Which of the following combinations of constraints has no feasible region?

• A. x + y 5 and x 2
• B. x 1 and y 1
• C. x 0 and y 0
• D. x + y 10 and x + y 5

Answer: (D)

Verified step by step guidance

1. Step 1: Understand what a feasible region is. In microeconomics and optimization problems, the feasible region is the set of all points (x, y) that satisfy all given constraints simultaneously.
2. Step 2: Analyze each pair of constraints to determine if there exists any (x, y) that satisfies both. For example, for constraints like \(x + y \leq 5\) and \(x \geq 2\), check if there is any overlap in the solution sets.
3. Step 3: For the pair \(x + y \leq 10\) and \(x + y \geq 5\), recognize that these constraints define a band or strip between two lines: one line where \(x + y = 10\) and another where \(x + y = 5\). The feasible region is all points between these two lines.
4. Step 4: For the pair \(x + y \leq 10\) and \(x + y \leq 5\), note that these two constraints cannot be satisfied simultaneously because \(x + y\) cannot be both less than or equal to 5 and greater than or equal to 10 at the same time. This means no points satisfy both constraints, so the feasible region is empty.
5. Step 5: Conclude that the combination of constraints with no feasible region is the one where the constraints contradict each other, such as \(x + y \leq 10\) and \(x + y \geq 5\) is feasible, but \(x + y \leq 10\) and \(x + y \leq 5\) is not. Carefully check the inequalities to identify contradictions.