The greatest and the least absolute value of z+1, where `|z+4| le 3` are respectively

Options

A. 6 and 0
B. 10 an 6
C. 4 and 3
D. None of these

Answer

To solve the problem of finding the greatest and least absolute value of \( z + 1 \) given the condition \( |z + 4| \leq 3 \), we can follow these steps:

### Step 1: Understand the condition
The condition \( |z + 4| \leq 3 \) represents the set of complex numbers \( z \) that are within a distance of 3 from the point -4 on the real line. This can be interpreted geometrically as a circle centered at -4 with a radius of 3.

### Step 2: Identify the circle's boundaries
The center of the circle is at \( -4 \) and the radius is \( 3 \). Therefore, the points on the circle can be described as:
- The leftmost point: \( -4 - 3 = -7 \)
- The rightmost point: \( -4 + 3 = -1 \)

Thus, the values of \( z \) lie within the interval \( [-7, -1] \).

### Step 3: Express \( z + 1 \)
Now, we need to find the absolute value of \( z + 1 \):
\[ |z + 1| = |(z + 4) - 3| \]
This means we need to consider how the values of \( z + 1 \) change as \( z \) varies within the interval \( [-7, -1] \).

### Step 4: Determine the range of \( z + 1 \)
Calculating the values at the endpoints:
- When \( z = -7 \):
\[ z + 1 = -7 + 1 = -6 \quad \Rightarrow \quad |z + 1| = 6 \]
- When \( z = -1 \):
\[ z + 1 = -1 + 1 = 0 \quad \Rightarrow \quad |z + 1| = 0 \]

### Step 5: Find the greatest and least absolute values
From the calculations:
- The least absolute value of \( z + 1 \) is \( 0 \).
- The greatest absolute value of \( z + 1 \) is \( 6 \).

### Conclusion
Thus, the greatest and least absolute values of \( z + 1 \) are:
- **Least absolute value**: \( 0 \)
- **Greatest absolute value**: \( 6 \)