How can I prove the triangle inequality and the given corollaries?

Question

How can I prove the following 3 claims? Or, if possible, could you provide suggestions for how I could prove them?

For real numbers $x$ and $y$:

• Claim 1. $|x| − |y| ≤ |x − y|$

• Claim 2. $|x-y| ≤ |x|+|y|$

• Claim 3. $|x| − |y| ≤ |x + y|$

Answer 1

It is enough to prove the second one (which is called the triangle inequality) because the remaining inequalities follows immediately.

Indeed,

1. Let's show that $|x|-|y|\leq |x-y|$. Using the triangle inequality we have, $$|x|=|(x-y)+y|\leq |x-y|+|y|.$$

2. Let's show that $|x|-|y|\leq |x+y|$. Using the triangle inequality we have, $$|x|=|(x+y)+(-y)|\leq |x+y|+|-y|=|x+y|+|y|.$$

In both of them I've used that $|a-b|\leq |a|+|b|$. So it suffices to prove the second one.