If x and y are positive integers that satisfy `1lexle4` and `3x+2yle13`, then the possible values of y are:

Options

A. 1,2,3,4,5
B. 1,2,3,4,5,6
C. `1/2,3,5`
D. 2,3,4,5

Answer

To solve the problem, we need to find the possible values of \( y \) given the conditions \( 1 < x < 4 \) and \( 3x + 2y \leq 13 \).

### Step-by-Step Solution:

1. **Identify the possible values for \( x \)**:
Since \( x \) is a positive integer and \( 1 < x < 4 \), the possible values for \( x \) are \( 2 \) and \( 3 \).

2. **Evaluate for \( x = 2 \)**:
- Substitute \( x = 2 \) into the inequality:
\[ 3(2) + 2y \leq 13 \]
- This simplifies to:
\[ 6 + 2y \leq 13 \]
- Rearranging gives:
\[ 2y \leq 7 \]
- Dividing by \( 2 \):
\[ y \leq 3.5 \]
- Since \( y \) must be a positive integer, the possible values for \( y \) are \( 1, 2, 3 \).

3. **Evaluate for \( x = 3 \)**:
- Substitute \( x = 3 \) into the inequality:
\[ 3(3) + 2y \leq 13 \]
- This simplifies to:
\[ 9 + 2y \leq 13 \]
- Rearranging gives:
\[ 2y \leq 4 \]
- Dividing by \( 2 \):
\[ y \leq 2 \]
- Since \( y \) must be a positive integer, the possible values for \( y \) are \( 1, 2 \).

4. **Combine the results**:
- From \( x = 2 \), the possible values for \( y \) are \( 1, 2, 3 \).
- From \( x = 3 \), the possible values for \( y \) are \( 1, 2 \).
- Therefore, the combined possible values for \( y \) are \( 1, 2, 3 \).

### Final Answer:
The possible values of \( y \) are \( 1, 2, 3 \).