TITLE: Absolute value and max/min function: why $a + b + |a - b|=2\max(a,b)$?
QUESTION [13 upvotes]: I am being told that $a + b + |a - b|$ is equal to $2\max(a,b)$.
What is the reasoning behind this?

REPLY [14 votes]: Intuitively, notice that $\frac{a + b}{2}$ is the midway point between $a$ and $b$, and $\frac{|a - b|}{2}$ is half the distance between the two numbers, so $$\frac{a + b}{2} + \frac{|a - b|}{2}$$ is the mid way point between the two points plus half of the distance between them, which brings you to the larger of the two numbers, thus $$\frac{a+b + |a-b|}{2} = \max(a,b).$$