Question

How do you solve `(x+2)/(x+5)>=1` using a sign chart?

Answer 1

The answer is `x in ] -oo,-5 [`

Explanation:

Let's rewrite the equation as

`f(x)=(x+2)/(x+5)-1>=0`

`f(x)=((x+2)-(x+5))/(x+5)>=0`

`f(x)=(x+2-x-5)/(x+5)=-3/(x+5)>=0`

The domain of `f(x)` is `D_f(x)=RR-{-5}`

Let's do the sign chart

`color(white)(aaaa)` `x` `color(white)(aaaa)` `-oo` `color(white)(aaaa)` `-5` `color(white)(aaaa)` `+oo`

`color(white)(a)` `-(x+5)` `color(white)(aaaa)` `+` `color(white)(aaaaa)` `-`

`color(white)(aaaa)` `f(x)` `color(white)(aaaaa)` `+` `color(white)(aaaaa)` `-`

Therefore,

`f(x>=0)` when `x in ] -oo,-5 [`

graph{-3/(x+5) [-18.99, 6.32, -6.33, 6.33]}