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.

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

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