The problem is to determine which expression results in the smallest value, given $a = -3$ and $b = 8$. The options provided involve absolute values:

1. $|a + b|$
2. $|a - b|$
3. $|a| - |b|$
4. $|a| + |b|$

Let's calculate each expression:

1. $|a + b| = |-3 + 8| = |5| = 5$
2. $|a - b| = |-3 - 8| = |-11| = 11$
3. $|a| - |b| = | -3 | - |8| = 3 - 8 = -5$ (absolute values cannot be negative, so consider it as -5)
4. $|a| + |b| = | -3 | + |8| = 3 + 8 = 11$

The smallest value is $|a| - |b| = -5$.

Thus, the correct answer is (C).

Would you like further clarification or an explanation? Here are five related questions:

1. What is the significance of absolute value in mathematics?
2. How do you handle inequalities involving absolute values?
3. How do negative values inside absolute values affect calculations?
4. Can you provide more examples of minimum and maximum absolute value problems?
5. Why does subtracting absolute values lead to negative results sometimes?

Tip: When dealing with absolute values, always consider the distance from zero and compare expressions step by step.