Complex number triangular inequality

Question

For two complex numbers $z_1$ and $z_2$, which of the following holds true?

$$|z_1|+|z_2|\geq|z_1+z_2|$$

$$|z_1|+|z_2|\geq|z_1-z_2|$$

Please explain why the other one is void, thanks.

Answer 1

They are both valid. First line is the standard triangle inequality that is usually represented in textbooks. The second line is just the first line with some $w_2 = -z_2$. Indeed,

$|z_1+w_2| \le |z_1|+|w_2|= |z_1| + |-z_2| = |z_1| + |z_2|$