Question

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

Answer 1

The answer is `x in [-1, 0 [`

Explanation:

We cannot cross over.

Let's rewrite the inequality

`(x+3)/x+2<=0`

`(x+3+2x)/(x)<=0`

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

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

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

Let's do the sign chart

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

`color(white)(aaaa)` `x+1` `color(white)(aaaa)` `-` `color(white)(aaaaaa)` `+` `color(white)(aa)` `∥` `color(white)(aa)` `+`

`color(white)(aaaa)` `x` `color(white)(aaaaaaaa)` `-` `color(white)(aaaaaa)` `-` `color(white)(aa)` `∥` `color(white)(aa)` `+`

`color(white)(aaaa)` `f(x)` `color(white)(aaaaaa)` `+` `color(white)(aaaaa)` `-` `color(white)(aa)` `∥` `color(white)(aa)` `+`

Therefore,

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