Find the equation solution $\left[ x\right] +\left[ \dfrac{9}{x}\right]=6$ solve this following equation
$$\left[ x\right] +\left[ \dfrac{9}{x}\right]=6$$
 the equation obvious have a root $x=3$. But how to solve other roots?

Obviously $x>0$, suppose that $\lfloor x \rfloor =k$ (clearly $k$ must be between $0$ and $5$), then the solutions are $\lfloor \frac{9}{x} \rfloor = 6-k\iff 6-k\leq \frac{9}{x} < 6-k+1\iff \frac{9}{6-k+1}< x \leq \frac{9}{6-k}$.
So the solution set is $\bigcup\limits_{k=0}^{5}( (\frac{9}{6-k+1},\frac{9}{6-k}]\cap [k,k+1))$.
This is just a simple and fast calculation now.
