Let $a,b,c,d\in\Bbb{R}$ such that $|c|\ne|d|$. Prove that $\frac{a+b}{c+d}\le\frac{|a|+|b|}{||c|-|d||}$

Question

I can figure that $a+b\le|a+b|\le|a|+|b|$ but I do not know how to deal with $||c|-|d||$

Answer 1

It is the reversed triangle inequality. For $x,y \in \mathbb{R}$, we have that $||x| -|y|| \leq |x - y|$.

See this post:

Reverse Triangle Inequality Proof