Question

The solution set of ${\log }_{(1-x)}(x-2)\ge -1$ is

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

Answer 1

The correct option is C $x=\varphi$

${\log }_{(1-x)}(x-2)\ge -1$
$\log$ function is defined when
$\left(i\right)1-x>0,1-x\ne 1$
$\Rightarrow x<1,x\ne 0$
$\left(ii\right)x-2>0$
$\Rightarrow x>2$
Clearly, no value of $x$ for which these inequalities are satisfied.
Thus, $x=\varphi$