Question

Solve the following inequalities: $\frac{1}{\left|x\right|-3}<\frac{1}{2}$

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

Answer 1

Answer: (B)

$(-\infty ,-5)\cup (5,\infty )$

Given, $\frac{1}{\left|x\right|-3}<\frac{1}{2}$
$=>\frac{1}{\left|x\right|-3}-\frac{1}{2}<0$
$=>\frac{2-\left|x\right|+3}{(\left|x\right|-3)\times 2}<0$
$=>5-\left|x\right|<0$
Now, when $x>0$ , we have $\left|x\right|=x$
So, $5-x<0$
$=>x>5$ -- (1)
And when $x<0$ , we have $\left|x\right|=-x$
So, $5-(-x)<0$
$5+x<0$
$=>x<-5$ -- (2)
From $\left(1\right),\left(2\right)$
$(-\infty ,-5)\cup (5,\infty )$