Let's evaluate each statement related to subtracting integers:
1. **A positive integer minus a positive integer will always have a positive solution.**
- This is **false**. For example, \(3 - 5 = -2\), which is negative.
2. **A negative integer minus a positive integer will always have a positive solution.**
- This is **false**. For example, \(-3 - 5 = -8\), which is negative.
3. **A negative integer minus a negative integer will always have a negative solution.**
- This is **false**. For example, \(-5 - (-3) = -5 + 3 = -2\), which is still negative, but \(-3 - (-5) = -3 + 5 = 2\), which is positive.
4. **A positive integer minus a negative integer will always have a positive solution.**
- This is **true**. For example, \(3 - (-2) = 3 + 2 = 5\), which is positive.
Based on the analysis, the last statement is the only one that is true:
**A positive integer minus a negative integer will always have a positive solution.**