<s>[INST] Solve for $x$: $x\lfloor x + 2 \rfloor +\lfloor 2x - 2 \rfloor +3x =12$ Solve for $x$: $$x\lfloor x + 2 \rfloor +\lfloor 2x - 2 \rfloor +3x =12$$
My attempt: I have changed this equation into the fractional part function. so that we know $0 \leq \{x\}<1$. I have final equation $x^2 +7x-x\{x+2\}-\{2x-2\}=14$. How to proceed now?
 [/INST] I would let $x=y+z$, with integer $y$ and $0\leq z<1,z=\{x\}$.  Then either
$$(y+z)(y+2)+2y-2+3y+3z=12$$
or 
$$(y+z)(y+2)+2y-1+3y+3z=12$$
So, for example, $z(y+5)$ is an integer, so only some fractions are allowed.
Also, 12 is between $y(y+2)+2y-2+3y$ and $(y+1)(y+2)+2y-2+3y+3$, for the first case, or something similar for the second case; and $y$ is an integer.
</s>