# Three-part inequalities

## Divide or multiply by negatives

The same rules apply to three part inequalities with regard to dividing or multiplying by a negative you flip both inequality signs.

**Examples**

1. Solve the following three-part inequality

`-5<=3\-2x<=5`

Subtract `-3` from all 3 sides to try and get `x` on its own.

`-5-3<=3-3-2x<=5-3`

`-8<=-2x<=2`

Divide all 3 sides by `-2` to get `x` on its own but you must flip the inequality sign too:

`(-8)/-2>=\(-2x)/-2>=2/-2`

`4>=x>=-1`

**Answer:** `4` is greater than or equal to `x` and `x` is less than or equal to `-1`.

2. Solve

`10>(2x-5)/-4>5`

Multiply all the sides by `-4` to get `x` on its own. Don’t forget to flip the inequality sign.

`-4times10<(-4times(2x-5))/-4<-4times5`

`-40<2x-5<-20`

Add `5` to each side to get `x` on its own.

`-40+5<2x-5+5<-20+5`

`-35<2x<-15`

Divide all three sides by `2` to get `x` on its own.

`(-35)/2<(2x)/2<-15/2`

`-17 1/2<\x<-7\1/2`