Q:

Absolute value : Freshman exercise

Prove that each of the group R :
$$|x+y| \leq\ |x| +  |y|$$
Of course I'm not a lazy person I tried solving before since the teacher gave this exercise to us in the  exam. I wanted to know the right answer because we still haven't corrected the test. 
Thanks and Sorry for any faults. This is my first question in this website. How would I know if it's duplicate?

A:

If $x+y\geq0$ then
$$|x+y|=x+y\leq|x|+|y|.$$
If $x+y<0$ then
$$|x+y|=-x-y\leq|x|+|y|.$$

A:

As both sides are non-negative numbers, it is the same to prove
\begin{align*}
\lvert x+y\rvert^2\le(\lvert x\rvert+\lvert y\rvert)^2&\iff (x+y)^2\le x^2+y^2+2\lvert x\rvert\,\lvert y\rvert\\&\iff 2xy\le 2\lvert x\rvert\,\lvert y\rvert\iff xy\le\lvert xy\rvert
\end{align*}
(We used  $\lvert x\rvert^2=x^2$).

