Question

How do you solve `(3x+5)/(6-2x)<=0`?

Answer 1

The answer is `x in ] -oo,-5/3 ] uu ] 3, +oo [`

Explanation:

Let `f(x)=(3x+5)/(6-2x)`

The domain of `f(x)` is `D_f(x)=RR-{3}`

Now, we can make the sign chart

`color(white)(aaaa)` `x` `color(white)(aaaa)` `-oo` `color(white)(aaaa)` `-5/3` `color(white)(aaaaaaa)` `3` `color(white)(aaaaaaaaaaa)` `+oo`

`color(white)(aaaa)` `3x+5` `color(white)(aaaa)` `-` `color(white)(aaaaaa)` `+` `color(white)(aa)` `∥` `color(white)(aaaa)` `+`

`color(white)(aaaa)` `6-2x` `color(white)(aaaa)` `+` `color(white)(aaaaaa)` `+` `color(white)(aa)` `∥` `color(white)(aaaa)` `-`

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

Therefore,

`f(x)<=0`, when `x in ] -oo,-5/3 ] uu ] 3, +oo [`