1. Solve the inequality: $\log _{|x-1|}\left(\frac{x-2}{x}\right)>1$. In the answer, indicate the largest negative integer that is a solution to this inequality.
1. Find the domain: $(x-2) x>0$. From this, $x<0$ or $x>2$. If $|x-1|<1$, then $0<x<2$. But if $|x-1|>1$, then $x<0$ or $x>2$. Therefore, the inequality is equivalent to the system:

$\left\{\begin{array}{c}|x-1|>1 \\ \frac{x-2}{x}>|x-1|\end{array} \Leftrightarrow\right.$ a) $\left\{\begin{array}{c}x<0 \\ \frac{x-2}{x}>1-x\end{array}\right.$ or b) $\left\{\begin{array}{c}x>2 \\ \frac{x-2}{x}>x-1\end{array}\right.$

From this, the solution to the first system a) is $-\sqrt{2}<x<0$, and the second system b) has no solutions. Answer: $\{-1\}$