I'm a little stuck with the following inequality and was wondering if anyone could help me out.

$\displaystyle \frac{\mbox{Re}(z_1+z_2)}{|z_3+z_4|} \leq \frac{|z_1|+|z_2|}{||z_3|-|z_4||} \ \ \mbox{where} \ \ z_3 \neq z_4$

$\displaystyle \frac{x_1+x_2}{|z_3+z_4|} \leq \frac{|z_1|+|z_2|}{||z_3|-|z_4||}$

$\displaystyle \frac{x_1+x_2}{|z_3+z_4|} \leq \frac{\sqrt{x_1^2+y_1^2}+\sqrt{x_2^2+y_2^2}}{||z_3 |-|z_4||}$

at which point I get stuck.

I was also thinking of using the property that:

$\displaystyle |z_3+z_4| \geq ||z_3|-|z_4|| \therefore \frac{1}{|z_3+z_4|} \leq \frac{1}{||z_3|-|z_4||}$ but the numerators are throwing me off.