Solve this floor function

Algebra Level 5

$\large \frac 1{\lfloor x \rfloor} + \frac 1{\lfloor 2 x \rfloor} = \{ 9x \} + \frac 13$

If the smallest positive solution of $x$ satisfying the equation above is of the form $\frac {m}{n}$ , where $m$ and $n$ are coprime positive integers, find $m-n$ .

Notations:

$\lfloor \cdot \rfloor$ denotes the floor function.
$\{ \cdot \}$ denotes the fractional part function.