Question

Find the value of $x$ which satisfies the equation, ${\log }_{0.5}\frac{3-x}{x+2}<0$

• A. $-2<x<\frac{1}{2}$
• B. $-4<x<1$
• C. $-3<x<2$
• D. none of these

Answer 1

The correct option is A $-2<x<\frac{1}{2}$

Domain is $\left(\frac{3-x}{x+2}\right)>0\Rightarrow x\in (-2,3)$
${\log }_{0.5}\frac{3-x}{x+2}<0$
$\Rightarrow \frac{3-x}{x+2}>{\left(0.5\right)}^0$, since base of log is less than 1.
$\Rightarrow \frac{3-x}{x+2}>1$
$\Rightarrow \frac{3-x}{x+2}-1>0$
$\Rightarrow \frac{3-x-x-2}{x+2}>0$
$\Rightarrow \frac{2x-1}{x+2}<0$
$\Rightarrow -2<x<\frac{1}{2}$