<p>If \(|z + 4| \leq 3\), then the maximum value of \(|z + 1|\) is</p>
Step-by-Step Solution
Key Concept: Interpret |z + 4| ≤ 3 as a closed disk centered at -4 with radius 3, then find the point in this disk farthest from -1 using the triangle inequality principle.
<p><strong>Step 1:</strong> Interpret the constraint: |z + 4| ≤ 3 means z lies in a closed disk centered at -4 with radius 3.</p><p><strong>Step 2:</strong> We want to maximize |z + 1|, which is the distance from z to the point -1.</p><p><strong>Step 3:</strong> The maximum distance from -1 to any point in the disk occurs along the line connecting -1 and the center -4, extended outward beyond the center.</p><p><strong>Step 4:</strong> Distance between -4 and -1 is |-4 - (-1)| = |-3| = 3.</p><p><strong>Step 5:</strong> Maximum value of |z + 1| = distance from -1 to center + radius = 3 + 3 = 6.</p><p>∴ Answer: C</p>
Correct Answer: C