marks are: Questions in this section run from $1$ to $4$ marks. One mark buys a single symbol — filling in $<$ or $>$ between two numbers, or naming the inequality that a shaded region shows. Solving a linear inequality is usually $2$ marks, one for the rearrangement and one for the accurate final inequality; adding ‘and represent the solution set on the number line’ makes it $3$ , with the extra mark sitting on the circles and the arrow. A quadratic inequality is $3$ or $4$ marks, and they are spread across the factorised form, the critical values and the correct interval — so write $(x + 2) (x - 7) < 0$ on the page even if you then pick the wrong interval, because that line has already earned something. On a region question the method marks live in the boundary lines: draw and label every line before you shade anything at all.
Command words to expect: Solve the inequalityGive your answer as an inequality in xRepresent the solution set on a number lineWrite down the inequality shown by the number lineWrite down all the integers that satisfyShow, by shading, the region that satisfiesLabel the region RFind a possible integer value of
Accuracy and rounding: Nothing rounds here — an inequality is exact — so the marks go on form instead. The answer has to be an inequality in the letter: $x \leq 5$ , never a bare $5$ and never $x = 5$ . Where the division does not go exactly, keep it exact: $3 x < 10$ is $x < \frac{10}{3}$ , not $x < 3.33$ . Writing the two sides the other way round is fine as long as the direction survives the swap, so $5 \geq x$ is accepted for $x \leq 5$ — but $x \geq 5$ is a different statement about a different half of the number line and scores nothing. When a question asks for integers, hand over the list itself, $- 7, - 6, - 5, - 4$ , rather than the inequality you read them from.
Calculator: A calculator is allowed in every paper of this qualification, on both Foundation and Higher tiers, and it is close to useless on this topic. It will divide $6$ by $- 2$ and return $- 3$ without ever mentioning that the inequality sign has to turn round at that same moment, and the sign is where the mark is. Its one honest use is at the end: substitute your test value into the original inequality and check that the statement it produces is genuinely true.

Check yourself

You should now be able to:

Read and write $>$ , $<$ , $\geq$ and $\leq$ , and use them to compare two numbers, negatives included.
Use an open circle for $<$ or $>$ and a filled circle for $\geq$ or $\leq$ , and match a number line to interval notation such as $(3, 7]$ .
Solve a linear inequality in one letter using the same inverse operations you would use on the matching equation.
Reverse the inequality sign whenever you multiply or divide both sides by a negative number.
Write the answer as an inequality in the letter, never as a single value and never as an equation.
Draw a solution set on a number line, and read one off a number line that has been drawn for you.
List every integer that satisfies a compound inequality such as $- 8 < n < - 3$ .
Draw the line $y = m x + c$ and decide which side of it an inequality describes, dashed for strict and solid for inclusive.
Identify or shade the region satisfying two or more linear inequalities at once, then confirm it by testing a point inside.
Solve a quadratic inequality on the Higher tier by factorising, finding the critical values and testing an interval, then show the result on a number line.

Specification coverage

The 7 things 4MA1 asks you to do

The assessable objectives for equations, formulae and identities — inequalities, in the order the specification lists them.

Solving linear inequalities
Understand and use the symbols >, <, >= and <=
Representing inequalities on a number line
Represent simple linear inequalities on rectangular Cartesian graphs
Identify regions on rectangular Cartesian graphs defined by simple linear inequalities
Solve quadratic inequalities in one unknown and represent the solution set on a number line

Show 1 more objectives

Identify harder examples of regions defined by linear inequalities

See who is stuck on inequalities before you teach it.

Create a class, share the 8-character code, and set inequalities as practice. Your students work through all 44 approved practice questions — not just the 6 on this page — and you see which objectives the class has not got yet. Free for individual students and teachers.

How Learnly works for teachers

Set up your class — free Practise on your own

What you need to be able to do already

Multiply and divide with negative numbers, and keep the sign right

Undo operations one at a time to get a letter on its own

Draw the line \(y = mx + c\), and say whether a point lies above it or below itGraphs

What is inequalities?

The notation, and how to read it

The words a question will use

Solution set

Strict inequality

Inclusive inequality

Compound inequality

Region

How to work out a inequalities

6 IGCSE Maths practice questions on inequalities, fully worked

Solve the double inequality: 3 < 2x + 5 ≤ 11. Give your answer as an inequality in x, e.g. x ≥ 4. [2 marks]

A number line shows the set of numbers greater than 3 and less than or equal to 7. Which interval notation correctly describes this set? [1 mark]

The temperature in a freezer must be kept below -5°C. Represent this inequality on the number line below. Give your answer as an inequality in x, e.g. x ≥ 4. [1 mark]

A taxi firm charges a flat fee of £3 and then £2 per mile. Jack wants the total cost to be less than £15. Which inequality represents this situation? total cost<15? [1 mark]

Work out the region that satisfies both inequalities: x > 3 and y ≤ 2x − 1 [2 marks]

Solve the quadratic inequality x2−5x<14 and represent the solution set on a number line. [4 marks]

Where marks actually get lost

Wrong inequality direction

Confusing open and closed circles

Misplacing negative numbers

Using wrong variable or no variable

Forgetting to flip inequality sign

Drawing arrow in wrong direction

Incorrect solution format

Incorrect sign when factoring quadratic

Wrong test point selection

Incorrect number line representation

How the marks are actually awarded

You should now be able to:

The 7 things 4MA1 asks you to do

See who is stuck on inequalities before you teach it.

A taxi firm charges a flat fee of £3 and then £2 per mile. Jack wants the total cost to be less than £15. Which inequality represents this situation? $total cost < 15$ ? [1 mark]

Solve the quadratic inequality $x^{2} - 5 x < 14$ and represent the solution set on a number line. [4 marks]