IF `|z+4|le3,` then the maximum value of `|z+1|` is

Options

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

Explanation

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

### Step-by-Step Solution:

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

2. **Rewriting the Condition:**
We can express this condition in terms of \(z\):
\[ -3 \leq z + 4 \leq 3 \]
This leads to:
\[ -3 - 4 \leq z \leq 3 - 4 \]
Simplifying gives:
\[ -7 \leq z \leq -1 \]

3. **Finding the Maximum Value of \(|z + 1|\):**
We want to find the maximum value of \(|z + 1|\). We can express this as:
\[ |z + 1| = |(z + 4) - 3| \]
Since \(|z + 4| \leq 3\), we can analyze the expression \(|(z + 4) - 3|\).

4. **Using Triangle Inequality:**
By the triangle inequality, we know:
\[ |z + 1| = |(z + 4) - 3| \leq |z + 4| + |3| \]
Given that \(|z + 4| \leq 3\), we can substitute:
\[ |z + 1| \leq 3 + 3 = 6 \]

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

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

---