Question

The set of real values of x for which $lo{g}_{0.2}\frac{x+2}{x}\le 1$ is

• A. $(-\infty ,-\frac{5}{2}]\cup (0,+\infty )$
• B. $[\frac{5}{2},+\infty )$
• C. $(-\infty ,-2)\cup (0,+\infty )$
• D. $Noneofthese$

Answer 1

The correct option is A $(-\infty ,-\frac{5}{2}]\cup (0,+\infty )$

$lo{g}_{0.2}\frac{x+2}{x}\le 1$ ....(i)
For log to be defined, $\frac{x+2}{x}>0\Rightarrow x>0$ or
x < -2
Now from(i), $lo{g}_{0.2}\frac{x+2}{x}\le lo{g}_{0.2}0.2$
$\Rightarrow \frac{x+2}{x}\ge 0.2$ .....(ii)
Case (i) x > 0
From (ii),$x+2\ge 0.2x$
$\Rightarrow 0.8x\ge -2$
$\Rightarrow x\ge -\frac{5}{2}.$
Case (ii) x < -2
From(ii), $x+2\le 0.2x\Rightarrow 0.8x\le -2\Rightarrow x\le -\frac{5}{2}$
$\Rightarrow xϵ(-\infty ,-\frac{5}{2}]\cup (0,\infty )$.