Mathematics

The key is to recognize that $|z|$ appears naturally when you use the triangle inequality on the expression $z + \frac{4}{z}$. Let $r = |z|$; then $\left|\frac{4}{z}\right| = \frac{4}{r}$. The constraint $\left| z + \frac{4}{z} \right| = 2$ limits how $z$ and $\frac{4}{z}$ can combine, which in turn constrains $r$. By considering when these two terms align or oppose each other, you'll find the extreme values of $|z|$.

The key is to recognize that $|z|$ appears naturally when you use the triangle inequality on the expression $z + \frac{4}{z}$. Let $r = |z|$; then $\left|\frac{4}{z}\right| = \frac{4}{r}$. The constraint $\left| z + \frac{4}{z} \right| = 2$ limits how $z$ and $\frac{4}{z}$ can combine, which in turn constrains $r$. By considering when these two terms align or oppose each other, you'll find the extreme values of $|z|$.