Ordering and Inequalities

Use the signs <, >, ≤ and ≥ to compare numbers, turn words like at least and at most into an inequality, and list the integers that fit an inequality such as −3 < n ≤ 2.

In the exam for Cambridge IGCSE Maths 0580, Edexcel IGCSE Maths 4MA1 · 5 real past-paper questions below

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What the Signs Mean

An inequality sign compares two numbers that are not equal. There are four signs to know, plus one for “not equal”.

Sign	Means
$<$	is less than
$>$	is more than
$\leq$	is less than or equal to
$\geq$	is more than or equal to
$\neq =$	is not equal to

Rule

The wide, open side of the sign always faces the bigger number. The point faces the smaller one.

Worked example

1
$7 > 3$ — Seven is more than three. The open side of the sign faces the bigger number, 7.
2
$3 < 7$ — The same fact read from the other side. The sign turns round, but its open side still faces 7.
3
$7 \neq = 3$ — Seven and three are different numbers, so “not equal” is also true. A crossed-out equals sign means not equal.

Key idea

The line under $\leq$ and $\geq$ is half an equals sign. It means the number itself is allowed too.

Put a Sign Between Two Numbers

The 1-mark question: "Write < or > between each pair of numbers." Picture the number line: the one further right is bigger.

Worked example

1
$- 5 □ - 2$ — Two negative numbers. Picture them on the number line.
2
$- 5 < - 2$ — −5 is further left, further below zero, so it is the smaller number.
3
$0.3 □ 0.25$ — Decimals with different numbers of places. Give them the same number of places first.
4
$0.3 = 0.30$ — A zero on the end does not change a decimal. Now both numbers have 2 decimal places.
5
$0.30 > 0.25$ — 30 hundredths is more than 25 hundredths, so 0.3 is bigger.
6
$\frac{2}{3} □ \frac{3}{5}$ — Fractions with different bottoms. Make the bottoms the same first.
7
$\frac{2}{3} = \frac{10}{15}, \frac{3}{5} = \frac{9}{15}$ — 15 is in the 3 and the 5 times tables. Multiply the top and bottom of the first by 5, and of the second by 3.
8
$\frac{10}{15} > \frac{9}{15}$ — Same bottoms, so compare the tops: 10 is more than 9. So the first fraction is bigger.

Common mistake

Writing $- 5 > - 2$ because 5 is bigger than 2. With negatives, the one nearer zero is bigger.

Compare Different Forms

Exams mix forms: a fraction against a percentage, or a power against a decimal. Change both into decimals, then compare. Sometimes the answer is =.

Worked example

1
$\frac{3}{8} = 3 \div 8 = 0.375$ — Change the fraction to a decimal: divide the top by the bottom.
2
$37.5% = 37.5 \div 100 = 0.375$ — Change the percentage to a decimal: divide by 100.
3
$\frac{3}{8} = 37.5%$ — Both are 0.375, so they are equal.
4
$0. 3^{2} = 0.3 \times 0.3 = 0.09$ — Work out the power first. Squaring a decimal below 1 makes it smaller.
5
$0. 3^{2} < 0.3$ — 0.09 is 9 hundredths and 0.3 is 30 hundredths, so the square is smaller.
6
$\frac{5}{6} = 5 \div 6 = 0.8333 \dots$ — Divide 5 by 6. The 3s go on for ever, so keep the dots.
7
$\frac{5}{6} > 0.83$ — Compare place by place. Both start 0.83, then $\frac{5}{6}$ carries on with more 3s, so it is bigger.

Common mistake

Rounding $\frac{5}{6}$ to 0.83 and writing =. 0.8333… is a little more than 0.83.

Key idea

“Ring every sign that is correct” can have two answers. 0.4 = 40%, which is more than 4%, so >, ≥ and ≠ are all correct between 0.4 and 4%.

Words Into Symbols

The 1-mark question: "Write an inequality for ...". Turn each phrase into a sign. Use a letter for the unknown.

Words	Symbol
more than 4	$x > 4$
at least 4 (4 or more)	$x \geq 4$
less than 4	$x < 4$
at most 4 (4 or less)	$x \leq 4$

Worked example

1
$ride: at least 120 cm$ — The rule on a fairground ride. Use $h$ for a child’s height in cm.
2
$h \geq 120$ — “At least” means 120 or more. 120 itself is allowed, so use the sign with the line under it.
3
$5 > x$ — Some answers are written with the letter on the right. This says 5 is more than $x$ .
4
$x < 5$ — Read it from the other side: swap the two sides and turn the sign round. Both say $x$ is below 5.

Common mistake

Writing $h > 120$ for "at least 120". A child exactly 120 cm tall can ride, so you need $\geq$ .

List the Integers

The 2-mark question: "n is an integer. $- 3 < n \leq 2$ . List the possible values of n." An integer is a whole number, and it can be negative.

Worked example

1
$- 3 < n \leq 2$ — The question. Read each end on its own.
2
$- 3$ — The left sign is <, with no line, so −3 itself is out. Start at the next integer up, −2.
3
$2$ — The right sign is ≤, with a line, so 2 is in. Stop at 2.
4
$n = - 2, - 1, 0, 1, 2$ — Write every whole number from −2 up to 2. Do not skip 0: it is an integer too.

Common mistake

Leaving out 0, or including −3. Write every whole number from one end to the other, then check both ends.

Read the Number Line

The 2-mark question: “Write down the inequality shown on the number line.” Here the line has a filled circle at −1 and an open circle at 4, joined by a line.

Worked example

1
$filled circle at - 1$ — A filled circle means −1 is included, so this end uses ≤.
2
$open circle at 4$ — An open circle means 4 is not included, so this end uses <.
3
$- 1 \leq x < 4$ — Write it the way the line reads: the smaller number on the left, $x$ in the middle, the bigger number on the right.

Common mistake

Using < at both ends. Check each circle on its own: filled means ≤, open means <.

Key idea

Each correct end can earn a mark on its own. If you are unsure, still write both ends.

Which Method? Read the Question

Look at what the question gives you and what it wants back.

You see	You do
Write < or > between	number line: right is bigger
A fraction, a percentage or a power	change both to decimals, then compare
"at least" or "at most"	use the sign with a line under it
"more than" or "less than"	use the sign with no line
n is an integer, list the values	whole numbers, check both ends
Inequality shown on a number line	filled circle ≤ or ≥, open circle < or >
Order a list of numbers	Ordering Values
Solve an inequality with x in it	Linear Inequalities

Key idea

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

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