The question is asking for which expression among the given options results in the smallest value, given $a = -3$ and $b = 8$. Let's analyze each option step by step:

Given:

$a = -3, \, b = 8$

Evaluate Each Option:

1. Option (A): $|a + b|$ $a + b = -3 + 8 = 5$ $|a + b| = |5| = 5$

2. Option (B): $|a - b|$ $a - b = -3 - 8 = -11$ $|a - b| = |-11| = 11$

3. Option (C): $|a| - |b|$ $|a| = |-3| = 3, \, |b| = |8| = 8$ $|a| - |b| = 3 - 8 = -5$ Since it’s the absolute value: $||a| - |b|| = |-5| = 5$

4. Option (D): $|a| + |b|$ $|a| = 3, \, |b| = 8$ $|a| + |b| = 3 + 8 = 11$

Summary of Values:

• (A): $|a + b| = 5$
• (B): $|a - b| = 11$
• (C): $|a| - |b| = 5$
• (D): $|a| + |b| = 11$

Conclusion:

The smallest value is 5, which is achieved by both Option (A) and Option (C). Based on the typical interpretation of such problems, Option (A) is generally considered correct, as it evaluates the sum of $a$ and $b$.

Let me know if you'd like additional clarifications or a detailed breakdown!

Related Questions:

1. How does the absolute value function affect mathematical operations like subtraction or addition?
2. What are other examples where multiple expressions yield the same minimum value?
3. How do changes in $a$ and $b$ values affect the outcomes in similar problems?
4. Why is absolute value used in mathematical problems like these?
5. Can a negative number in any option lead to a smaller value than absolute comparisons?

Tip:

When solving absolute value problems, always calculate expressions inside the absolute value first before applying the function.