Elementary inequality involving modulus

Question

Is is true that $$\lvert a+b\rvert^2\geq (\lvert a\rvert-\lvert b\rvert)^2, \quad \forall a,b\in\mathbb{R} ?$$ Is there any related result, related to this inequality?

Answer 1

$|a+b|=|a-(-b)|\geq |a|-|-b|=|a|-|b|$. (I assume you've already proved $|x-y|\geq |x|-|y|$ for $x,y\in\mathbb{R}$)