Dan Bought $x$ Pounds Of Potatoes For \$0.85 Per Pound And $y$ Pounds Of Grapes For \$1.29 Per Pound. The Total Cost Was Less Than \$5. Which Inequality Represents His Purchase?A. $1.29x + 0.85y \ \textless \ 5$ B.

Introduction

In this article, we will explore the concept of inequalities and how they can be used to represent real-world problems. We will use the example of Dan's purchase of potatoes and grapes to illustrate how an inequality can be formed to represent a given situation.

The Problem

Dan bought $x$ pounds of potatoes for $0.85 per pound and $y$ pounds of grapes for $1.29 per pound. The total cost was less than $5. We need to find the inequality that represents Dan's purchase.

Step 1: Identify the Variables

The variables in this problem are $x$ and $y$, which represent the number of pounds of potatoes and grapes purchased, respectively.

Step 2: Identify the Constraints

The constraint in this problem is that the total cost of the purchase was less than $5. This can be represented as an inequality.

Step 3: Form the Inequality

To form the inequality, we need to multiply the price of each item by the number of pounds purchased and add the two products together. This will give us the total cost of the purchase.

The cost of potatoes is $0.85 per pound, so the total cost of potatoes is $0.85x$. The cost of grapes is $1.29 per pound, so the total cost of grapes is $1.29y$.

The total cost of the purchase is the sum of the cost of potatoes and the cost of grapes, which is $0.85x + 1.29y$.

Since the total cost was less than $5, we can write the inequality as $0.85x + 1.29y < 5$.

Step 4: Simplify the Inequality

We can simplify the inequality by multiplying both sides by 100 to get rid of the decimals.

$85x + 129y < 500$

Conclusion

The inequality that represents Dan's purchase is $85x + 129y < 500$. This inequality states that the total cost of the purchase is less than $500.

Answer

The correct answer is A. $1.29x + 0.85y \ \textless \ 5$ is not correct, the correct answer is $85x + 129y < 500$.

Discussion

In this article, we used the example of Dan's purchase of potatoes and grapes to illustrate how an inequality can be formed to represent a given situation. We identified the variables, constraints, and formed the inequality using the given information.

The inequality $85x + 129y < 500$ represents the situation where the total cost of the purchase is less than $500. This is a classic example of a linear inequality, which is a fundamental concept in mathematics.

Real-World Applications

Inequalities are used in many real-world applications, such as finance, economics, and engineering. For example, in finance, inequalities are used to model the behavior of financial markets and to make predictions about future stock prices.

In economics, inequalities are used to model the behavior of consumers and to make predictions about future demand for goods and services.

In engineering, inequalities are used to model the behavior of physical systems and to make predictions about future performance.

Conclusion

In conclusion, inequalities are a fundamental concept in mathematics that have many real-world applications. The example of Dan's purchase of potatoes and grapes illustrates how an inequality can be formed to represent a given situation.

The inequality $85x + 129y < 500$ represents the situation where the total cost of the purchase is less than $500. This is a classic example of a linear inequality, which is a fundamental concept in mathematics.

References

• [1] "Linear Inequalities" by Math Open Reference
• [2] "Inequalities in Finance" by Investopedia
• [3] "Inequalities in Economics" by Economics Help

Further Reading

• "Linear Algebra and Its Applications" by Gilbert Strang
• "Calculus: Early Transcendentals" by James Stewart
• "Mathematics for Economists" by Carl P. Simon and Lawrence Blume
  Dan's Purchase Inequality: Q&A ================================

Introduction

In our previous article, we explored the concept of inequalities and how they can be used to represent real-world problems. We used the example of Dan's purchase of potatoes and grapes to illustrate how an inequality can be formed to represent a given situation.

In this article, we will answer some frequently asked questions about Dan's purchase inequality.

Q: What is the inequality that represents Dan's purchase?

A: The inequality that represents Dan's purchase is $85x + 129y < 500$. This inequality states that the total cost of the purchase is less than $500.

Q: What are the variables in this inequality?

A: The variables in this inequality are $x$ and $y$, which represent the number of pounds of potatoes and grapes purchased, respectively.

Q: What is the constraint in this inequality?

A: The constraint in this inequality is that the total cost of the purchase was less than $5.

Q: How did you form the inequality?

A: To form the inequality, we multiplied the price of each item by the number of pounds purchased and added the two products together. This gave us the total cost of the purchase.

Q: Can you simplify the inequality?

A: Yes, we can simplify the inequality by multiplying both sides by 100 to get rid of the decimals. This gives us $85x + 129y < 500$.

Q: What is the significance of this inequality?

A: This inequality represents a real-world situation where the total cost of a purchase is less than a certain amount. It is a classic example of a linear inequality, which is a fundamental concept in mathematics.

Q: How can this inequality be used in real-world applications?

A: This inequality can be used in many real-world applications, such as finance, economics, and engineering. For example, in finance, inequalities are used to model the behavior of financial markets and to make predictions about future stock prices.

Q: What are some common mistakes to avoid when working with inequalities?

A: Some common mistakes to avoid when working with inequalities include:

• Not identifying the variables and constraints correctly
• Not forming the inequality correctly
• Not simplifying the inequality correctly
• Not understanding the significance of the inequality

Q: How can I practice working with inequalities?

A: You can practice working with inequalities by:

• Solving inequality problems
• Graphing inequality functions
• Using inequality formulas to model real-world situations
• Working with different types of inequalities, such as linear and quadratic inequalities

Conclusion

In conclusion, Dan's purchase inequality is a classic example of a linear inequality that represents a real-world situation. By understanding how to form and simplify inequalities, you can apply this concept to many real-world applications.

References

• [1] "Linear Inequalities" by Math Open Reference
• [2] "Inequalities in Finance" by Investopedia
• [3] "Inequalities in Economics" by Economics Help

Further Reading

• "Linear Algebra and Its Applications" by Gilbert Strang
• "Calculus: Early Transcendentals" by James Stewart
• "Mathematics for Economists" by Carl P. Simon and Lawrence Blume