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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. \( |a + b| \)**.

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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.

If a and b are real numbers such that a > 0 and b < 0, then which of the following is equivalent to |a - |b||?

Learn how to simplify the absolute value expression |a - |b|| where a > 0 and b < 0. This problem covers key algebraic concepts such as absolute values, real numbers, and how to simplify nested absolute expressions.

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