Question

Solve the equation $\left|\frac{x}{x-1}\right|+\left|x\right|=\frac{x^2}{|x-1|}$. Number of non negative integer values excluded from the solution to the given equation is

Answer 1

Let $f\left(x\right)=\frac{x}{x-1}andg\left(x\right)=x$ then $f\left(x\right)+g\left(x\right)=\frac{x}{x-1}+x$
$=\frac{x^2}{x-1}$
$\therefore$ The given equation can be reduced in the form $\left|f\left(x\right)\right|\left|g\left(x\right)\right|=|f\left(x\right)+g\left(x\right)|$
Hence, $f\left(x\right).g\left(x\right)\ge 0$
$\Rightarrow \frac{x^2}{x-1}\ge 0$
By wavy curve method,
$\therefore x\in (1,\infty )\cup \left\{0\right\}$