Question

If ${\log }_{0.5}\frac{3-x}{x+2}<0$, then the number of integral values of x is

• A. 1
• B. 2
• C. 4
• D. 3

Answer 1

The correct option is B 2

$\mathrm{Given}\mathrm{that},{\log }_{0.5}\frac{3-x}{x+2}<0\mathrm{For}{\log }_{0.5}\frac{3-x}{x+2}<0\mathrm{to}\mathrm{be}\mathrm{defined}\frac{3-x}{x+2}>0\Rightarrow \frac{x-3}{x+2}<0\Rightarrow -2<x<3....\left(1\right]{\log }_{0.5}\frac{3-x}{x+2}<0\Rightarrow \frac{3-x}{x+2}>1\Rightarrow \frac{3-x}{x+2}-1>0\Rightarrow \frac{3-x-x-2}{x+2}>0\mathrm{or},\frac{2x-1}{x+2}<0\mathrm{or},-2<x<\frac{1}{2}...\left(2\right]\mathrm{From}\left(1\right]\mathrm{and}\left(2\right]\mathrm{we}\mathrm{get}x\in \left(-2,\frac{1}{2}\right)\mathrm{The}\mathrm{integral}\mathrm{values}\mathrm{of}x\mathrm{are}-1\mathrm{and}0.$