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Cambridge IGCSE Mathematics 0580 · Algebra and graphs

Inequalities

Reading, writing and drawing inequalities, including on a number line, and listing the integers that satisfy them. Extended students also solve linear inequalities, show inequalities in $x$ and $y$ as regions on a graph, and write down the inequalities that define a region.

C2.6E2.6Core and ExtendedFree lesson

What the syllabus says

C2.6 (Core)

Represent and interpret inequalities, including on a number line.

Notes and examples

When representing and interpreting inequalities on a number line:
open circles should be used to represent strict inequalities ( $<$ , $>$ )
closed circles should be used to represent inclusive inequalities ( $⩽$ , $⩾$ )
e.g. $- 3 ⩽ x < 1$

E2.6 (Extended)

1 Represent and interpret inequalities, including on a number line.
2 Construct, solve and interpret linear inequalities.
3 Represent and interpret linear inequalities in two variables graphically.
4 List inequalities that define a given region.

Notes and examples

When representing and interpreting inequalities on a number line:
open circles should be used to represent strict inequalities ( $<$ , $>$ )
closed circles should be used to represent inclusive inequalities ( $⩽$ , $⩾$ ).
e.g. $- 3 ⩽ x < 1$
Examples include:
$3 x < 2 x + 4$
$- 3 ⩽ 3 x - 2 < 7$ .
The following conventions should be used:
broken lines should be used to represent strict inequalities ( $<$ , $>$ )
solid lines should be used to represent inclusive inequalities ( $⩽$ , $⩾$ )
shading should be used to represent unwanted regions (unless otherwise directed in the question).
e.g.
Linear programming problems are not included.

Quoted from the Cambridge IGCSE Mathematics 0580 syllabus for 2025 to 2027.

Inequality symbols

An inequality compares two quantities that need not be equal. There are four symbols:

$x > 3$ : $x$ is greater than 3;
$x < 3$ : $x$ is less than 3;
$x ⩾ 3$ : $x$ is greater than or equal to 3;
$x ⩽ 3$ : $x$ is less than or equal to 3.

The narrow end of the symbol points at the smaller quantity, so $x > 3$ and $3 < x$ say the same thing. The strict symbols $<$ and $>$ leave out the boundary value; $⩽$ and $⩾$ include it. So 3 satisfies $x ⩾ 3$ but not $x > 3$ .

A double inequality such as $- 3 ⩽ x < 1$ says two things at once: $x$ is at least $- 3$ , and $x$ is less than 1. Read it from the middle outwards. It describes every number from $- 3$ up to, but not including, 1, so $- 3$ , $- 1.5$ , 0 and 0.99 all satisfy it, but 1 does not.

Everyday wording translates directly: "at least 18" is $⩾ 18$ , "no more than 40" is $⩽ 40$ , "under 12" or "less than 12" is $< 12$ , and "over 100" is $> 100$ .

Inequalities on a number line

On a number line an inequality is drawn as a line over all the values it includes, with a circle at each boundary:

an open circle (not filled in) for a strict inequality, $<$ or $>$ , because the boundary value is not included;
a closed circle (filled in) for $⩽$ or $⩾$ , because the boundary value is included.

A one-sided inequality such as $x > 2$ has a single circle and an arrow showing that it goes on forever in one direction.

The number line below shows $- 3 ⩽ x < 1$ : a closed circle at $- 3$ , an open circle at 1 and a line joining them. To read a number line, work the other way: look at each circle to decide between $<$ and $⩽$ , then write the inequality with $x$ in the middle.

Listing integer values

Questions often say " $n$ is an integer" and ask you to list the values that satisfy an inequality. Integers are whole numbers: positive, negative or zero. Work along the number line from the lower boundary to the upper one, and check each end carefully:

$- 3 ⩽ n < 1$ gives $- 3, - 2, - 1, 0$ . The value $- 3$ is included; 1 is not.
$- 3 < n ⩽ 1$ gives $- 2, - 1, 0, 1$ .

Do not forget 0, and remember that among negative numbers $- 2$ is greater than $- 3$ . If the boundary is not a whole number, as in $x > - 4.5$ , the smallest integer is the first whole number to the right of it on the number line, which is $- 4$ .

Worked example 1

The number line shows an inequality. (a) Write down the inequality, in terms of $x$ . (b) $n$ is an integer that satisfies the inequality. List the possible values of $n$ .

(a) The circle at $- 1$ is open, so $- 1$ is not included: $x > - 1$ .
The circle at 3 is closed, so 3 is included: $x ⩽ 3$ .
Together: $- 1 < x ⩽ 3$ .
(b) The integers greater than $- 1$ , up to and including 3, are 0, 1, 2 and 3.

Answer: (a) $- 1 < x ⩽ 3$ (b) $0, 1, 2, 3$

Check yourself

$n$ is an integer and $- 2 < n ⩽ 3$ . List all the possible values of $n$ .

Answer it in the portal, where your answer is checked as you type →

Check yourself

Write down the inequality shown on the number line, in terms of $x$ .

Answer it in the portal, where your answer is checked as you type →

Solving linear inequalities Extended

Extended candidates solve inequalities as well as read them. A linear inequality is solved exactly like a linear equation, keeping the inequality sign where the equals sign would be. You may add or subtract the same amount on both sides, and multiply or divide both sides by the same positive number.

There is one new rule: multiplying or dividing both sides by a negative number reverses the inequality sign. To see why, start with a true statement, $2 < 5$ , and multiply both sides by $- 1$ . You get $- 2$ and $- 5$ , and on the number line $- 2$ is to the right of $- 5$ , so $- 2 > - 5$ . The sign had to turn round for the statement to stay true.

You can often avoid the rule by collecting the letter on the side where it stays positive. For $5 - 2 x > 11$ , add $2 x$ to both sides and subtract 11: $- 6 > 2 x$ , so $- 3 > x$ , which is $x < - 3$ . Dividing by $- 2$ directly gives the same answer, as long as you reverse the sign.

The syllabus example $3 x < 2 x + 4$ needs only one step: subtract $2 x$ from both sides to get $x < 4$ . A double inequality such as $- 3 ⩽ 3 x - 2 < 7$ is solved by doing the same thing to all three parts: add 2 to get $- 1 ⩽ 3 x < 9$ , then divide by 3 to get $- \frac{1}{3} ⩽ x < 3$ .

The solution of an inequality is a range of values, so write it as an inequality, for example $x < - 3$ , not as a single number.

Worked example 2 Extended

Solve $7 - 3 x ⩽ 2 x + 22$ .

Add $3 x$ to both sides: $7 ⩽ 5 x + 22$ .
Subtract 22 from both sides: $- 15 ⩽ 5 x$ .
Divide by 5, which is positive, so the sign stays the same: $- 3 ⩽ x$ .
Write it with $x$ first: $x ⩾ - 3$ .

Answer: $x ⩾ - 3$

Worked example 3 Extended

(a) Solve $1 < 2 x + 5 ⩽ 13$ . (b) List the integer values of $x$ that satisfy the inequality.

(a) Subtract 5 from all three parts: $- 4 < 2 x ⩽ 8$ .
Divide all three parts by 2: $- 2 < x ⩽ 4$ .
(b) $x$ is greater than $- 2$ and at most 4, so the integers are $- 1, 0, 1, 2, 3, 4$ .

Answer: (a) $- 2 < x ⩽ 4$ (b) $- 1, 0, 1, 2, 3, 4$

Check yourself Extended

Solve $7 x - 3 ⩾ 4 x + 9$ .

Answer it in the portal, where your answer is checked as you type →

Check yourself Extended

Solve $4 - 5 x > 19$ .

Answer it in the portal, where your answer is checked as you type →

Inequalities in two variables: regions on a graph Extended

An inequality in $x$ and $y$ , such as $y ⩾ 1$ or $y < 2 x + 1$ , is satisfied by a whole region of the coordinate grid, not just a line. To show it:

Draw the boundary line by replacing the inequality sign with $=$ . For $y < 2 x + 1$ , draw $y = 2 x + 1$ .
Use a broken line for $<$ or $>$ , because points on the line are not included, and a solid line for $⩽$ or $⩾$ , because they are.
Decide which side is wanted by testing a point that is not on the line, usually $(0, 0)$ . For $y < 2 x + 1$ : $0 < 2 (0) + 1$ is true, so the side containing the origin is the wanted side.
Shade the unwanted region, unless the question tells you otherwise. The region left unshaded is the answer, and with several inequalities it is easy to see where they all hold.

A vertical line has an equation $x = k$ and a horizontal line has $y = k$ . So $x < 1$ is everything to the left of the broken line $x = 1$ , and $y ⩾ 1$ is everything on or above the solid line $y = 1$ .

Linear programming, finding the greatest or least value of an expression over a region, is not part of this syllabus.

Worked example 4 Extended

On a grid, show the region that satisfies $x < 3$ , $y ⩾ 1$ and $y ⩽ x + 2$ . Shade the unwanted regions.

$x < 3$ : draw $x = 3$ as a broken vertical line, because the inequality is strict. The wanted side is to the left, so shade to the right.
$y ⩾ 1$ : draw $y = 1$ as a solid horizontal line. The wanted side is above, so shade below.
$y ⩽ x + 2$ : draw the solid line $y = x + 2$ through $(0, 2)$ and $(3, 5)$ . Test $(0, 0)$ : $0 ⩽ 2$ is true, so the origin side is wanted; shade above the line.
The unshaded triangle with corners $(- 1, 1)$ , $(3, 1)$ and $(3, 5)$ is the required region. Points on the line $x = 3$ are not in it.

Answer: The unshaded triangle shown in the diagram

Finding the inequalities that define a region Extended

Sometimes the region is drawn for you and you must write down the inequalities that define it. Work one boundary line at a time:

Find the equation of the line. Vertical and horizontal lines are $x = k$ and $y = k$ ; for a sloping line find the gradient $m$ and the $y$ -intercept $c$ , and write $y = m x + c$ .
Choose the direction: pick a point inside the region and substitute it. If $(1, 2)$ is inside and the line is $y = x + 3$ , then $2 < 1 + 3$ , so the inequality is $y < x + 3$ or $y ⩽ x + 3$ .
Choose strict or inclusive from the line itself: broken means $<$ or $>$ , solid means $⩽$ or $⩾$ .

There is one inequality for each boundary line, so a triangular region needs three.

Worked example 5 Extended

The unshaded region in the diagram is defined by three inequalities. Write down the three inequalities.

The solid horizontal line is $y = 1$ , and the region is above it: $y ⩾ 1$ .
The broken vertical line is $x = 4$ , and the region is to the left of it: $x < 4$ .
The solid sloping line passes through $(1, 1)$ and $(4, 7)$ , so its gradient is $\frac{7 - 1}{4 - 1} = 2$ . Then $1 = 2 (1) + c$ gives $c = - 1$ , so the line is $y = 2 x - 1$ .
Test $(3, 2)$ , which is inside the region: $2 ⩽ 2 (3) - 1 = 5$ is true. The line is solid, so the inequality is $y ⩽ 2 x - 1$ .

Answer: $y ⩾ 1$ , $x < 4$ , $y ⩽ 2 x - 1$

Key points

$<$ and $>$ are strict: the boundary is not included. $⩽$ and $⩾$ include it.
On a number line: open circle for strict, closed circle for inclusive.
When listing integers, check both ends separately, and remember 0.
Extended: solve an inequality like an equation, but reverse the sign when you multiply or divide by a negative number.
Extended: for a region, broken line for strict, solid line for inclusive, test a point, and shade the unwanted side.

Where marks are lost

Using a closed circle for $<$ or an open circle for $⩽$ on a number line.
Including the boundary of a strict inequality when listing integers, or leaving out the boundary of an inclusive one.
Writing "at least 25" as $t > 25$ : "at least" includes 25, so it is $t ⩾ 25$ .
Extended: dividing by a negative number and forgetting to reverse the sign.
Extended: giving the answer as $x = 4$ when the solution is $x < 4$ .
Extended: drawing every boundary line solid, or shading the wanted region when the question asks you to shade the unwanted one.

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An original lesson written to the Cambridge IGCSE Mathematics 0580 syllabus (2025 to 2027). The worked examples and questions are our own, not Cambridge questions. Back to every IGCSE Mathematics 0580 lesson.

Inequalities

Inequality symbols

Inequalities on a number line

Listing integer values

Worked example 1

Check yourself

Check yourself

Solving linear inequalities Extended

Worked example 2 Extended

Worked example 3 Extended

Check yourself Extended

Check yourself Extended

Inequalities in two variables: regions on a graph Extended

Worked example 4 Extended

Finding the inequalities that define a region Extended

Worked example 5 Extended

Key points

Where marks are lost

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Sequences

Equations

Proportion

Indices II