Solve this floor function

Algebra Level 5

1x+12x={9x}+13\large \frac 1{\lfloor x \rfloor} + \frac 1{\lfloor 2 x \rfloor} = \{ 9x \} + \frac 13

If the smallest positive solution of xx satisfying the equation above is of the form mn\frac {m}{n}, where mm and nn are coprime positive integers, find mnm-n.

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