Say I want to find $m\in\mathbb{N}$ such that for some $u\in\mathbb{R}$ we have that $(u-\frac{1}{m})^3>2$. Expanding we obtain $u^3-\frac{3u^2}{m}+\frac{3u}{m^2}-\frac{1}{m^3}>2$. Now in every example I see in my textbook they will do something like $u^3-\frac{3u^2}{m}+\frac{3u}{m^2}>u^3-\frac{3u^2}{m}+\frac{3u}{m^2}-\frac{1}{m^3}>2$ and continue simplifying until it's easy to solve for $m$.
This makes absolutely no sense to me since if we find $m$ from the larger expression it is my current opinion that this $m$ will perhaps be to small to satisfy the smaller equation.
DON'T EXPAND! Take the 3rd root.
$(u-\frac{1}{m})^3>2$
$(u-\frac{1}{m}) > 2^{\frac 1 3}$
$ u - 2^{\frac 1 3} > \frac{1}{m}$
$ m > \frac{1}{u - 2^{\frac 1 3}}$
(assuming $m > 0$)