The greatest value of `| z + 1| if | z + 4 | le 3 ` is

Options

A. 4
B. 5
C. 6
D. none

Answer

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

### Step 1: Understand the constraint
The expression \( |z + 4| \leq 3 \) describes a circle in the complex plane. Specifically, it represents all points \( z \) that are within or on the boundary of a circle centered at \( -4 \) (in the complex plane) with a radius of \( 3 \).

### Step 2: Rewrite the expression we want to maximize
We want to maximize \( |z + 1| \). This can be rewritten as \( |(z + 4) - 3| \). This means we are looking for the distance from the point \( -1 \) to the points \( z \) that lie within or on the circle defined by \( |z + 4| \leq 3 \).

### Step 3: Identify the center and radius of the circle
The center of the circle is at \( -4 \) and the radius is \( 3 \). The farthest point from \( -1 \) on this circle will be in the direction away from \( -1 \).

### Step 4: Calculate the distance from the center of the circle to the point of interest
The distance from the center of the circle \( -4 \) to the point \( -1 \) is:
\[ |-4 - (-1)| = |-4 + 1| = |-3| = 3 \]

### Step 5: Determine the farthest point on the circle from \( -1 \)
Since the distance from the center of the circle to \( -1 \) is \( 3 \) and the radius of the circle is also \( 3 \), the farthest point on the circle from \( -1 \) will be:
\[ 3 + 3 = 6 \]

### Step 6: Conclusion
Thus, the greatest value of \( |z + 1| \) given the constraint \( |z + 4| \leq 3 \) is \( 6 \).

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

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