Comparing Numbers: the Signs <, >, ≤ and ≥

What the four signs say and how to read them, how to compare integers with them, and how to write and read statements such as −2 ≤ n < 3.

Before you start

The integers are $\ldots, -3, -2, -1, 0, 1, 2, 3, \ldots$ Those greater than 0 are positive, those less than 0 are negative, and 0 is neither.

On a number line the integers are marked at equal spaces, with the positive integers to the right of 0 and the negative integers to the left.

A number further to the right is greater: $-3$ is to the right of $-8$ , so $-3$ is greater than $-8$ .

The Four Signs

To compare two numbers, we can write a sign between them in place of words. The sign $>$ means "is greater than" and $<$ means "is less than". Two more signs, $\le$ and $\ge$ , also allow the two numbers to be equal.

Sign	Read as	In everyday words	True statements
$<$	is less than	fewer than, below	$-8 < -3$
$>$	is greater than	more than, above	$2 > -5$
$\le$	is less than or equal to	at most, no more than, not more than	$2 \le 3$ and $3 \le 3$
$\ge$	is greater than or equal to	at least, not less than	$5 \ge 3$ and $3 \ge 3$

A statement with a sign is read from left to right: $-8 < -3$ is read "negative eight is less than negative three". The open end of $<$ or $>$ faces the greater number, so $7 > 4$ and $4 < 7$ say the same thing.

Comparing Integers with < and >

Rule

On a number line, the number further to the right is the greater. Write the sign so that its open end faces that number: $-3$ is to the right of $-8$ , so $-3 > -8$ , or $-8 < -3$ .

Every negative number is less than 0 and less than every positive number: $-6 < 0$ and $-6 < 2$ . Of two negative numbers, the one further from zero is the smaller: $-20$ is further from zero than $-11$ , so $-20 < -11$ .

Example 1: Comparing two integers

Write $<$ or $>$ between the two numbers of each pair.
(a) $-4$ and 2
(b) $-6$ and $-9$
(c) 0 and $-1$

Solution

(a) $-4$ is to the left of 2, so $-4$ is less than 2.

(b) $-6$ is to the right of $-9$ , so $-6$ is greater than $-9$ . Both are negative, and $-9$ is further from zero.

Each statement can be turned round: $2 > -4$ , $-9 < -6$ and $-1 < 0$ say the same three things.

(a) $-4 < 2$ (b) $-6 > -9$ (c) $0 > -1$

Three or more numbers that are in order can be joined in one statement, with the same sign in every space.

Example 2: Joining numbers in order

(a) Arrange $-2$ , 3, $-5$ and 0 in ascending order (from the least to the greatest), and join them with $<$ .
(b) Write the same four numbers in descending order (from the greatest to the least), joined with $>$ .

Solution

Mark the four numbers on a number line.

(a) Ascending order is their order from left to right. Each number is less than the next one, so $<$ goes in every space.

(b) Descending order is their order from right to left. Each number is greater than the next one, so $>$ goes in every space.

A statement like this is read in pairs: $-5 < -2 < 0$ says that $-5 < -2$ and that $-2 < 0$ .

(a) $-5 < -2 < 0 < 3$ (b) $3 > 0 > -2 > -5$

Or Equal To: the Signs ≤ and ≥

The signs $<$ and $>$ compare two numbers that are different. The line under $\le$ and $\ge$ adds the words "or equal to".

What "or equal to" means

A statement with $\le$ or $\ge$ is true when either of its two parts is true. $2 \le 3$ is true because 2 is less than 3, and $3 \le 3$ is true because 3 is equal to 3. But $3 < 3$ is false, and $4 \le 3$ is false.

So a statement with $\le$ is false only when the first number is greater than the second, and a statement with $\ge$ is false only when the first number is less than the second.

Example 3: True or false

Is each statement true or false?
(a) $-5 \ge -5$
(b) $-4 \le -7$
(c) $-3 \ge -2$
(d) $-1 \le 4$

Solution

(a) $-5$ is equal to $-5$ , and $\ge$ allows "equal to".

(b) $-4$ is to the right of $-7$ , so $-4$ is greater than $-7$ . It is neither less than $-7$ nor equal to it.

(d) $-1$ is to the left of 4, so $-1$ is less than 4. One of the two parts of "less than or equal to" is true, and one is enough.

(a) True (b) False (c) False (d) True

From Words to Signs, and Back

A letter can stand for a number that is not known. $x \le 3$ is read " $x$ is less than or equal to 3": $x$ may be 3 or any number less than 3.

Choosing the sign

Decide whether the words say greater or less.
Ask whether the number named is itself allowed. If it is, use $\ge$ or $\le$ . If it is not, use $>$ or $<$ .

The words	Is 5 itself allowed?	Statement
more than 5, greater than 5, above 5	No	$x > 5$
fewer than 5, less than 5, below 5	No	$x < 5$
at least 5, not less than 5	Yes	$x \ge 5$
at most 5, no more than 5, not more than 5	Yes	$x \le 5$

Example 4: Writing a statement with a sign

Write each sentence as a statement using $<$ , $>$ , $\le$ or $\ge$ .
(a) The temperature of a store room, $t$ °C, must not exceed 30 °C.
(b) To watch a film rated NC16, your age, $a$ years, must be at least 16.
(c) The temperature inside a freezer, $f$ °C, is below $-15$ °C.

Solution

(a) "Must not exceed 30" means 30 or anything less.

(b) "At least 16" means 16 or anything greater.

(a) $t \le 30$ (b) $a \ge 16$ (c) $f < -15$

Example 5: Writing a statement in words

Write in words what each statement says.
(a) A car that keeps to the speed limit on an expressway travels at $v$ km/h, and $v \le 90$ .
(b) A child who may go on a water slide is $h$ cm tall, and $h \ge 120$ .
(c) A pupil who fails a test has a mark of $m$ , and $m < 50$ .

Solution

(a) $\le$ is "less than or equal to": 90 is allowed, and so is anything less.

(b) $\ge$ is "greater than or equal to": 120 is allowed, and so is anything greater.

(a) The speed of the car is at most 90 km/h, so the speed limit is 90 km/h. (b) The child is at least 120 cm tall. (c) The mark is less than 50, so a mark of 50 is not a fail.

Statements with Two Signs

Often a number has to lie between two ends, like a temperature that must be neither too low nor too high.

Joining two statements

A statement that a number is greater than (or equal to) one end can be joined with a statement that the same number is less than (or equal to) a larger end. The letter goes in the middle: $-2 \le n$ and $n < 3$ together are $-2 \le n < 3$ . The smaller end is written on the left, as on a number line, so each of the two signs is $<$ or $\le$ .

Example 6: The integers that fit a statement

List all the integers $n$ for which $-2 \le n < 3$ .

Solution

The statement is read " $n$ is greater than or equal to $-2$ and less than 3". It joins $-2 \le n$ and $n < 3$ .

$-2 \le n$ says the same as $n \ge -2$ , so $-2$ is included. $n < 3$ , so 3 is left out.

The integers are $-2$ , $-1$ , 0, 1 and 2.

Example 7: The least and the greatest integer

(a) List the positive integers $n$ for which $n \le 4$ .
(b) Write down the least integer $n$ for which $n > -7$ .
(c) Write down the greatest integer $n$ for which $n \le -2$ .

Solution

(a) $n$ is 4 or less, so 4 is included. The positive integers start at 1.

(b) The integers greater than $-7$ are $-6, -5, -4, \ldots$ , going on without end to the right. $-7$ itself is left out, so the least of them is $-6$ .

(c) The integers less than or equal to $-2$ are $-2, -3, -4, \ldots$ , going on without end to the left. $-2$ is included, and it is the one furthest to the right.

(a) 1, 2, 3, 4 (b) $-6$ (c) $-2$

Example 8: A temperature range

A freezer for ice cream is set so that its temperature, $f$ °C, is at least $-24$ °C and below $-18$ °C.
(a) Write this as one statement with two signs.
(b) Which of these temperatures are allowed?
$-25$ °C, $-24$ °C, $-20$ °C, $-18$ °C

Solution

(a) "At least $-24$ " is $f \ge -24$ , which says the same as $-24 \le f$ . "Below $-18$ " is $f < -18$ . $-24$ is the smaller end, so it is written on the left.

(b) $-25$ is to the left of $-24$ , so it is less than $-24$ . It is not allowed. $-24$ is allowed, because $\le$ allows "equal to".

$-20$ is to the right of $-24$ and to the left of $-18$ , so it is allowed. $-18$ is not less than $-18$ , so it is not allowed.

The number line below shows the four temperatures that were tested, not only the ones that fit. Above each point is "yes" if the temperature is allowed and "no" if it is not.

(a) $-24 \le f < -18$ (b) $-24$ °C and $-20$ °C

Common Mistakes

Writing $-8 > -3$ because 8 is greater than 3. $-8$ is to the left of $-3$ on the number line, so $-8 < -3$ .
Leaving out the "equal to" number. In $n \le 3$ , $n$ can be 3. In $n < 3$ , it cannot.
Mixing up "at least" and "at most". "At least 16" means 16 or more: $a \ge 16$ . "At most 16" means 16 or less: $a \le 16$ .
Reading $-2 \le n$ as " $n$ is less than or equal to $-2$ ". The open end of the sign faces $n$ , so $n$ is greater than or equal to $-2$ .
Joining numbers with signs that point different ways. $3 > -5 < -2$ does not put the three numbers in order. Write $-5 < -2 < 3$ .

Practice

Work each question on paper before you open its solution.

Question 1

Write $<$ or $>$ between the two numbers of each pair.
(a) $-7$ and $-2$
(b) 0 and $-9$
(c) $-15$ and 4
(d) $-1$ and $-100$

ShowHide solution

(a) $-7$ is to the left of $-2$ .

(b) 0 is to the right of every negative number.

(d) Both numbers are negative, and $-100$ is further from zero, so it is the smaller.

(a) $-7 < -2$ (b) $0 > -9$ (c) $-15 < 4$ (d) $-1 > -100$

Question 2

(a) Write $-1 > -7$ the other way round, using $<$ .
(b) Arrange 5, $-6$ , 0 and $-1$ in ascending order, and join them with $<$ .
(c) Arrange $-4$ , 2, $-9$ and 7 in descending order, and join them with $>$ .

ShowHide solution

(a) The greater number, $-1$ , stays at the open end of the sign.

(b) From left to right on the number line, the numbers are $-6$ , $-1$ , 0 and 5.

(c) From right to left on the number line, the numbers are 7, 2, $-4$ and $-9$ . Of the two negative numbers, $-9$ is further from zero, so it is the smaller.

(a) $-7 < -1$ (b) $-6 < -1 < 0 < 5$ (c) $7 > 2 > -4 > -9$

Question 3

Is each statement true or false?
(a) $-3 \le -3$
(b) $-10 \ge -1$
(c) $0 \le -4$
(d) $-2 \ge -6$

ShowHide solution

(a) The two numbers are equal, and $\le$ allows "equal to".

(b) $-10$ is to the left of $-1$ , so it is less than $-1$ .

(d) $-2$ is to the right of $-6$ , so it is greater than $-6$ .

(a) True (b) False (c) False (d) True

Question 4

Which of the signs $<$ , $>$ , $\le$ and $\ge$ can go in the box to make a true statement? Give every sign that can.
(a) $-6 \; \square \; {-6}$
(b) $-9 \; \square \; {-2}$
(c) $3 \; \square \; {-3}$

ShowHide solution

(a) The two numbers are equal, so only the signs that allow "equal to" make a true statement.

(b) $-9$ is to the left of $-2$ , so it is less than $-2$ . Both $<$ and $\le$ are true, because $\le$ needs only one of its two parts.

(a) $\le$ and $\ge$ (b) $<$ and $\le$ (c) $>$ and $\ge$

Question 5

Write each sentence as a statement using $<$ , $>$ , $\le$ or $\ge$ .
(a) More than 200 people came to a school concert. $p$ people came.
(b) Frozen food must be kept at $-18$ °C or colder. It is kept at $t$ °C.
(c) A holiday camp takes children who are at least 7 years old and at most 12 years old. A child at the camp is $a$ years old.

ShowHide solution

(a) "More than 200" does not allow 200 itself.

(b) Colder means a lower temperature, so $t$ is $-18$ or less.

(c) "At least 7" gives $a \ge 7$ , which says the same as $7 \le a$ . "At most 12" gives $a \le 12$ . The two are joined into one statement.

(a) $p > 200$ (b) $t \le -18$ (c) $7 \le a \le 12$

Question 6

Write each sentence as a statement using $<$ , $>$ , $\le$ or $\ge$ .
(a) The speed limit on a road is 50 km/h. A driver who keeps to it drives at $s$ km/h.
(b) A van may enter a car park only if its height, $h$ cm, is no more than 220 cm.
(c) A school bus carries fewer than 40 pupils. It carries $n$ pupils.
(d) To pass a test, a pupil's mark, $m$ , must be not less than 50.

ShowHide solution

(a) A speed limit is the greatest speed allowed: 50 km/h is allowed, and so is any lower speed.

(b) "No more than 220" means 220 or anything less.

(d) "Not less than 50" means 50 or anything greater. It says the same as "at least 50".

(a) $s \le 50$ (b) $h \le 220$ (c) $n < 40$ (d) $m \ge 50$

Question 7

Write in words what each statement says.
(a) The load in a lift is $w$ kg, and $w \le 800$ .
(b) The temperature at the top of a mountain is $T$ °C, and $T < 0$ .
(c) The mark of a pupil who passes a test marked out of 100 is $m$ , and $50 \le m \le 100$ .

ShowHide solution

(a) $\le$ allows 800 and anything less.

(b) $<$ does not allow 0 itself.

(a) The load is at most 800 kg. (b) The temperature is below 0 °C. (c) The mark is at least 50 and at most 100.

Question 8

On a dive for beginners, the height of a diver is $h$ m, where sea level is 0 m and $-12 \le h < 0$ .
(a) Write the statement in words.
(b) Which of these heights are allowed on the dive?
$-15$ m, $-12$ m, $-5$ m, 0 m, 3 m

ShowHide solution

(a) The statement joins $-12 \le h$ and $h < 0$ . $-12 \le h$ says the same as $h \ge -12$ .

(b) $-15$ is to the left of $-12$ , so it is less than $-12$ . It is not allowed.

$-12$ is allowed, because $\le$ allows "equal to". $-5$ is to the right of $-12$ and to the left of 0, so it is allowed.

0 is not less than 0, and 3 is greater than 0. Neither is allowed.

(a) $h$ is greater than or equal to $-12$ and less than 0. The diver is below sea level, and at most 12 m below it. (b) $-12$ m and $-5$ m

Question 9

(a) List all the integers $n$ for which $-4 < n \le 1$ .
(b) List all the negative integers $n$ for which $n \ge -3$ .
(c) Write down the greatest integer $n$ for which $n < -5$ .

ShowHide solution

(a) $n$ is greater than $-4$ , so $-4$ is left out. $n$ is less than or equal to 1, so 1 is included.

(b) $n$ is $-3$ or greater, so $-3$ is included. $n$ is negative, so it is to the left of 0.

(c) The integers less than $-5$ are $-6, -7, -8, \ldots$ They are all to the left of $-5$ , and the one furthest to the right is $-6$ .

(a) $-3$ , $-2$ , $-1$ , 0, 1 (b) $-3$ , $-2$ , $-1$ (c) $-6$

Question 10

(a) List all the integers $n$ for which $-5 < n < 0$ , and show them on a number line.
(b) How many integers $n$ are there for which $-3 \le n \le 3$ ?
(c) The points on the number line below are all the integers $n$ for which $a \le n < b$ , where $a$ and $b$ are integers. Write down $a$ and $b$ .

ShowHide solution

(a) Both signs are $<$ , so $-5$ and 0 are both left out. The number line at the top of this solution shows the integers that are left.

(b) Both signs are $\le$ , so $-3$ and 3 are both included. The integers are $-3$ , $-2$ , $-1$ , 0, 1, 2 and 3.

(c) The least point is $-6$ , and $a \le n$ includes $a$ , so $a$ is $-6$ . The greatest point is $-2$ , and $n < b$ leaves $b$ out, so $b$ is the next integer to the right of $-2$ .

Check: $-6 \le n < -1$ gives $-6$ , $-5$ , $-4$ , $-3$ and $-2$ , which are the five points.

(a) $-4$ , $-3$ , $-2$ , $-1$ (b) 7 integers (c) $a = -6$ and $b = -1$

Question 11

A test is marked out of 100, and every mark is a whole number. Grade B is given for a mark of at least 60 but less than 70. A pupil with grade B has a mark of $m$ .
(a) Write a statement with two signs for $m$ .
(b) Which of these marks are given grade B?
59, 60, 65, 69, 70
(c) Write down the greatest mark that is given grade B.
(d) Use your answer to (c) to write another statement for $m$ that uses $\le$ twice.

ShowHide solution

(a) "At least 60" is $m \ge 60$ , which says the same as $60 \le m$ . "Less than 70" is $m < 70$ .

(b) 59 is less than 60, so it is left out. 60 is included, because $\le$ allows "equal to". 65 and 69 are greater than 60 and less than 70. 70 is not less than 70.

(d) For whole numbers, "less than 70" says the same as "at most 69".

(a) $60 \le m < 70$ (b) 60, 65 and 69 (c) 69 (d) $60 \le m \le 69$

Key Points

$<$ is "less than" and $>$ is "greater than". The open end of the sign faces the greater number, so $-8 < -3$ and $-3 > -8$ say the same thing.
On a number line, the number further to the right is the greater. Of two negative numbers, the one further from zero is the smaller.
Numbers in order can be joined in one statement: $-5 < -2 < 3$ .
$\le$ is "less than or equal to" (at most). $\ge$ is "greater than or equal to" (at least).
A statement with $\le$ or $\ge$ is true when either part is true: $4 \le 4$ is true and $-3 \ge -2$ is false.
$-2 \le n < 3$ joins $-2 \le n$ and $n < 3$ . The integers that fit it are $-2$ , $-1$ , 0, 1 and 2.