Решите пожалуйста неравенство.

Метод рационализации.

$log_{x} \frac{2x-1}{x+4} \geq 0\; \; \Leftrightarrow \; \; \left\{\begin{array}{c}x\ \textgreater \ 0\; ,\; x\ne 1\\\frac{2x-1}{x+4}\ \textgreater \ 0\\(x-1)(\frac{2x-1}{x+4}-1) \geq 0\end{array}\right \\\\\\\left\{\begin{array}{c}x\ \textgreater \ 0,\; x\ne 1\\x\in (-\infty ,-4)\cup (\frac{1}{2},+\infty )\\(x-1)\cdot \frac{2x-1-x-4}{x+4} \geq 0\end{array}\right \; \; \left\{\begin{array}{cc}x\in (\frac{1}{2},1)\cup (1,+\infty )\\\frac{(x-1)(x-5)}{x+4} \geq 0\end{array}\right$

$\left\{\begin{array}{cc}x\in (\frac{1}{2},1)\cup (1,+\infty )\\x\in (-4,1\, ]\cup [\, 5,+\infty )\\\end{array}\right \; \; \Rightarrow \; \; x\in (\frac{1}{2},1)\cup [\, 5,+\infty )$