Felix bought x pounds of grapes that cost $1.25 per pound and y boxes of cereal that cost $2.50 per box. He spent less than $10. Which graph represents this scenario?
The correct Answer and Explanation is:
To solve this problem, we first need to create an inequality based on the costs:
- Felix bought x pounds of grapes costing $1.25 per pound, which means the cost for the grapes is (1.25x).
- He also bought y boxes of cereal costing $2.50 per box, which means the cost for the cereal is (2.50y).
The total cost for both the grapes and the cereal is (1.25x + 2.50y). According to the problem, the total amount Felix spent is less than $10, so we have the inequality:
[
1.25x + 2.50y < 10
]
Now, we can simplify this inequality by dividing the entire equation by 1.25 to make it easier to work with:
[
x + 2y < 8
]
This inequality represents a region on the graph where the combination of x (pounds of grapes) and y (boxes of cereal) must satisfy the condition that the total cost is less than $10.
Key Elements of the Graph:
- The graph will have x on the horizontal axis (representing pounds of grapes) and y on the vertical axis (representing boxes of cereal).
- The region of interest lies below the line (x + 2y = 8), which represents the boundary where the total cost is exactly $10.
- Any points below this line represent combinations of grapes and cereal where the total cost is less than $10.
After evaluating the graphs, the correct one is the one that shows a line for (x + 2y = 8) with the region below the line shaded, indicating the points where the total cost is less than $10.