Question

The solution set of the inequality ${\log }_{0.5}\sqrt{\frac{x-4}{x-3}}<{\log }_{0.5}2$ is

• A. $(0,3)$
• B. $(0,\frac{8}{3})$
• C. $(\frac{8}{3},3)$
• D. $(\frac{8}{3},3)\cup (4,\infty )$

Answer 1

The correct option is C $(\frac{8}{3},3)$

Given,
${\log }_{0.5}\sqrt{\frac{x-4}{x-3}}<{\log }_{0.5}2$
$\log$ function is defined when $\sqrt{\frac{x-4}{x-3}}>0$
$\Rightarrow x\in (-\infty ,3)\cup (4,\infty )$

Now, $\frac{1}{2}{\log }_{0.5}\left(\frac{x-4}{x-3}\right)<{\log }_{0.5}2$
$\Rightarrow {\log }_{0.5}\left(\frac{x-4}{x-3}\right)<{\log }_{0.5}4$
$\Rightarrow \frac{x-4}{x-3}>4$
$\Rightarrow \frac{x-4-4x+12}{x-3}>0$
$\Rightarrow \frac{3x-8}{x-3}<0$
$\Rightarrow x\in (\frac{8}{3},3)$