If `x + 5` is the mean proportion between `x + 2` and `x + 9`: find the value of `x`.

Answer

To solve the problem, we need to find the value of \( x \) given that \( x + 5 \) is the mean proportion between \( x + 2 \) and \( x + 9 \).

### Step-by-Step Solution:

1. **Understanding Mean Proportion**:
The mean proportion \( B \) between two quantities \( A \) and \( C \) is defined by the equation:
\[ B^2 = A \times C \]
In our case, \( A = x + 2 \), \( B = x + 5 \), and \( C = x + 9 \).

2. **Setting Up the Equation**:
According to the definition of mean proportion:
\[ (x + 5)^2 = (x + 2)(x + 9) \]

3. **Expanding Both Sides**:
- Left Side:
\[ (x + 5)^2 = x^2 + 10x + 25 \]
- Right Side:
\[ (x + 2)(x + 9) = x^2 + 9x + 2x + 18 = x^2 + 11x + 18 \]

4. **Setting the Expanded Forms Equal**:
Now we have:
\[ x^2 + 10x + 25 = x^2 + 11x + 18 \]

5. **Eliminating \( x^2 \)**:
Subtract \( x^2 \) from both sides:
\[ 10x + 25 = 11x + 18 \]

6. **Rearranging the Equation**:
Move all terms involving \( x \) to one side and constant terms to the other side:
\[ 10x - 11x = 18 - 25 \]
This simplifies to:
\[ -x = -7 \]

7. **Solving for \( x \)**:
Multiply both sides by -1:
\[ x = 7 \]

### Final Answer:
The value of \( x \) is \( 7 \).

---