If x is a perfect square integer such that `7 lt (2x - 3) lt 17`, then the value of x is :

Options

A. 25
B. 16
C. 9
D. 4

Answer

To solve the problem, we need to find the value of \( x \) that is a perfect square integer and satisfies the inequality \( 7 < (2x - 3) < 17 \).

### Step-by-Step Solution:

1. **Set Up the Inequality**:
We start with the given inequality:
\[ 7 < 2x - 3 < 17 \]

2. **Add 3 to All Parts of the Inequality**:
To isolate the term with \( x \), we add 3 to each part of the inequality:
\[ 7 + 3 < 2x - 3 + 3 < 17 + 3 \]
This simplifies to:
\[ 10 < 2x < 20 \]

3. **Divide All Parts by 2**:
Next, we divide the entire inequality by 2 to solve for \( x \):
\[ \frac{10}{2} < \frac{2x}{2} < \frac{20}{2} \]
This gives us:
\[ 5 < x < 10 \]

4. **Identify Perfect Square Integers**:
Now, we need to find perfect square integers within the range \( 5 < x < 10 \). The perfect squares less than 10 are:
- \( 1^2 = 1 \)
- \( 2^2 = 4 \)
- \( 3^2 = 9 \)
- \( 4^2 = 16 \) (not in range)

The only perfect square that fits our criteria is:
\[ 9 \]

5. **Conclusion**:
Therefore, the value of \( x \) that is a perfect square integer and satisfies the inequality is:
\[ \boxed{9} \]