To solve the problem, we need to analyze the condition given for the complex number \( z \).
### Step 1: Define the complex number
Let \( z = x + iy \), where \( x = \text{Re}(z) \) and \( y = \text{Im}(z) \).
### Step 2: Write the given condition
We are given the condition:
\[
| \text{Re}(z) | + | \text{Im}(z) | = 4
\]
This can be rewritten as:
\[
|x| + |y| = 4
\]
### Step 3: Understand the geometric interpretation
The equation \( |x| + |y| = 4 \) represents a diamond (or rhombus) shape in the coordinate plane with vertices at the points \( (4, 0) \), \( (0, 4) \), \( (-4, 0) \), and \( (0, -4) \).
### Step 4: Determine the maximum and minimum values of \( |z| \)
The modulus of the complex number \( z \) is given by:
\[
|z| = \sqrt{x^2 + y^2}
\]
To find the maximum and minimum values of \( |z| \), we need to consider the points on the diamond.
1. **Maximum value of \( |z| \)**:
The maximum distance from the origin to any point on the diamond occurs at the vertices of the diamond. The maximum distance is:
\[
|z|_{\text{max}} = \sqrt{4^2 + 0^2} = 4
\]
This occurs at the points \( (4, 0) \), \( (0, 4) \), \( (-4, 0) \), and \( (0, -4) \).
2. **Minimum value of \( |z| \)**:
The minimum distance occurs at the points where the line \( x + y = 4 \) intersects the line \( y = -x \) (the line \( x + y = 0 \)). The intersection point can be found by solving the equations:
\[
x + y = 4 \quad \text{and} \quad y = -x
\]
Substituting \( y = -x \) into \( x + y = 4 \):
\[
x - x = 4 \implies 0 = 4 \quad \text{(no solution)}
\]
Instead, we can find the minimum distance from the origin to the lines defined by \( |x| + |y| = 4 \). The minimum distance occurs at the points where the lines are closest to the origin, which can be calculated using the formula for the distance from a point to a line.
The minimum distance from the origin to the line \( x + y = 4 \) can be calculated using the formula:
\[
d = \frac{|Ax_0 + By_0 + C|}{\sqrt{A^2 + B^2}}
\]
where \( A = 1, B = 1, C = -4 \) and \( (x_0, y_0) = (0, 0) \):
\[
d = \frac{|1(0) + 1(0) - 4|}{\sqrt{1^2 + 1^2}} = \frac{4}{\sqrt{2}} = 2\sqrt{2}
\]
### Step 5: Conclusion
Thus, the possible values of \( |z| \) range from \( 2\sqrt{2} \) to \( 4 \). Therefore, the value of \( |z| \) cannot be less than \( 2\sqrt{2} \) and cannot exceed \( 4 \).
### Final Answer
The value of \( |z| \) cannot be less than \( 2\sqrt{2} \) or greater than \( 4 \).
