Question

How do you solve `(x+3)^2/x<=0` using a sign chart?

Answer 1

The answer is `x in ] -oo,0 [`

Explanation:

Let `f(x)=(x+3)^2/x`

The domain of `f(x)` is `D_f(x)=RR-{0}`

`AAx in D_f(x), (x+3)^2>0`

We can make the sign chart

`color(white)(aaaa)` `x` `color(white)(aaaa)` `-oo` `color(white)(aaaaaaa)` `0` `color(white)(aaaaa)` `+oo`

`color(white)(aaaa)` `x` `color(white)(aaaaaaaa)` `-` `color(white)(aaaa)` `color(red)(∥)` `color(white)(a)` `+`

`color(white)(aaaa)` `f(x)` `color(white)(aaaaaa)` `-` `color(white)(aaa)` `color(red)(∥)` `color(white)(a)` `+`

Therefore,

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