The problem asks to find the expression equivalent to $|a - |b||$ given that $a > 0$ and $b < 0$.

Let's break this down step by step:

1. Since $b < 0$, the absolute value of $b$ is $|b| = -b$. This is because the absolute value of a negative number is its positive counterpart.

2. The expression $|a - |b||$ then becomes $|a - (-b)| = |a + b|$.

Thus, the expression $|a - |b||$ simplifies to $|a + b|$.

Answer:

The correct choice is C. $|a + b|$.

Would you like further explanation on this? Here are some related questions:

1. What is the geometric interpretation of absolute value expressions like $|a - b|$?
2. How does the sign of $b$ affect the expression $|a - |b||$?
3. How would the answer change if both $a$ and $b$ were positive?
4. Can you explain the steps to simplify nested absolute values in more complex cases?
5. How can properties of absolute values be applied to solve inequalities?

Tip: When simplifying expressions with absolute values, carefully consider the sign of the variable inside to determine whether it should remain as-is or be negated.