Find the value of
`sqrt (( 2x - 5) ^(2)) + 2 sqrt (( x - 1) ^(2)) if 1 lt x lt 2.`

Options

A. 1
B. 2
C. 3
D. 4

Answer

To solve the problem, we need to evaluate the expression \( \sqrt{(2x - 5)^2} + 2\sqrt{(x - 1)^2} \) for \( 1 < x < 2 \).

### Step-by-Step Solution:

1. **Understanding the Square Root**:
The expression \( \sqrt{(a)^2} \) is equal to \( |a| \) (the absolute value of \( a \)). Thus, we can rewrite the expression:
\[ \sqrt{(2x - 5)^2} = |2x - 5| \quad \text{and} \quad \sqrt{(x - 1)^2} = |x - 1| \]
Therefore, the expression becomes:
\[ |2x - 5| + 2|x - 1| \]

2. **Analyzing the Absolute Values**:
- For \( 1 < x < 2 \):
- \( x - 1 > 0 \) implies \( |x - 1| = x - 1 \).
- Now, let's evaluate \( 2x - 5 \):
- When \( x = 1 \), \( 2x - 5 = 2(1) - 5 = -3 \) (negative).
- When \( x = 2 \), \( 2x - 5 = 2(2) - 5 = -1 \) (still negative).
- Thus, for \( 1 < x < 2 \), \( 2x - 5 < 0 \) implies \( |2x - 5| = -(2x - 5) = 5 - 2x \).

3. **Substituting Back into the Expression**:
Now substituting the absolute values back into the expression:
\[ |2x - 5| + 2|x - 1| = (5 - 2x) + 2(x - 1) \]

4. **Simplifying the Expression**:
\[ = 5 - 2x + 2x - 2 = 5 - 2 = 3 \]

5. **Conclusion**:
Therefore, the value of the expression \( \sqrt{(2x - 5)^2} + 2\sqrt{(x - 1)^2} \) for \( 1 < x < 2 \) is:
\[ \boxed{3} \]