Let  z
  be the complex number that satisfies  |z+z¯|+|z−z¯|=2z+9
 . Find  z

Given the problem, we are to find the complex number \( z \) that satisfies the equation:

\[
|z + \overline{z}| + |z - \overline{z}| = 2z + 9
\]

### Step 1: Express \( z \) in terms of real and imaginary parts.
Let \( z = x + iy \), where \( x \) is the real part and \( y \) is the imaginary part. The conjugate \( \overline{z} \) is given by:

\[
\overline{z} = x - iy
\]

### Step 2: Simplify \( z + \overline{z} \) and \( z - \overline{z} \).
\[
z + \overline{z} = (x + iy) + (x - iy) = 2x
\]
\[
z - \overline{z} = (x + iy) - (x - iy) = 2iy
\]

### Step 3: Compute the magnitudes \( |z + \overline{z}| \) and \( |z - \overline{z}| \).
\[
|z + \overline{z}| = |2x| = 2|x|
\]
\[
|z - \overline{z}| = |2iy| = 2|y|
\]

### Step 4: Substitute into the given equation.
The equation now becomes:
\[
2|x| + 2|y| = 2z + 9
\]

### Step 5: Express \( z \) as \( x + iy \) in the equation.
Since \( z = x + iy \), the equation can be written as:
\[
2|x| + 2|y| = 2(x + iy) + 9
\]

### Step 6: Compare real and imaginary parts.
Let's separate the equation into real and imaginary components:

**Real part:**
\[
2|x| = 2x + 9
\]

**Imaginary part:**
\[
2|y| = 2iy
\]

### Step 7: Analyze the real part equation.
From \( 2|x| = 2x + 9 \), we have two cases based on the sign of \( x \):

- **Case 1: \( x \geq 0 \)**
  \[
  2x = 2x + 9 \implies 0 = 9
  \]
  This is not possible, so \( x \) cannot be non-negative.

- **Case 2: \( x < 0 \)**
  \[
  2(-x) = 2x + 9 \implies -4x = 9 \implies x = -\frac{9}{4}
  \]

### Step 8: Analyze the imaginary part equation.
For the imaginary part, \( y \) must satisfy \( 2|y| = 0 \), implying \( y = 0 \).

### Step 9: Combine the results.
Thus, the complex number \( z \) is:
\[
z = -\frac{9}{4} + 0i = -\frac{9}{4}
\]

### Final Answer:
The complex number \( z \) that satisfies the given equation is:
\[
\boxed{-\frac{9}{4}}
\]

Would you like more details on any part of this solution? Here are some related questions:

1. What are the properties of complex conjugates?
2. How do we find the magnitude of a complex number?
3. Why does \( y \) equal zero in this problem?
4. How does the sign of \( x \) affect the equation in this case?
5. Can there be another possible solution for \( z \)?

**Tip:** When dealing with equations involving absolute values, always consider different cases based on the signs of the expressions inside the absolute value.

Learn how to solve the complex equation |z + z¯| + |z - z¯| = 2z + 9 step-by-step. This problem involves understanding complex numbers, absolute values, and equations. Suitable for college-level students and anyone interested in advanced algebra.

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