Penelope went to the store to buy x pounds of broccoli for $1.10 per pound and y cans of soup for $2.50 each. In total, she spent less than $10. The inequality relating the purchases she made and the total purchase price is 1.10x + 2.50y < 10. Which are reasonable solutions for this situation? Check all that apply. (–1, 4) (0, 2) (3, 2.5) (2, 4) (0.5, 3.78) (1.5, 3)
The Correct Answer and Explanation is :
To determine which solutions are reasonable for Penelope’s shopping situation, we need to analyze the inequality (1.10x + 2.50y < 10). Here, (x) represents the pounds of broccoli purchased, and (y) represents the number of cans of soup.
First, let’s rearrange the inequality:
[
1.10x + 2.50y < 10
]
To find valid values for (x) and (y), we can evaluate the given points to see if they satisfy this inequality:
- Point (-1, 4):
[
1.10(-1) + 2.50(4) = -1.10 + 10 = 8.90 < 10 \quad \text{(Valid)}
] - Point (0, 2):
[
1.10(0) + 2.50(2) = 0 + 5 = 5 < 10 \quad \text{(Valid)}
] - Point (3, 2.5):
[
1.10(3) + 2.50(2.5) = 3.30 + 6.25 = 9.55 < 10 \quad \text{(Valid)}
] - Point (2, 4):
[
1.10(2) + 2.50(4) = 2.20 + 10 = 12.20 < 10 \quad \text{(Invalid)}
] - Point (0.5, 3.78):
[
1.10(0.5) + 2.50(3.78) = 0.55 + 9.45 = 10.00 < 10 \quad \text{(Invalid)}
] - Point (1.5, 3):
[
1.10(1.5) + 2.50(3) = 1.65 + 7.50 = 9.15 < 10 \quad \text{(Valid)}
]
Based on this evaluation, the valid solutions that satisfy the inequality (1.10x + 2.50y < 10) are:
- (-1, 4)
- (0, 2)
- (3, 2.5)
- (1.5, 3)
Reasoning:
In the context of this problem, negative quantities of broccoli or non-integer cans of soup do not make practical sense, yet mathematically, (-1, 4) is valid. The other points represent feasible combinations of broccoli and soup that stay under the $10 limit. For instance, buying 0 pounds of broccoli and 2 cans of soup gives a total of $5, which is a reasonable purchase.
Understanding these calculations helps ensure Penelope stays within her budget while shopping for groceries. In real-life situations, constraints like budget, quantity limitations, and pricing can often be analyzed using such inequalities, emphasizing the importance of evaluating solutions within context.