I am trying to solve

$\displaystyle log_{0.5}\frac{x+3}{x-1} \geq 0. $

$\displaystyle \frac{log\frac{x+3}{x-1}}{log_{0.5}} \geq 0$

Because log 0.5 is always negative, the entire numerator must be negative as well for the entire expression to be positive. Therefore,

$\displaystyle 0 < \frac{x+3}{x-1} \leq 1. $

For $\displaystyle 0 < \frac{x+3}{x-1} $ I reached the conclusion that $\displaystyle x < -3 $ or $\displaystyle x > 1$

But I'm stuck at $\displaystyle \frac{x+3}{x-1} \leq 1$

if I multiply the denominator over it will be x + 3<x -1, which makes no sense to me

Please, if someone could point out where I made my error, I've spent a lot of time on this question and the due date is approaching fast. Thank you!