If `2 < x < 5 and 3 < y < 5`, which of the following best describes `x - y`?

Options

A. `-3 lt x - y lt 2`
B. `-3 lt x - y lt 5`
C. `0 lt x - y lt 5`
D. `3 lt x - y lt 5`

Answer

To solve the problem of finding the range of \( x - y \) given the inequalities \( 2 < x < 5 \) and \( 3 < y < 5 \), we will follow these steps:

### Step 1: Identify the minimum value of \( x - y \)

To find the minimum value of \( x - y \), we need to minimize \( x \) and maximize \( y \):
- The minimum value of \( x \) is just above 2 (let's denote it as \( x \to 2^+ \)).
- The maximum value of \( y \) is just below 5 (let's denote it as \( y \to 5^- \)).

Thus, we can calculate:
\[ x - y \text{ (minimum)} = (2^+) - (5^-) \approx 2 - 5 = -3 \]

### Step 2: Identify the maximum value of \( x - y \)

To find the maximum value of \( x - y \), we need to maximize \( x \) and minimize \( y \):
- The maximum value of \( x \) is just below 5 (let's denote it as \( x \to 5^- \)).
- The minimum value of \( y \) is just above 3 (let's denote it as \( y \to 3^+ \)).

Thus, we can calculate:
\[ x - y \text{ (maximum)} = (5^-) - (3^+) \approx 5 - 3 = 2 \]

### Step 3: Combine the results

From the calculations above, we have:
- The minimum value of \( x - y \) is slightly greater than \(-3\).
- The maximum value of \( x - y \) is slightly less than \(2\).

Therefore, we can express the range of \( x - y \) as:
\[ -3 < x - y < 2 \]

### Final Answer

The best description of \( x - y \) is:
\[ -3 < x - y < 2 \]