Question

The set of all the values of x for which $lo{g}_{\frac{1}{4}}(2-x)>lo{g}_{\frac{1}{4}}\left(\frac{2}{x+1}\right)$ is

• A. $(1,5)$
• B. $(-1,2)$
• C. $(-1,0)\cup (1,\infty )$
• D. None of these

Answer 1

The correct option is C $(-1,0)\cup (1,\infty )$

$\log \frac{1}{4}(2-x)>\log \frac{1}{4}\left(\frac{2}{x+1}\right)$

$2-x<\frac{2}{x+1}$

$\frac{2}{x+1}+x-2>0$

$\frac{2+x^2-x-2}{x+1}>0$

$\frac{x(x-1)}{(x+1)}>0$

$x\in (-1,0)U(1,0)$