Question

How do you solve `6/(x+1)<-3`?

Answer 1

The answer is `=x in (-3,-1)`

Explanation:

We cannot do crossing over.

Let's simplify the inequality

`6/(x+1)<-3`

`6/(x+1)+3<0`

`(6+3(x+1))/(x+1)<0`

`(6+3x+3)/(x+1)<0`

`(3x+9)/(x+1)<0`

`(3(x+3))/(x+1)<0`

Let `f(x)=(3(x+3))/(x+1)`

We can build the sign chart

`color(white)(aaaa)` `x` `color(white)(aaaa)` `-oo` `color(white)(aaaa)` `-3` `color(white)(aaaaaaa)` `-1` `color(white)(aaaaaaa)` `+oo`

`color(white)(aaaa)` `x+3` `color(white)(aaaaaa)` `-` `color(white)(aaaa)` `+` `color(white)(aaaa)` `||` `color(white)(aaaa)` `+`

`color(white)(aaaa)` `x+1` `color(white)(aaaaaa)` `-` `color(white)(aaaa)` `-` `color(white)(aaaa)` `||` `color(white)(aaaa)` `+`

`color(white)(aaaa)` `f(x)` `color(white)(aaaaaaa)` `+` `color(white)(aaaa)` `-` `color(white)(aaaa)` `||` `color(white)(aaaa)` `+`

Therefore,

`f(x)<0` when `x in (-3,-1)`