Inequalities

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

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).

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 y \textless 2 on a number line.

y is less than (<) 2, which means an open circle at 2 must be used. y 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.