<p>If \(z\) is any complex number such that \(|z + 4| \leq 3\), then find the greatest value of \(|z + 1|\).</p>
Step-by-Step Solution
Key Concept: The set of points satisfying |z + 4| ≤ 3 forms a closed disk centered at -4 with radius 3. The maximum value of |z + 1| occurs at the point on this disk farthest from -1.
<p><strong>Step 1:</strong> Interpret the constraint geometrically. The condition |z + 4| ≤ 3 represents all complex numbers z in a closed disk centered at the point -4 with radius 3.</p><p><strong>Step 2:</strong> We need to find the maximum value of |z + 1|, which represents 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 when z lies on the boundary of the disk (circle |z + 4| = 3) on the line passing through -1 and -4, on the far side from -1.</p><p><strong>Step 4:</strong> The distance between the centers is |-4 - (-1)| = |-3| = 3.</p><p><strong>Step 5:</strong> The maximum value of |z + 1| is the distance between -1 and -4, plus the radius of the disk: max|z + 1| = 3 + 3 = 6.</p><p>∴ Answer: <strong>6</strong></p>
Correct Answer: 6