Question

Solve the inequality: ${\log }_{1-x}(x-2)\ge -1$

Answer 1

Given inequality

${\log }_{1-x}(x-2)\ge -1$

The left hand side of inequality is defined for $x$ which satisfy the following
$1-x>0,x-2>0,1-x\ne 1$

$\Rightarrow x<1,x>2,x\ne 0$
Clearly, there is no single value of $x$ for which these inequalities are satisfied.
Thus set of its solution is empty