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?

Notice that $x-1 < \lfloor x \rfloor \leq x$, so $x^3-x+1 > x^3- \lfloor x \rfloor \geq x^3-x$, and $x^3-x+1 > 3 \geq x^3-x$. The first part of the inequality shows that $x<2$, and the second shows that $x \geq 1$. This means that $\lfloor x \rfloor=1$, so $x^3-1=3$ and $x=\sqrt[3]4$.
