Question

How do you solve `(3x+6)/(2x-12)<=0` using a sign chart?

Answer 1

The solution is `x in [-2,6)`

Explanation:

Let `f(x)=(3x+6)/(2x-12)=(3(x+2))/(2(x-6))`

Let's build the sign chart

`color(white)(aaaa)` `x` `color(white)(aaaa)` `-oo` `color(white)(aaaaaa)` `-2` `color(white)(aaaaaaa)` `6` `color(white)(aaaaaa)` `+oo`

`color(white)(aaaa)` `x+2` `color(white)(aaaaaa)` `-` `color(white)(aaa)` `0` `color(white)(aaa)` `+` `color(white)(aaaaaa)` `+`

`color(white)(aaaa)` `x-6` `color(white)(aaaaaa)` `-` `color(white)(aaa)` `0` `color(white)(aaa)` `-` `color(white)(aa)` `||` `color(white)(aaa)` `+`

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

Therefore,

`f(x)<=0` when `x in [-2,6)`

graph{(3x+6)/(2x-12) [-22.8, 22.83, -11.4, 11.4]}