Question

Solve: $2\left[x\right]=\left\{4\right\}+x$, where [ ] represents greatest integer function and { } represents fractional part function, for possible values of x

Answer 1

$2\left[x\right]=4\left\{x\right\}+x$

Let $x=u+v$

where $v$ is the fractional part between $0$ and $1$ and $u$ is an integer.

$\Rightarrow 2u=4v+u+v$

$\Rightarrow u=5v$

As $u$ is an integer and $v$ is a fraction.

$\Rightarrow 0\le v=\frac{u}{5}<1$

$\Rightarrow 0\le u\le 4$

$x=u+v=u+\frac{u}{5}=\frac{6u}{5}$

$\Rightarrow x=0,\frac{6}{5},\frac{12}{5},\frac{18}{5},\frac{24}{5}$