Example + Given the function $f(x)=\frac{1}{x+1}+\log _{\frac{1}{\top}} \frac{x}{2-x}$, solve the inequality $f\left[x\left(x-\frac{1}{2}\right)\right]>\frac{1}{2}$.
Analysis This problem involves a combination of fractional and logarithmic inequalities. At first glance, it seems difficult to solve, but the key to solving this problem is recognizing that the function is monotonically decreasing within its domain.

Solution It is easy to prove that the function $y=f(x)$ is monotonically decreasing in its domain $(0,2)$, and $f(1)=\frac{1}{2}$. Therefore, the original inequality $f\left[x\left(x-\frac{1}{2}\right)\right]>f(1)$ is equivalent to $\left\{\begin{array}{l}x\left(x-\frac{1}{2}\right)<1 \\ 0<x\left(x-\frac{1}{2}\right)<2\end{array} \Rightarrow\left\{x \left\lvert\, \frac{1}{2}<x<\frac{1+\sqrt{17}}{4}\right.\right\}\right.$.