# Three-part inequalities on a number line

For an expression like

`-1<=x<5`

Split this into two i.e.

`-1<=x`

And

`x<5`

`-1<=x` this reads as `-1` is less than or equal to `x`

`x<5` this reads `x` is less than `5`

These can be shown on a number line as

Or as:

i.e. without an arrow.

**Example 1**

Write down the inequality shown in the diagram below:

Because `x` is the line between the two circles this can be read as:

The ring above `-5` shows that `-5` is less than `x`

`(-5<\x)`

The ring above `+5` shows that `x` is less than `5`

`(x<\5)`

Joining the two together

`-5<\x<5`

**Answer:**

`-5<\x<5`

**Example 2**

Write down the inequality shown in the diagram below:

Because `x` is the line between the two inequalities this can be read as

The ring above `-4` shows that `-4` is less than `x`

`(-4<\x)`

The circle above the `3` shows that `x` is less than or equal to `3`

`(x<=3)`

Joining the two together

`-4<\x<=3`

**Answer:**

`-4<\x<=3`

# Three-part inequalities on a number line

For an expression like

`-1<=x<5`

Split this into two i.e.

`-1<=x`

And

`x<5`

`-1<=x` this reads as `-1` is less than or equal to `x`

`x<5` this reads `x` is less than `5`

These can be shown on a number line as:

If you join these arrows up

**BUT NOTE: **

We normally leave the `x` out

**Example 1**

Write down the inequality shown in the diagram below:

Because `x` is the line between the two circles this can be read as:

The ring above `-5` shows that `-5` is less than `x`

`(-5<\x)`

The ring above `+5` shows that `x` is less than `5`

`(x<\5)`

Joining the two together

`-5<\x<5`

**Answer:**

`-5<\x<5`

**Example 2**

Write down the inequality shown in the diagram below:

Because `x` is the line between the two inequalities this can be read as

The ring above `-4` shows that `-4` is less than `x`

`(-4<\x)`

The circle above the `3` shows that `x` is less than or equal to `3`

`(x<=3)`

Joining the two together

`-4<\x<=3`

**Answer:**

`-4<\x<=3`