I'm trying to learn Real Analysis on my own, but I found that i'm a bit rusty with the elementary stuff.
How do I solve equations like $|x| + |x+1| = 1$ and $|x-1| + |x+1| = 2$? I don't want the whole solution, because I have a feeling that what I'm really looking for is a property of the absolute value which I can't remember.
Also, there is another exercise, that given the Archimedean property (for every $x \in \mathbb{R}$ there is a number $[x] \in \mathbb{Z}$ and you know the rest) prove that for $x, y \in \mathbb{R}$, x greater than 0, than there $\exists n \in \mathbb{N} : nx > y$.
Edit: Given the fact the floor function exists and knowing that $x \in R_+$ and $y \in R$ prove that there $\exists n \in N : nx > y$.