The theorem states that for all real numbers \(x\) and \(y\), if \(x + y \ge 0\), then \(|x + y| = x + y\). This is proven using the definition of absolute value, which states that \(|a| = a\) if \(a \ge 0\). Given that \(a = x + y\) and \(a \ge 0\), it directly follows that \(|x + y| = x + y\). The proof relies solely on the properties of absolute values and the conditions provided.
PREREQUISITESStudents of mathematics, particularly those studying real analysis or algebra, as well as educators looking for clear proofs of fundamental theorems involving absolute values.
Deveno said:Let $a = x+y$. What is $|a|$?
tmt said: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.