When planning for a party, one caterer recommends the amount of meat be 2 pounds fewer than $\frac{1}{3}$ the total number of guests. The customer decides they want at least that amount of meat. Which graph represents the overall equation represented by this scenario (all points may not apply to the scenario)?
Answer (2)
The amount of meat m is related to the number of guests g by the equation m = 3 1 g − 2 .
The customer wants at least that amount of meat, leading to the inequality m ≥ 3 1 g − 2 .
The graph represents the region above the line m = 3 1 g − 2 .
The graph represents the amounts of meat that satisfy the customer's requirement of having at least the recommended amount.
Explanation
Problem Analysis Let's analyze the problem. We are given a scenario where the amount of meat, m , is related to the number of guests, g , by the equation m = 3 1 g − 2 . The customer wants at least that amount of meat, so we have the inequality m ≥ 3 1 g − 2 . We need to identify the graph that represents this inequality.
Identifying the Graph The inequality m ≥ 3 1 g − 2 represents the region above the line m = 3 1 g − 2 . This is a linear inequality. The graph should show a line with a slope of 3 1 and a y-intercept of -2, and the shaded region should be above the line.
Interpreting the Graph The graph represents the amount of meat m as a function of the number of guests g . The line m = 3 1 g − 2 divides the graph into two regions. The region above the line represents the amounts of meat that satisfy the customer's requirement of having at least the recommended amount.
Conclusion Therefore, the graph that represents the scenario is the one showing the region m ≥ 3 1 g − 2 .
Examples
Imagine you're planning a pizza party. A caterer suggests ordering a certain number of pizzas based on the number of guests. The caterer recommends the number of pizzas be 3 fewer than half the number of guests. You want to make sure you have at least that many pizzas. This scenario can be represented by the inequality p ≥ 2 1 g − 3 , where p is the number of pizzas and g is the number of guests. Understanding this inequality helps you determine the minimum number of pizzas to order to satisfy your guests.
The equation representing the relationship between meat and guests is m = 3 1 g − 2 . The customer wants at least this amount, leading to the inequality m ≥ 3 1 g − 2 , which is represented graphically by the area above the line. Therefore, the correct graph will show this line with the region above it shaded, indicating acceptable quantities of meat.
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