<s>[INST] Solve for equations involving floor function 
Solve for real $x$: $$\frac{1}{\lfloor x \rfloor} + \frac{1}{\lfloor 2x \rfloor} = x - \lfloor x \rfloor + \frac{1}{3}$$

Hello! I hope everybody is doing well. I was not able to solve the above problem. And this problem becomes even more difficult with the fact that $x $ is not necessarily a positive real.
Here is what I did: 

Let $x=p+r, 0 \leq r <1$ Now we divide into cases whether $r $ is less than $0.5 $ or not and I reach here(1st part): $0 \frac{3}{2p} - \frac13 < \frac12$. Now I could have done that $\frac95<2\le [x]\le4<\frac92 $ but since $x$ is not necessarily positive, I guess this is not always true.

Any help would be appreciated. Thanks. 
 [/INST] I would approach the problem in this way.
Let's put
$$
{1 \over {\left\lfloor x \right\rfloor }} + {1 \over {\left\lfloor {2x} \right\rfloor }} = x - \left\lfloor x \right\rfloor  + {1 \over 3} = \left\{ x \right\} + {1 \over 3} = t
$$
Since the fractional part of $x$ has values in the range $[0, \, 1)$ we have the the RHS is
$$
0 \le \left\{ x \right\} = t - {1 \over 3} < 1\quad  \Rightarrow \quad {1 \over 3} \le t < {4 \over 3}
$$
At the same time, we know that
$$
x - 1 < \left\lfloor x \right\rfloor  \le x
$$
thus for the LHS we get this set of inequalities
$$
\left\{ \matrix{
  {1 \over 3} \le {1 \over {\left\lfloor x \right\rfloor }} + {1 \over {\left\lfloor {2x} \right\rfloor }} = t < {4 \over 3} \hfill \cr 
  {1 \over x} + {1 \over {2x}} \le {1 \over {\left\lfloor x \right\rfloor }} + {1 \over {\left\lfloor {2x} \right\rfloor }} < {1 \over {x - 1}} + {1 \over {2x - 1}} \hfill \cr}  \right.
$$
Translating this into inequalities for $x$, we shall have that
$$
\left\{ \matrix{
  {1 \over 3} < {1 \over {x - 1}} + {1 \over {2x - 1}} \hfill \cr 
  {1 \over x} + {1 \over {2x}} < {4 \over 3} \hfill \cr}  \right.
$$
Solving the quadrics give
$$
\left\{ \matrix{
  x \in \left( {1/2,\;3 - \sqrt {22} /2} \right) \cup \left( {1,\;3 + \sqrt {22} /2} \right) \hfill \cr 
  x \in \left( { - \infty ,\;0} \right) \cup \left( {9/8,\;\infty } \right) \hfill \cr}  \right.\quad  \Rightarrow \quad x \in \left( {9/8,\;3 + \sqrt {22} /2} \right) \approx \left( {1.1,\;5.34} \right)
$$
So we have just to find the possible solutions for 
$$
\left\{ x \right\} = {1 \over {\left\lfloor x \right\rfloor }} + {1 \over {\left\lfloor {2x} \right\rfloor }} - {1 \over 3}
 \quad \left| {\;x \in \left[ {1,\;3/2} \right)\,\; \cup \;\left[ {3/2,\;2} \right)\; \cup \; \cdots \,\; \cup \;\left[ {5,\;11/2} \right)} \right.
$$
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