Univalent triangle inequality

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$|Z_1| = | \frac{v(1+\alpha) + \sqrt{v^2(1+\alpha)^2-4\alpha}}{2}|$

Triangle inequality |x+y|=|x|+|y|

Where x= $\frac{v(1+\alpha)}{2}$ and $y= \frac{\sqrt{v^2(1+\alpha)^2-4\alpha}}{2}$

I've been stuck on this problem now for a couple of days and I'm having a difficult time proving this case.

I'm trying to prove that $|Z_1| \leq 1$ while using triangle inequality.

Where in Case I: $−1\leq \alpha \leq 0$ and $0<v<1$

I was wondering if anybody can assistance me on this problem. I want to thank you ahead of time for your cooperation.