How to write an expression in an equivalent form without absolute values? The question I have in front of me is the very first problem in Trench's Introduction to Real Analysis:

Write the following expression in equivalent form not involving absolute values:

$a+b+|a-b|$

Looking at the provided answer in the back of the book its:

$2\max(a,b)$

In the chapter I can't find any examples with solutions involving the min/max functions, so feel at a complete loss as to how to get to the above answer.
All you have to do is write down what the absolute value means.

The definition we have is
$$|x|:=\begin{cases}x&amp;\text{ if }x\ge0\\-x&amp;\text{ if }x&lt;0\end{cases}$$

So now the equation we have means

$$\begin{align}
a+b+|a-b|&amp;=\begin{cases}a+b+a-b&amp;\text{ if }a-b\ge0\\a+b-(a-b)&amp;\text{ if }a-b&lt;0\end{cases}\\
\\&amp;=\begin{cases}2a&amp;\text{ if }a\ge b\\2b&amp;\text{ if }a&lt;b\end{cases}
\end{align}$$

Now all that is required is to recognize
$$\max(a,b)=\begin{cases}a&amp;\text{ if }a\ge b\\b&amp;\text{ if }a&lt;b\end{cases}$$
giving you an answer of 
$$a+b+|a-b|=2\max(a,b)$$
There is also a trick. $\sqrt{x}$ means normally the positive square root. Thus, we can use $\sqrt{x^2}=|x|$:

$$a+b+|a-b|=a+b+\sqrt{(a-b)^2}$$