The Correct Answer and Explanation is:
The correct reasonable solutions are (0, 2), (3, 2.5), and (1.5, 3).
To determine the reasonable solutions, we must check two conditions for each ordered pair (x, y). First, since x and y represent pounds of items being purchased, neither value can be negative. This is the “reasonable” part of the question. Second, the pair must satisfy the inequality 1.10x + 2.50y < 10, meaning the total cost is less than $10.
Let’s test each option:
- (-1, 4): This is not a reasonable solution. Although it mathematically satisfies the inequality (1.10(-1) + 2.50(4) = 8.90, which is less than 10), one cannot buy a negative amount of an item.
- (0, 2): This is a reasonable solution. Both values are non-negative. The cost is 1.10(0) + 2.50(2) = 0 + 5.00 = $5.00. Since $5.00 is less than $10, this option is correct.
- (3, 2.5): This is a reasonable solution. Both values are positive. The cost is 1.10(3) + 2.50(2.5) = 3.30 + 6.25 = $9.55. Since $9.55 is less than $10, this option is correct.
- (2, 4): This is not a solution. The cost is 1.10(2) + 2.50(4) = 2.20 + 10.00 = $12.20. This amount is not less than $10, so it violates the inequality.
- (0.5, 3.78): This is not a solution. The cost is 1.10(0.5) + 2.50(3.78) = 0.55 + 9.45 = $10.00. The inequality requires the total to be strictly less than $10, and $10.00 is equal to, not less than, $10.
- (1.5, 3): This is a reasonable solution. Both values are positive. The cost is 1.10(1.5) + 2.50(3) = 1.65 + 7.50 = $9.15. Since $9.15 is less than $10, this option is correct.
