Absolute value and max/min function: why $a + b + |a - b|=2\max(a,b)$? I am being told that $a + b + |a - b|$ is equal to $2\max(a,b)$.
What is the reasoning behind this?
 A: Adding $|a-b|$ to the smaller of the numbers makes it equal to the larger; then adding the original larger number yields twice that.
A: $$a+b+|a-b|=\begin{cases}a+b+a-b =2a & \text{when} \:a\ge b\\ a+b+b-a = 2b & \text{when} \:b>a\end{cases}$$
Hence we can say that $$a+b+|a-b|=2 \max(a,b)$$
A: The set $\{a,b\}$ will have maximum either $a$ or $b$.  You can argue the equality you want to show by first assuming $b$ is the maximum (then do the same argument if $a$ is the maximum).  So suppose $a < b$ (this same argument would work in reverse if $b < a$).
$|a - b|$ is the distance between $a$ and $b$, i.e., how many units you have to walk from the smaller, $a$, to get to the larger, $b$.
Now, if you take $a$ and add $|a - b|$, i.e., you add the number of units you need to get from $a$ to $b$, where do you end up?  Well, at $b$ of course!  So $a + |a - b| = b$, where $b$ is the maximum of $\{a,b\}$.
Ok, so $$\begin{split} a + b + |a - b| &= (a + |a - b|) + b \\ &= b + b \\ &= \max\{a, b\} + \max\{a, b\} \end{split}$$
A: 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 $$\frac{a+b + |a-b|}{2} = \max(a,b).$$
A: By considering cases $a>b$ and $a\le b$, it's easy to see that
$$
|a-b| = \max(a,b) - \min(a,b), \tag{$note\colon |a-b| = |b-a|$}
$$
and
$$
a+b = \max(a,b) + \min(a,b).
$$
Now add them.
