A third triangle inequality?

Question

I was studying about complex numbers when I encountered this expression in my notebook, $$|a+b| \geq ||a|-|b||$$ It's different from the two triangle inequalities I already knew i.e. $$|a+b| \leq |a|+|b|$$ and $$|a-b| \geq ||a|-|b||$$ where $a$ and $b$ are any two complex numbers and $|.|$ represents the modulus function.

I couldn't find this inequality on the internet and even tried to prove it myself but don't know how to proceed.

So my question is

1. Is this expression right or I had just made some mistake copying it from the board?
2. If it's right, how to prove it?

Thanks for help:)

Answer 1

Take $-b$ instead of $b$ in your last equation (you can do that as your inequality holds for every complex number).

This means that the last inequality is equivalent to the one you're asking about.

Answer 2

No, it's not different.

$$|\pm x\pm y|=|(\pm x)+(\pm y)|\ge||\pm x|-|\pm y||=||x|-|y||.$$