If ` 1 lt x lt 2` , then the value of ` sqrt(( x - 1)^(2)) + sqrt( ( x - 3)^(2))` is

Options

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

Answer

To solve the problem, we need to evaluate the expression \( \sqrt{(x - 1)^2} + \sqrt{(x - 3)^2} \) given the condition \( 1 < x < 2 \).

### Step-by-Step Solution:

1. **Understanding the Square Root**:
The expression involves square roots of squares. We know that:
\[ \sqrt{(a)^2} = |a| \]
Therefore, we can rewrite the expression as:
\[ \sqrt{(x - 1)^2} + \sqrt{(x - 3)^2} = |x - 1| + |x - 3| \]

2. **Evaluating \( |x - 1| \)**:
Since \( x \) is between 1 and 2, we have:
\[ x - 1 > 0 \quad \text{(because \( x > 1 \))} \]
Thus:
\[ |x - 1| = x - 1 \]

3. **Evaluating \( |x - 3| \)**:
Here, since \( x < 2 \), we have:
\[ x - 3 < 0 \quad \text{(because \( x < 2 < 3 \))} \]
Thus:
\[ |x - 3| = -(x - 3) = 3 - x \]

4. **Combining the Results**:
Now we can substitute back into our expression:
\[ |x - 1| + |x - 3| = (x - 1) + (3 - x) \]

5. **Simplifying the Expression**:
Combine the terms:
\[ (x - 1) + (3 - x) = x - 1 + 3 - x = 2 \]

### Conclusion:
The value of the expression \( \sqrt{(x - 1)^2} + \sqrt{(x - 3)^2} \) when \( 1 < x < 2 \) is:
\[ \boxed{2} \]