Question

The number of real solution(s) of the equation $2\left[x\right]=x+\left\{x\right\}$ is
($[.]$ denotes the greatest integer function and $\{.\}$ denotes the fractional part function)

• A. $1$
• B. $2$
• C. more than $2$ but finite
• D. infinite

Answer 1

The correct option is B $2$

$2\left[x\right]=x+\left\{x\right\}$
$\Rightarrow 2\left[x\right]=\left[x\right]+2\left\{x\right\}[\because x=[x]+\{x\left\}\right]$
$\Rightarrow \left\{x\right\}=\frac{\left[x\right]}{2}\cdots \left(1\right)$

As $0\le \left\{x\right\}<1$,
$0\le \frac{\left[x\right]}{2}<1\Rightarrow 0\le \left[x\right]<2$
$\therefore \left[x\right]=0,1$

If $\left[x\right]=0\Rightarrow x\in [0,1)$
Using equation $\left(1\right)$,
$\left\{x\right\}=0\Rightarrow x=0$

If $\left[x\right]=1\Rightarrow x\in [1,2)$
Using equation $\left(1\right)$, we get
$\left\{x\right\}=\frac{1}{2}\Rightarrow x=\frac{3}{2}$