Question

How do you solve `3/(x+1)<=3` using a sign chart?

Answer 1

The solution is `x in (-oo,-1) uu [0,+oo)`

Explanation:

We cannot do crossing over

Let's rearrange the inequality

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

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

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

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

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

Let's build the sign chart

`color(white)(aaaa)` `x` `color(white)(aaaa)` `-oo` `color(white)(aaaaaa)` `-1` `color(white)(aaaaaaaa)` `0` `color(white)(aaaaa)` `+oo`

`color(white)(aaaa)` `-3x` `color(white)(aaaaaa)` `+` `color(white)(aaaa)` `||` `color(white)(aaaa)` `+` `color(white)(aa)` `0` `color(white)(aa)` `-`

`color(white)(aaaa)` `x+1` `color(white)(aaaaa)` `-` `color(white)(aaaa)` `||` `color(white)(aaaa)` `+` `color(white)(aa)` `0` `color(white)(aa)` `+`

`color(white)(aaaa)` `f(x)` `color(white)(aaaaaa)` `-` `color(white)(aaaa)` `||` `color(white)(aaaa)` `+` `color(white)(aa)` `0` `color(white)(aa)` `-`

Therefore,

`f(x)<=0` when `x in (-oo,-1) uu [0,+oo)` graph{3/(x+1)-3 [-19.3, 21.26, -12.45, 7.82]}