The set of all values of \(x\) satisfying the inequality \(\log _{\left(x+\frac{1}{x}\right)}\left[\log _{2}\left(\frac{x-1}{x+2}\right)\right]>0\) is

The base x+1/x must be positive and not 1, so x > 0, which makes the base >= 2. The inequality then needs log2((x-1)/(x+2)) > 1, i.e. (x-1)/(x+2) > 2, i.e. x < -5, which contradicts x > 0. So no solution.