Question

The set of all the solutions of the inequality ${\log }_{1-x}(x-2)\ge -1$ is

• A. $(-\infty ,0)$
• B. $(2,\infty )$
• C. $(-\infty ,1)$
• D. $\varphi$

Answer 1

The correct option is D $\varphi$

${\log }_{1-x}(x-2)\ge -1$
Above equation is valid when,
$S_1=\{x-2>0\}$
$S_2=\{1-x>0,1-x\ne 1\}$
Intersection of $S_1$ and $S_2$ is $x=\left\{\varnothing \right\}$.
Ans: D