An inequality is a statement that two quantities are not necessarily equal and says which one is larger. It uses one of four symbols. The symbols are strict: equality is not allowed. The symbols = also allow the two sides to be equal. **Symbol** | **Meaning** | **Words often used** | **Example** a>b | a is greater than b | more than, above, exceeds, over | 7>2 a= b | a is greater than or equal to b | at least, no less than, a minimum of | 5 5 a 1. An equation and an inequality behave differently. The equation x+3=7 has exactly one solution, x=4. The inequality x+37. Is x=4 a solution of x+3140 for “a rider’s height, h cm, must be at least 140 cm”. A rider who is exactly 140 cm tall is allowed, so the statement is h 140. Another is to compare negative numbers by their size without the sign: -8 is less than -5 because it lies further left.

Inequalities and their symbols

Secondary 2

An inequality is a statement that two quantities are not necessarily equal and says which one is larger. It uses one of four symbols. The symbols $<$ and $>$ are strict: equality is not allowed. The symbols $\le$ and $\ge$ also allow the two sides to be equal.

Symbol	Meaning	Words often used	Example
$a>b$	$a$ is greater than $b$	more than, above, exceeds, over	$7>2$
$a<b$	$a$ is less than $b$	less than, below, under, fewer than	$-3<1$
$a\ge b$	$a$ is greater than or equal to $b$	at least, no less than, a minimum of	$5\ge5$
$a\le b$	$a$ is less than or equal to $b$	at most, no more than, does not exceed, a maximum of	$4\le9$

Why: numbers increase from left to right along a number line, so $a<b$ means $a$ lies to the left of $b$. On the line above, $-4$ lies to the left of $-1$, so $-4<-1$, even though $4>1$.

An equation and an inequality behave differently. The equation $x+3=7$ has exactly one solution, $x=4$. The inequality $x+3<7$ is true for $x=3$, $x=0$, $x=-2.5$ and every other number less than $4$, so it has infinitely many solutions.

Method

Choose a letter for the quantity and state its unit.
Decide whether the boundary value itself is allowed. “At least” and “at most” allow it; “more than”, “less than”, “above” and “below” do not.
Write the letter first, then the symbol, then the boundary value.
To test whether a number is a solution, substitute it and decide whether the statement is true.

Example

A lift can carry at most $13$ passengers. If $n$ is the number of passengers, then $n\le13$.

A cabin-bag weighing more than $7$ kg is charged extra. If the bag has mass $m$ kg and is charged extra, then $m>7$.

Is $x=4$ a solution of $x+3<7$? $4+3=7$, and $7<7$ is false, so $x=4$ is not a solution. It is a solution of $x+3\le7$, because $7\le7$ is true.

Check: a common error is to write $h>140$ for “a rider’s height, $h$ cm, must be at least $140$ cm”. A rider who is exactly $140$ cm tall is allowed, so the statement is $h\ge140$. Another is to compare negative numbers by their size without the sign: $-8$ is less than $-5$ because it lies further left.

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Symbol	Meaning	Words often used	Example
\(a>b\)	\(a\) is greater than \(b\)	more than, above, exceeds, over	\(7>2\)
\(a<b\)	\(a\) is less than \(b\)	less than, below, under, fewer than	\(-3<1\)
\(a\ge b\)	\(a\) is greater than or equal to \(b\)	at least, no less than, a minimum of	\(5\ge5\)
\(a\le b\)	\(a\) is less than or equal to \(b\)	at most, no more than, does not exceed, a maximum of	\(4\le9\)