<s>[INST] What is the number of solution of the equation $\ x^3 - \lfloor x\rfloor = 3$? What is the number of solution of the equation $\ x^3 - \lfloor x\rfloor = 3$ ? (where $\lfloor x \rfloor\ $ is the greatest integer $\le x$)
I tried plotting the graphs of these equations on Desmos graph calculator and that they intersect each other in between x = 1 and 2 but I couldn't figure out a way to get to this conclusion on my own.
Is there any way by which I can determine where these functions intersect?
 [/INST] Here is another answer I came up with:
Notice that $\lfloor x\rfloor$ and $3$ are integers, so $x^3$ is an integer, say $x=\sqrt[3]{y^3+z}$, where $0\leq z<(y+1)^3-y^3$, and y and z are integers. Now we have $y^3+z-y=3$, or $(y-1)y(y+1)=3-z.$ Since $z\geq0$, $(y-1)y(y+1)\leq 3$ and $y\leq1$. However, notice that if $y<-1$, the inequality $z<(y+1)^3-y^3$ fails, so $y\geq-1$, $(y-1)y(y+1)=0$, $z=3$, and $y=1$ to make the inequality true, so  $x=\sqrt[3]4$.
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