Key Concepts and Formulas

• Geometric Interpretation: $|z - z_0|$ represents the distance between complex numbers $z$ and $z_0$ in the Argand plane. $|z - z_0| \le r$ represents all points $z$ within or on a circle of radius $r$ centered at $z_0$.
• Triangle Inequality: For complex numbers $a$ and $b$, $|a + b| \le |a| + |b|$. Equality holds if and only if $a$ and $b$ have the same argument or either is zero.

Step-by-Step Solution

• Step 1: Interpret the Given Condition Geometrically The condition $|z + 4| \le 3$ can be rewritten as $|z - (-4)| \le 3$. This represents all complex numbers $z$ lying within or on a circle centered at $-4$ (i.e., the point $(-4, 0)$ in the complex plane) with a radius of $3$.

  • Why? Rewriting in the form $|z - z_0| \le r$ allows us to directly identify the center and radius, which are crucial for the geometric approach.

• Step 2: Interpret the Expression to Maximize Geometrically We want to find the maximum value of $|z + 1|$, which can be rewritten as $|z - (-1)|$. This represents the distance between the complex number $z$ and the point $-1$ (i.e., the point $(-1, 0)$ in the complex plane).

  • Why? This allows us to frame the problem as finding the point $z$ within the circle defined in Step 1 that is farthest from the point $-1$.

• Step 3: Geometric Solution - Find the Point z for Maximum Distance The maximum distance between the point $-1$ and any point $z$ within the circle will occur when $z$ lies on the circle's boundary and is collinear with the center of the circle and the point $-1$, on the opposite side of the center from $-1$.

  • Why? This is a fundamental geometric principle: to maximize the distance from a point to a circle, we must extend a line from the point through the center to the far side of the circle.

• Step 4: Geometric Solution - Calculate the Maximum Distance The distance between the center of the circle, $-4$, and the point $-1$ is $|-4 - (-1)| = |-3| = 3$. The maximum distance from $-1$ to any point $z$ within the circle is the sum of this distance and the radius of the circle: $3 + 3 = 6$.

  • Why? We are simply adding the distance from the fixed point to the circle's center to the radius to find the maximum distance to a point on the circle.

• Step 5: Algebraic Solution - Apply the Triangle Inequality We want to maximize $|z + 1|$. Rewrite $z + 1$ as $(z + 4) - 3$. Then, by the triangle inequality, $|(z + 4) - 3| \le |z + 4| + |-3|$.

  • Why? Rewriting $z+1$ in terms of $z+4$ allows us to use the given condition $|z+4| \le 3$. The triangle inequality gives us an upper bound for $|z+1|$.

• Step 6: Algebraic Solution - Substitute the Given Condition and Simplify Since $|z + 4| \le 3$, we have $|z + 1| \le |z + 4| + 3 \le 3 + 3 = 6$. Therefore, $|z + 1| \le 6$.

  • Why? Substituting the given condition into the inequality allows us to find the maximum possible value of $|z+1|$.

• Step 7: Algebraic Solution - Verify the Equality Condition The maximum value of 6 is achieved when $|z+4| = 3$ and the arguments of $(z+4)$ and $-3$ are equal. This occurs when $z+4 = -3$, which gives $z = -7$. When $z = -7$, $|z+1| = |-7+1| = |-6| = 6$.

  • Why? We need to ensure that the upper bound we found using the triangle inequality is actually attainable. We do this by finding a value of $z$ that satisfies both the given condition and achieves the maximum value.

Common Mistakes & Tips

• Center Identification: Be careful with the sign when identifying the center from $|z - z_0|$. In $|z + 4|$, the center is at $-4$, not $4$.
• Equality Condition: Remember that the triangle inequality becomes an equality when the complex numbers involved have the same argument (direction).

Summary

The problem asks for the maximum value of $|z+1|$ given $|z+4| \le 3$. Both geometric and algebraic approaches show that the maximum value is 6. The geometric approach visualizes the problem as finding the farthest point within a circle from a given point. The algebraic approach uses the triangle inequality to find an upper bound for $|z+1|$ and then verifies that this upper bound is attainable.

The final answer is $\boxed{6}$. which corresponds to option (A).