If `x > 2 and x < 3`, then which of the following is positive?
I. `(x - 2)(x - 3)`
II. `(2 - x)(x - 3)`
III. `(2 - x)(3 - x)`

Options

A. I only
B. II only
C. III only
D. I and II only

Answer

To determine which of the given expressions is positive under the conditions \( x > 2 \) and \( x < 3 \), we will analyze each expression step by step.

### Given Conditions:
1. \( x > 2 \)
2. \( x < 3 \)

### Step 1: Analyze the inequalities
From the inequalities, we can derive:
- From \( x > 2 \):
- \( x - 2 > 0 \) (which means \( x - 2 \) is positive)
- \( 2 - x < 0 \) (which means \( 2 - x \) is negative)

- From \( x < 3 \):
- \( x - 3 < 0 \) (which means \( x - 3 \) is negative)
- \( 3 - x > 0 \) (which means \( 3 - x \) is positive)

### Step 2: Evaluate each expression

#### Expression I: \( (x - 2)(x - 3) \)
- We know:
- \( x - 2 > 0 \) (positive)
- \( x - 3 < 0 \) (negative)

The product of a positive number and a negative number is negative:
\[ (x - 2)(x - 3) < 0 \]
Thus, Expression I is **negative**.

#### Expression II: \( (2 - x)(x - 3) \)
- We know:
- \( 2 - x < 0 \) (negative)
- \( x - 3 < 0 \) (negative)

The product of two negative numbers is positive:
\[ (2 - x)(x - 3) > 0 \]
Thus, Expression II is **positive**.

#### Expression III: \( (2 - x)(3 - x) \)
- We know:
- \( 2 - x < 0 \) (negative)
- \( 3 - x > 0 \) (positive)

The product of a negative number and a positive number is negative:
\[ (2 - x)(3 - x) < 0 \]
Thus, Expression III is **negative**.

### Conclusion:
- Expression I: Negative
- Expression II: Positive
- Expression III: Negative

The only positive expression is **Expression II**.

### Final Answer:
The correct option is that **only Expression II is positive**.

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