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 $$x \geq 0$$:

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.

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.