Proving absolute value equations

Question

How do I prove that for all real numbers $x$ and $y$ I have

|x + y| ≤ |x| + |y|

||x| − |y|| ≤ |x − y|

|xy| = |x||y|

I know the definition of absolute value tells me: for any real number $x$ we define $|x| = x$ if $x ≥ 0$ or $−x$ if $x < 0$