Proving absolute values theorems

[tmt1]

For all real numbers $x$ and $y$ , if $x + y >= 0$ then $|x + y| = x + y$. How would I prove this?

My textbook just assumes this to be true.

 

[Deveno]

Let $a = x+y$. What is $|a|$?

 

[tmt1]

Let $a = x+y$. What is $|a|$?

$|x + y|$

 

[I like Serena]

For all real numbers $x$ and $y$ , if $x + y >= 0$ then $|x + y| = x + y$. How would I prove this?

My textbook just assumes this to be true.

The absolute value of a real number is defined as:
$$|a| = \begin{cases}a&\text{if } a\ge 0 \\ -a & \text{if } a<0\end{cases}$$

Since it is already given that $a = x+y \ge 0$, it follows from the definition that $|x+y|=x+y$.