Prove a complex inequality

Question

I need to prove that \begin{gather*} | z^4 + z + 1 | \geq |z^4| - |z| -1 \end{gather*} with $z = (r\exp (it))$ and $r \geq 2^\frac{1}{3}$

It seems like I only have to use the reversed triangle inequality, then it will be very easy. But it is explicitly given that $r \geq 2^\frac{1}{3}$. Is this inequality untrue when $r\leq 2^\frac{1}{3}$? I lack intution here.

Answer 1

It is true in general: $$ |z^4|=|z^4+z+1-z-1|\leq|z^4+z+1|+|-z|+|-1|=|z^4+z+1|+|z|+1. $$ Perhaps, the other information is relevant for something else?

Answer 2

Using triangle inequality i.e. $$|a|-|b|\leq |a+b|$$ In this case when you apply this inequality it yields $$|z^4+z+1|\geq |z^4+z|-|1|\geq |z^4|-|z|-|1|=|z|^4-|z|-1$$