MHB Reverse triangle inequality with a + sign

[ognik]

Thought I knew this, but am confused by the following example:
Show $ |z^3 - 5iz + 4| \ge 8 $
The example goes on: $ |z^3 - 5iz + 4| \ge ||z^3 - 5iz| - |4|| $, using the reverse triangle inequality
It's probably right, but I don't get why the +4 can just be made into a -4 ?

 

[Euge]

The reverse triangle inequality for the complex numbers states $||a|-|b|| \le |a-b|$ for all complex $a$ and $b$. To see this, fix $a, b\in \Bbb C$. Then $|a| = |(a - b) + b| \le |a-b| + |b|$; here, the triangle inequality was applied in the last step. So $|a| - |b| \le |a-b|$. Similarly $|b| - |a| \le |b - a| = |a - b|$. Hence $||a| - |b|| \le |a-b|$.

If we replace $b$ by $-b$ in the reverse triangle inequality above we obtain $||a| -|-b|| \le |a-(-b)|$, i.e., $||a| - |b|| \le |a+b|$. So by letting $a = z^3 - 5iz$ and $b = 4$, we get $||z^3 - 5iz| - |4|| \le |z^3-5iz + 4|$, as desired.

 

[ognik]

Thanks Euge, perfectly clear now.