3. For which values of the variable $x$ is the value of the function

$$
f(x)=\log _{\frac{1}{2}}\left(\frac{x}{x+2}-\frac{1}{x}\right)
$$

negative?
3. The base of the logarithm is $\frac{1}{2}, 01$. We transform and simplify the inequality $\frac{-3 x-2}{x(x+2)}>0$. We sketch the graph of the function $g(x)=\frac{-3 x-2}{x(x+2)}-$ the zero is $x=-\frac{2}{3}$, and the poles are $x_{1}=0, x_{2}=-2$. Finally, we read off for which $x$ it is positive. The solution is $x \in(-\infty,-2) \cup\left(-\frac{2}{3}, 0\right)$.

![](https://cdn.mathpix.com/cropped/2024_06_07_cd02ddf507985fa73630g-11.jpg?height=83&width=1642&top_left_y=1329&top_left_x=207)

![](https://cdn.mathpix.com/cropped/2024_06_07_cd02ddf507985fa73630g-11.jpg?height=68&width=1636&top_left_y=1391&top_left_x=207)

![](https://cdn.mathpix.com/cropped/2024_06_07_cd02ddf507985fa73630g-11.jpg?height=54&width=1639&top_left_y=1441&top_left_x=206)

Determination of the pole.......................................................................................................................................

Graphical representation ....................................................................................................

![](https://cdn.mathpix.com/cropped/2024_06_07_cd02ddf507985fa73630g-11.jpg?height=66&width=1639&top_left_y=1592&top_left_x=208)