# Two-part inequalities

## Divide or multiply by negatives

**Examples**

1. Solve

`-12x<=4`

Divide both sides by `-12` to get `x` on its own.

`(-12x)/-12>=(4)/-12`

`x>=-(1)/3`

(Notice how we reversed the inequality)

2. Solve

`11-2x>=3x+16`

Take `11` from both sides

`-11+11-2x>=3x+16-11`

`-2x>=3x+16-11`

`-2x>=3x+5`

Take `3x` from both sides

`-2x-3x>=3x-3x+5`

`-5x>=5`

DIVIDE BY `-5` (careful)

`(-5x)/-5<=(5)/-5`

`x<=-1`

(Notice how we reversed the inequality)

3. Solve

`-2(4x +1)<10`

Start by multiplying out using BIDMAS

Therefore

`-2times4x+(-2)times1<10`

`-8x-2<10`

`-8x<12`

Divide by `-8` to get `x` on its own. **But** don’t forget to flip the sign.

`(-8x)/-8>(12)/-8`

`x>\-12/8`

And now simplify the fraction

`x>\-3/2`

`x>\-1\1/2`

4. Solve

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

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

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

`2x-5<-20`

Add `5` to both sides to get `x` on its own.

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

`2x<-15`

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

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

`x<\-7\1/2`