Rewriting $a+b + |a- b|$ 
Quoting: Write the following expression in equivalent forms not involving absolute values: $$a+b + |a- b|$$

long time I did not look at this. Here are all the cases I see existing:
Case 1: when $a \geq 0$, $b\geq0$, and $a> b$
Case 2: when $a \geq 0$, $b\geq0$, and $b> a$
Case 3: when $a \geq 0$, $b\leq0$
Case 4: when $a \leq 0$, $b\geq0$
Case 5: when $a \leq 0$, $b\leq0$, and $a> b$
Case 6: when $a \leq 0$, $b\leq0$, and $b> a$
am i going in the right direction? or is there another way more efficient?
any input is much appreciated

The form of the formula varies depending on the sign of the expression $(a-b)$ under the module, therefore:
$$a+b + |a - b| = 
\begin{cases}
2b,\text{ if } a < b\\
2a,\text{ if } a\ge b.
\end{cases}$$
