Применим метод рационализации
0,2x\ne 1} \atop {x>0,x\ne 1,x>\frac{2}{5}}} \right. \\\\(log_{2x}(5x-2)-1)(x-1) \geq 0\\\\(2x-1)(5x-2-2x)(x-1) \geq 0\\\\(2x-1)(3x-2)(x-1) \geq 0\\\\---(\frac{1}{2})+++(\frac{2}{3})---(1)+++\\\\x\in (\frac{1}{2},\frac{2}{3}\, ]U(1,+\infty)" alt="log_{x}log_{2x}(5x-2) \geq 0\; ,\; \; ODZ:\; \; \left \{ {{2x>0,2x\ne 1} \atop {x>0,x\ne 1,x>\frac{2}{5}}} \right. \\\\(log_{2x}(5x-2)-1)(x-1) \geq 0\\\\(2x-1)(5x-2-2x)(x-1) \geq 0\\\\(2x-1)(3x-2)(x-1) \geq 0\\\\---(\frac{1}{2})+++(\frac{2}{3})---(1)+++\\\\x\in (\frac{1}{2},\frac{2}{3}\, ]U(1,+\infty)" align="absmiddle" class="latex-formula">