Question

How do you solve `(3x+2)/(x-4)<0`?

Answer 1

The solution is `x in (-2/3,4)`

Explanation:

Let `f(x)=(3x+2)/(x-4)`

We build a sign chart

`color(white)(aaaa)` `x` `color(white)(aaaa)` `-oo` `color(white)(aaaa)` `-2/3` `color(white)(aaaaaaa)` `4` `color(white)(aaaaaaaa)` `+oo`

`color(white)(aaaa)` `3x+2` `color(white)(aaaaa)` `-` `color(white)(aaaaa)` `+` `color(white)(aaa)` `||` `color(white)(aaaa)` `+`

`color(white)(aaaa)` `x-4` `color(white)(aaaaaa)` `-` `color(white)(aaaaa)` `-` `color(white)(aaa)` `||` `color(white)(aaaa)` `+`

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

Therefore,

`f(x)<0`, when `x in (-2/3,4)`

graph{(3x+2)/(x-4) [-28.86, 28.85, -14.43, 14.45]}