Case Study

Algebra Level 4

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

Find the number of real solutions to the equation above.

Notations:

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