If `|z+4| leq 3` then the maximum value of `|z+1|` is

Options

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

Explanation

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 given condition**:
The inequality \( |z + 4| \leq 3 \) describes a circle in the complex plane centered at \(-4\) with a radius of \(3\). This means that any complex number \(z\) that satisfies this condition lies within or on the boundary of this circle.

2. **Rewriting the expression**:
We want to find the maximum value of \( |z + 1| \). We can rewrite this expression:
\[ |z + 1| = |(z + 4) - 3| \]

3. **Applying the triangle inequality**:
Using the triangle inequality, we have:
\[ |z + 1| = |(z + 4) - 3| \leq |z + 4| + |3| \]
Since \( |z + 4| \leq 3 \), we can substitute this into the inequality:
\[ |z + 1| \leq |z + 4| + 3 \leq 3 + 3 = 6 \]

4. **Finding the maximum value**:
Thus, the maximum value of \( |z + 1| \) is \(6\).

5. **Conclusion**:
Therefore, the maximum value of \( |z + 1| \) is \(6\).

### Final Answer:
The maximum value of \( |z + 1| \) is \(6\).

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