If `3ltx+1lt8 and 2ltylt9`, which of the following best describes the range of values of `y-x`?

Options

A. `-7lty-xlt5`
B. `-5lty-xlt7`
C. `0lty-xlt2`
D. `2lty-xlt7`

Answer

To solve the problem, we need to find the range of values for \( y - x \) given the inequalities \( 3 < x + 1 < 8 \) and \( 2 < y < 9 \).

### Step 1: Determine the range of \( x \)
From the inequality \( 3 < x + 1 < 8 \), we can isolate \( x \) by subtracting 1 from all parts of the inequality:

\[ 3 - 1 < x + 1 - 1 < 8 - 1 \]

This simplifies to:

\[ 2 < x < 7 \]

So, the range of \( x \) is \( (2, 7) \).

### Step 2: Determine the range of \( y \)
The inequality for \( y \) is given as:

\[ 2 < y < 9 \]

So, the range of \( y \) is \( (2, 9) \).

### Step 3: Find the range of \( y - x \)
To find the range of \( y - x \), we need to consider the minimum and maximum values of \( y - x \) based on the ranges we have determined.

1. **Minimum value of \( y - x \)**:
- The minimum value of \( y \) is just above 2 (let's denote it as \( 2^+ \)).
- The maximum value of \( x \) is just below 7 (let's denote it as \( 7^- \)).
- Therefore, the minimum value of \( y - x \) is:

\[ y - x \text{ (min)} = 2^+ - 7^- \approx 2 - 7 = -5 \]

2. **Maximum value of \( y - x \)**:
- The maximum value of \( y \) is just below 9 (let's denote it as \( 9^- \)).
- The minimum value of \( x \) is just above 2 (let's denote it as \( 2^+ \)).
- Therefore, the maximum value of \( y - x \) is:

\[ y - x \text{ (max)} = 9^- - 2^+ \approx 9 - 2 = 7 \]

### Step 4: Combine the results
Thus, the range of \( y - x \) is:

\[ (-5, 7) \]

### Conclusion
The range of values of \( y - x \) is \( (-5, 7) \).