Question

The number of solutions of $2\left[x\right]=x+2\left\{x\right\}$ is/are

Answer 1

$2\left[x\right]=x+2\left\{x\right\}\Rightarrow 2\left[x\right]=\left[x\right]+\left\{x\right\}+2\left\{x\right\}\Rightarrow \left\{x\right\}=\frac{\left[x\right]}{3}$

As $0\le \left\{x\right\}<1$
$\Rightarrow 0\le \frac{\left[x\right]}{3}<1\Rightarrow 0\le \left[x\right]<3$

Case $1:$
If $\left[x\right]=0$
$\Rightarrow \left\{x\right\}=0$
So, $x=0$

Case $2:$
If $\left[x\right]=1$
$\Rightarrow \left\{x\right\}=\frac{1}{3}$
So, $x=\frac{4}{3}$

Case $3:$
If $\left[x\right]=2$
$\Rightarrow \left\{x\right\}=\frac{2}{3}$
So, $x=\frac{8}{3}$

$\therefore x=0,\frac{4}{3},\frac{8}{3}$ are the required solutions