Question

How do you solve `(x+3)/(x-1)>=0` using a sign chart?

Answer 1

The solution is `x in (-oo,-3] uu(1,+oo)`

Explanation:

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

Let's build the sign chart

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

`color(white)(aaaa)` `x+3` `color(white)(aaaaa)` `-` `color(white)(aaa)` `0` `color(white)(aaa)` `+` `color(white)(aaaaa)` `+`

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

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

Therefore,

`f(x)>=0`, when `x in (-oo,-3] uu(1,+oo)`

graph{(x+3)/(x-1) [-16.02, 16.01, -8.01, 8.01]}