Solve this floor function

Algebra Level 4

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

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

\[\] Notations:

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