**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 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 \(x \geq 0\):

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.

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.

- Question
What inequality is shown by this number line?

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.