These are not true.

I assume this is the MC III Algebra course, the Chapter on complex numbers.  The triangle inequality

$$|z_1 + z_2| \leq |z_1| + |z_2|$$

can be proved by the geometric interpretation of these numbers on the complex plane.  The other inequality $$|z_1 - z_2| \geq | |z_1| - |z_2| |$$ is a direct consequence of the one above.