Inequalities

Inequalities are the relationships between two expressions which are not equal to one another. The symbols used for inequalities are <, >, ≤, ≥ and ≠.

\(7 \textgreater x\) reads as '7 is greater than \(x\)' (or '\(x\) is less than 7', reading from right to left).

\(x \leq -4\) reads as '\(x\) is less than or equal to -4' (or '-4 is greater than or equal to \(x\)', reading from right to left).

\(x \neq 5 \) reads as ‘\(x\) is not equal to 5.’

Inequalities on a number line

Inequalities can be shown on a number line.

Open circles are used for numbers that are less than or greater than (< or >). Closed circles are used for numbers that are less than or equal to and greater than or equal to (≤ or ≥).

For example, this is the number line for the inequality \(x \geq 0\):

Number line, showing x is greater than 0

The symbol used is greater than or equal to (≥) so a closed circle must be used at 0. \(x\) is greater than or equal to 0, so the arrow from the circle must show the numbers that are larger than 0.

Example

Show the inequality \(x \textless 2\) on a number line.

\(x\) is less than (<) 2, which means an open circle at 2 must be used. \(x\) is less than 2, so an arrow below the values of 2 must be drawn in.

Number line, showing x is less than 0
Question

What inequality is shown by this number line?

Number line showing that x is greater than -5 and less than 4

There is a closed circle at -5 with the line showing the numbers that are greater than -5.

This means \(-5 \leq x\) (writing the \(x\) on the right-hand side).

There is also an open circle at 4, with the numbers less than 4 indicated. This means \(x \textless 4\) (writing the \(x\) on the left-hand side).

The line between these two points means that \(x\) satisfies both inequalities, so a double inequality must be created.

Putting \(x\) in the middle of the two inequalities gives \(-5 \leq x \textless 4\).

\(x\) is greater than or equal to -5 and \(x\) is less than 4.