- Identify the cost per unit for potatoes (\(\$0.85\)) and grapes (\(\$1.29\)).
- Represent the total cost as the sum of the product of quantity and unit price: \(0.85x + 1.29y\).
- Apply the constraint "less than \(\$5\)" using the inequality symbol \(<\).
- The correct inequality is \(\boxed{0.85x + 1.29y < 5}\).
Explanation
- Analyzing the Purchase Costs
Dan purchased \(x\) pounds of potatoes at \(\$0.85\) per pound and \(y\) pounds of grapes at \(\$1.29\) per pound. The goal is to determine the correct mathematical inequality representing the total cost being less than \(\$5\). Analysis of the costs yields that the total cost is the sum of the individual costs of each item purchased, which is given by \(0.85x + 1.29y\). Given the constraint that this total cost is less than \(\$5\), we must express this relationship using the less-than symbol, denoted by \(<\).
- Formulating the Inequality and Selecting the Correct Option
We have identified that the cost for potatoes is \(0.85x\) and the cost for grapes is \(1.29y\). Summing these gives the total cost:
\[0.85x + 1.29y\]
The problem states that this total cost is strictly less than \(\$5\). Therefore, we translate this into the inequality:
\[0.85x + 1.29y < 5\]
Comparing this result to the provided options:
- \(1.29x + 0.85y < 5\)
- \(1.29x + 0.85y > 5\)
- \(0.85x + 1.29y > 5\)
- \(0.85x + 1.29y < 5\)
The inequality that matches our derivation is \(0.85x + 1.29y < 5\).
Examples
This type of linear inequality is useful in everyday shopping scenarios where you have a set budget. For example, if you are planning to buy ingredients for a recipe but have a limited amount of money, you can create a similar inequality to ensure your total spending stays under your limit. By assigning variables to the quantities of different items and multiplying them by their respective unit prices, you can build a budget model to keep your expenses in check.