If is any complex number such that `|z+4|lt=3,` then find the greatest value of `|z+1|dot`

Explanation

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

### Step-by-Step Solution:

1. **Understanding the Given Condition**:
The condition \( |z + 4| \leq 3 \) describes a circle in the complex plane centered at \( -4 \) (which is the point \(-4 + 0i\)) with a radius of \( 3 \).

2. **Expressing \( |z + 1| \)**:
We want to express \( |z + 1| \) in terms of \( |z + 4| \):
\[ |z + 1| = |(z + 4) - 3| = |(z + 4) + (-3)| \]

3. **Applying the Triangle Inequality**:
By the triangle inequality, we have:
\[ |a + b| \leq |a| + |b| \]
where \( a = z + 4 \) and \( b = -3 \). Thus,
\[ |z + 1| \leq |z + 4| + |-3| = |z + 4| + 3 \]

4. **Using the Given Condition**:
Since \( |z + 4| \leq 3 \), we can substitute this into our inequality:
\[ |z + 1| \leq |z + 4| + 3 \leq 3 + 3 = 6 \]

5. **Finding the Maximum Value**:
Therefore, the greatest value of \( |z + 1| \) is:
\[ |z + 1| \leq 6 \]
Hence, the greatest value of \( |z + 1| \) is \( 6 \).

### Conclusion:
The greatest value of \( |z + 1| \) is \( 6 \).