x and y are integers such that x + y is less than 11 and x is greater than 6. Under these conditions, what is the smallest possible value of the expression x − y?

Introduction / Context:
This question tests reasoning with integer inequalities and expressions. You are given constraints on x and y, both integers, and asked to find the smallest possible value of x minus y. Instead of guessing pairs randomly, you can treat this as an optimization problem subject to simple inequality constraints and reason about how to make x − y as small as possible.

Given Data / Assumptions:

• x and y are integers.
• x + y < 11.
• x > 6.
• We want the smallest possible value of x − y.
• No other constraints, such as positivity of y, are given.

Concept / Approach:
To minimize x − y, we want y to be as large as possible relative to x because x − y decreases when y increases. However, the sum constraint x + y < 11 limits how large y can be for a given x. Since x must be greater than 6 and integer valued, we can consider each possible value of x and determine the maximum y allowed by the inequality. Then we compute x − y for each case and choose the smallest result. This is a small, systematic search guided by the inequalities.

Step-by-Step Solution:
Step 1: Since x is an integer and x > 6, the smallest possible value of x is 7. Step 2: The condition x + y < 11 implies y < 11 − x. Step 3: For x = 7, y must satisfy y < 11 − 7 = 4, so the largest integer y can be is 3. Step 4: Then x − y for x = 7 and y = 3 is 7 − 3 = 4. Step 5: Next, consider x = 8. Then y < 11 − 8 = 3, so the largest integer y can be is 2. Then x − y = 8 − 2 = 6. Step 6: For x = 9, y < 11 − 9 = 2, so y can be at most 1. Then x − y = 9 − 1 = 8. Step 7: As x increases further, the maximum possible y decreases, which makes x − y larger. Step 8: Therefore the smallest possible value of x − y occurs at the smallest allowed x and the largest allowed y, which we found to be x = 7 and y = 3. Step 9: The minimum value of x − y under all given constraints is 4.

Verification / Alternative check:
You can verify by considering whether negative values of x − y are possible. For x − y to be negative or very small, y would have to be greater than x. But if y were greater than x and x is at least 7, then x + y would be greater than 14, which would violate the condition x + y < 11. Thus y cannot exceed x under the given constraints, and x − y is always positive. Among the tested pairs, 7 and 3 give the smallest positive difference 4. No other pair with x greater than 7 gives a smaller difference.

Why Other Options Are Wrong:
Values such as 1, 0, or negative 2 would require y to be equal to or greater than x, which makes the sum x + y too large to satisfy the inequality. The value 24 corresponds to some pair with a much larger difference, which is not minimal. Only 4 is consistent with the constraints and arises from a valid integer pair (x, y) = (7, 3).

Common Pitfalls:
A common mistake is to assume that x and y must both be less than 11, which is not stated; only their sum is constrained. Another error is to forget that x must be greater than 6, leading students to test values like x = 5 or 6. Some students also overlook the fact that maximizing y under the sum constraint helps minimize x − y. A structured search starting from the smallest possible x and largest compatible y avoids these issues.

Final Answer:
The smallest possible value of x − y is 4.