If `|z+4| le 3`, then the maximum value of `|z+1|` is `:`

Options

A. 0
B. 4
C. 6
D. 10

Answer

To solve the problem, we need to find the maximum value of \( |z + 1| \) given the condition \( |z + 4| \leq 3 \).

### Step-by-Step Solution:

1. **Understanding the Condition**:
We start with the inequality \( |z + 4| \leq 3 \). This means that the complex number \( z + 4 \) lies within or on the boundary of a circle with radius 3 centered at the point -4 on the real axis.

2. **Rewriting the Inequality**:
The inequality \( |z + 4| \leq 3 \) can be rewritten in terms of real numbers:
\[ -3 \leq z + 4 \leq 3 \]
By subtracting 4 from all parts of the inequality, we get:
\[ -3 - 4 \leq z \leq 3 - 4 \]
Simplifying this gives:
\[ -7 \leq z \leq -1 \]

3. **Finding the Expression for \( |z + 1| \)**:
We need to express \( |z + 1| \):
\[ |z + 1| = |(z + 4) - 3| = |(z + 4) - 3| \]
Here, we can substitute the bounds of \( z \) into the expression \( z + 1 \):
- When \( z = -7 \):
\[ |z + 1| = |-7 + 1| = |-6| = 6 \]
- When \( z = -1 \):
\[ |z + 1| = |-1 + 1| = |0| = 0 \]

4. **Determining the Maximum Value**:
From the calculations above, we see that the maximum value of \( |z + 1| \) occurs when \( z = -7 \):
\[ \text{Maximum value of } |z + 1| = 6 \]

### Final Answer:
Thus, the maximum value of \( |z + 1| \) is \( \boxed{6} \).