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 expressed as:
\[
|x| + |y| = 4
\]
### Step 3: Analyze the equation geometrically
The equation \( |x| + |y| = 4 \) represents a diamond (or rhombus) shape in the coordinate plane with vertices at the points \( (4, 0) \), \( (-4, 0) \), \( (0, 4) \), and \( (0, -4) \).
### Step 4: Find the modulus of \( z \)
The modulus of \( z \) is given by:
\[
|z| = \sqrt{x^2 + y^2}
\]
### Step 5: Determine the maximum and minimum values of \( |z| \)
To find the maximum and minimum values of \( |z| \), we need to consider the distances from the origin to the points on the diamond defined by \( |x| + |y| = 4 \).
1. **Maximum value of \( |z| \)**:
The maximum distance from the origin to the vertices of the diamond occurs at the points \( (4, 0) \) and \( (0, 4) \):
\[
|z|_{\text{max}} = \sqrt{4^2 + 0^2} = 4
\]
or
\[
|z|_{\text{max}} = \sqrt{0^2 + 4^2} = 4
\]
2. **Minimum value of \( |z| \)**:
The minimum distance from the origin to the diamond occurs along the line \( x + y = 4 \). The perpendicular distance from the origin to this line can be calculated using the formula for the distance from a point to a line:
\[
\text{Distance} = \frac{|Ax_0 + By_0 + C|}{\sqrt{A^2 + B^2}}
\]
where \( A = 1, B = 1, C = -4 \), and \( (x_0, y_0) = (0, 0) \):
\[
\text{Distance} = \frac{|0 + 0 - 4|}{\sqrt{1^2 + 1^2}} = \frac{4}{\sqrt{2}} = 2\sqrt{2}
\]
### Step 6: Conclusion on the range of \( |z| \)
Thus, the values of \( |z| \) lie in the range:
\[
2\sqrt{2} \leq |z| \leq 4
\]
### Step 7: Identify the value that cannot be \( |z| \)
Now, we need to check which of the given options cannot be \( |z| \). Since \( |z| \) must be between \( 2\sqrt{2} \) and \( 4 \), we convert \( 2\sqrt{2} \) to a decimal:
\[
2\sqrt{2} \approx 2.828
\]
And \( 4 \) is obviously \( 4.000 \).
If we analyze the options, any value less than \( 2\sqrt{2} \) or greater than \( 4 \) cannot be \( |z| \).
### Final Answer
The value that cannot be \( |z| \) is \( \sqrt{7} \) since \( \sqrt{7} \approx 2.645 \), which is less than \( 2\sqrt{2} \).
