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 greater 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`