<s>[INST] Number of roots of $\lfloor\frac x3\rfloor=\frac x2$ How many roots does the equation $$\left\lfloor\frac x3\right\rfloor=\frac x2$$ have?

*

*$1$

*$2$

*$3$

*infinitely many

I checked that $x=0$ and $x=-2$ are the answers, so I think the answer is $(2)$. but I don't know how to solve the problem in general.
 [/INST] 
$$\left\lfloor \frac{x}{3} \right\rfloor = \frac{x}{2}$$

Hint Since $\left\lfloor \frac{x}{3} \right\rfloor$ is an integer, any solution $x$ must be a multiple of $2$. Now, for $x > 0$ we have $$\left\lfloor \frac{x}{3} \right\rfloor \leq \frac{x}{3} < \frac{x}{2} ,$$ and for $x \leq -6$ we have
$$\left\lfloor \frac{x}{3} \right\rfloor > \frac{x}{3} - 1 \geq \frac{x}{2} ,$$ leaving only a few possibilities to check manually.
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