Question

The number of solutions of the equation $\left[3x\right]-[x+1]=3x$, where $[.]$ denotes greatest integer function, is equal to

Answer 1

$\left[3x\right]-[x+1]=3x$
$\Rightarrow -[x+1]=3x-\left[3x\right]$
$\Rightarrow -[x+1]=\left\{3x\right\}$
where $\{.\}$ denotes fractional integer function
Now, $0\le \left\{3x\right\}<1$
$\therefore -[x+1]=0\Rightarrow [x+1]=0$
$\Rightarrow 0\le x+1<1$
$-1\le x<0\Rightarrow -3\le 3x<0$
$\Rightarrow \left[3x\right]=-3,-2,-1$
Now,
$\left[3x\right]-0=3x$
$\Rightarrow 3x=-3,-2,-1$
$x=-\frac{3}{3},-\frac{2}{3},-\frac{1}{3}$
$x=-1,-\frac{2}{3},-\frac{1}{3}$