|x+y|: let x,y>=0, then |x+y|=|x|+|y|.
Let x,y<0, |x+y|=-(x+y); |x|=-x, |y|=-y; so |x+y|=-x-y=|x|+|y|.
Let x<0, y>0 and x+y>0, so |x+y|=x+y; |x|=-x and |y|=y, x+y<y-x (x is negative so -x is positive). |x+y|<|x|+|y|.
Let x<0, y>0 and x+y<0, so |x+y|=-x-y; |x|=-x and |y|=y, -x-y<y-x. |x+y|<|x|+|y|.
The symmetry of the expressions means that x and y are interchangeable, so |x+y|<=|x|+|y|.