Question

The number of real values of $ x $ which satisfy the equation $ |\frac{x}{x-1}| + |x| = \frac{x}{|x-1|} $ is

• A. $2$
• B. $1$
• C. $infinite$
• D. $zero$

Answer 1

Answer: (B)

Given $\left|\frac{x}{x-1}\right| + \left|x\right| = \frac{x}{\left|x-1\right|} $
$ \left(i\right) when x > 1$,
$ \frac{x}{x-1} + x = \frac{x}{x-1}$
$\Rightarrow x= 0$, does not exist
$ \left(ii\right)$ When $0 \le x < 1$
$ \frac{x}{1- x}+x = \frac{x}{ 1-x} $
$\Rightarrow x = 0 $
$\left(iii\right)$ when $-\infty < x < 0 $
$ \therefore \frac{-x}{-\left(x-1\right)} -x = \frac{x}{-\left(x-1\right)} $
$ \Rightarrow \frac{2x}{\left(x-1\right)} -x = 0 $
$ \Rightarrow x\left[\frac{2-x+1}{x-1}\right] = 0$
$x = 0, x = 3$, does not exist
Hence, only one solution exist.