Prove or disprove an inequality involving complex numbers

Question

Consider following inequality:

$|z_1 + z_2|\ge|z_1| - |z_2| \ \ \ \ \ \ z_1,z_2 \in \mathbb{C}$

Is that true always? I tried many examples and it was correct but I don't know how to prove that if it's right.

Answer 1

Use the fact that$$\lvert z_1\rvert=\bigl\lvert z_1+z_2+(-z_2)\bigr\rvert\leqslant\lvert z_1+z_2\rvert+\lvert z_2\rvert.$$

Answer 2

Obviously it's true, since $$|z_1+z_2|+|-z_2|\geq|z_1|$$$$|z_1+z_2|\geq|z_1|-|z_2|$$

Answer 3

You should use the Reverse Triangle Inequality, say $|z_1\pm z_2|\ge \big||z_1|-|z_2|\big|$.

Then for any real number $|x|\ge x$.