A Grocery Bag Contains $x$ Apples, Each Weighing $\frac{1}{3}$ Of A Pound, And \$y$[/tex\] Pounds Of Grapes. The Total Weight Of The Grocery Bag Is Less Than 5 Pounds. Which Graph Represents The Possible Numbers Of

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Introduction

When it comes to understanding the relationship between different variables, graphing can be a powerful tool. In this article, we will explore a scenario where a grocery bag contains apples and grapes, and we need to determine the possible numbers of apples and grapes based on their weights. We will use algebraic equations to represent the situation and then graph the possible combinations of apples and grapes.

The Problem

A grocery bag contains $x$ apples, each weighing $\frac{1}{3}$ of a pound, and $y$ pounds of grapes. The total weight of the grocery bag is less than 5 pounds. We can represent this situation using the following inequality:

13x+y<5\frac{1}{3}x + y < 5

Understanding the Inequality

To understand the inequality, let's break it down into two parts:

  • \frac{1}{3}x$ represents the total weight of the apples in the bag.

  • y$ represents the weight of the grapes in the bag.

The inequality states that the sum of the weight of the apples and the weight of the grapes is less than 5 pounds.

Graphing the Inequality

To graph the inequality, we can start by graphing the equation $\frac{1}{3}x + y = 5$. This will give us a line that represents the boundary of the inequality.

import matplotlib.pyplot as plt
import numpy as np

# Define the x and y values
x = np.linspace(0, 15, 100)
y = 5 - (1/3)*x

# Create the plot
plt.plot(x, y)
plt.xlabel('x (weight of apples)')
plt.ylabel('y (weight of grapes)')
plt.title('Graph of the Equation')
plt.grid(True)
plt.show()

Graphing the Inequality

Now that we have the graph of the equation, we can shade the region below the line to represent the inequality.

import matplotlib.pyplot as plt
import numpy as np

# Define the x and y values
x = np.linspace(0, 15, 100)
y = 5 - (1/3)*x

# Create the plot
plt.plot(x, y)
plt.xlabel('x (weight of apples)')
plt.ylabel('y (weight of grapes)')
plt.title('Graph of the Inequality')
plt.fill_between(x, y, color='blue', alpha=0.3)
plt.grid(True)
plt.show()

Understanding the Graph

The graph represents the possible combinations of apples and grapes in the grocery bag. The x-axis represents the weight of the apples, and the y-axis represents the weight of the grapes. The shaded region below the line represents the possible combinations of apples and grapes that satisfy the inequality.

Conclusion

In this article, we used algebraic equations to represent the situation of a grocery bag containing apples and grapes. We then graphed the inequality to determine the possible combinations of apples and grapes. The graph represents the possible numbers of apples and grapes in the grocery bag, and it can be used to make informed decisions about the contents of the bag.

Discussion

The graph can be used to answer questions such as:

  • What is the maximum weight of the apples that can be in the bag?
  • What is the minimum weight of the grapes that can be in the bag?
  • What is the maximum weight of the grapes that can be in the bag?
  • What is the minimum weight of the apples that can be in the bag?

These questions can be answered by analyzing the graph and using the information it provides.

Final Thoughts

Graphing can be a powerful tool for understanding complex relationships between variables. In this article, we used graphing to determine the possible combinations of apples and grapes in a grocery bag. The graph represents the possible numbers of apples and grapes in the bag, and it can be used to make informed decisions about the contents of the bag.

References

Further Reading

Code

import matplotlib.pyplot as plt
import numpy as np

# Define the x and y values
x = np.linspace(0, 15, 100)
y = 5 - (1/3)*x

# Create the plot
plt.plot(x, y)
plt.xlabel('x (weight of apples)')
plt.ylabel('y (weight of grapes)')
plt.title('Graph of the Inequality')
plt.fill_between(x, y, color='blue', alpha=0.3)
plt.grid(True)
plt.show()
```<br/>
# A Grocery Bag Contains Apples and Grapes: Understanding the Possible Numbers of Apples and Grapes - Q&A

## Introduction

In our previous article, we explored a scenario where a grocery bag contains apples and grapes, and we needed to determine the possible numbers of apples and grapes based on their weights. We used algebraic equations to represent the situation and then graphed the inequality to determine the possible combinations of apples and grapes. In this article, we will answer some frequently asked questions about the scenario.

## Q&A

### Q1: What is the maximum weight of the apples that can be in the bag?

A1: The maximum weight of the apples that can be in the bag is 15 pounds. This is because the inequality $\frac{1}{3}x + y < 5$ can be rewritten as $x < 15$, which means that the weight of the apples cannot exceed 15 pounds.

### Q2: What is the minimum weight of the grapes that can be in the bag?

A2: The minimum weight of the grapes that can be in the bag is 0 pounds. This is because the inequality $\frac{1}{3}x + y < 5$ can be rewritten as $y > -\frac{1}{3}x + 5$, which means that the weight of the grapes must be greater than 0 pounds.

### Q3: What is the maximum weight of the grapes that can be in the bag?

A3: The maximum weight of the grapes that can be in the bag is 5 pounds. This is because the inequality $\frac{1}{3}x + y < 5$ can be rewritten as $y < 5 - \frac{1}{3}x$, which means that the weight of the grapes cannot exceed 5 pounds.

### Q4: What is the minimum weight of the apples that can be in the bag?

A4: The minimum weight of the apples that can be in the bag is 0 pounds. This is because the inequality $\frac{1}{3}x + y < 5$ can be rewritten as $x > -3y + 15$, which means that the weight of the apples must be greater than 0 pounds.

### Q5: Can the bag contain only apples?

A5: Yes, the bag can contain only apples. This is because the inequality $\frac{1}{3}x + y < 5$ can be rewritten as $x < 15$, which means that the weight of the apples can be 15 pounds or less, and the weight of the grapes can be 0 pounds.

### Q6: Can the bag contain only grapes?

A6: No, the bag cannot contain only grapes. This is because the inequality $\frac{1}{3}x + y < 5$ can be rewritten as $y < 5 - \frac{1}{3}x$, which means that the weight of the grapes cannot exceed 5 pounds, and the weight of the apples must be greater than 0 pounds.

### Q7: What is the relationship between the weight of the apples and the weight of the grapes?

A7: The weight of the apples and the weight of the grapes are inversely related. This is because the inequality $\frac{1}{3}x + y < 5$ can be rewritten as $x < 15 - 3y$, which means that as the weight of the apples increases, the weight of the grapes decreases, and vice versa.

## Conclusion

In this article, we answered some frequently asked questions about the scenario of a grocery bag containing apples and grapes. We used algebraic equations to represent the situation and then graphed the inequality to determine the possible combinations of apples and grapes. The graph represents the possible numbers of apples and grapes in the bag, and it can be used to make informed decisions about the contents of the bag.

## Discussion

The graph can be used to answer questions such as:

*   What is the maximum weight of the apples that can be in the bag?
*   What is the minimum weight of the grapes that can be in the bag?
*   What is the maximum weight of the grapes that can be in the bag?
*   What is the minimum weight of the apples that can be in the bag?

These questions can be answered by analyzing the graph and using the information it provides.

## Final Thoughts

Graphing can be a powerful tool for understanding complex relationships between variables. In this article, we used graphing to determine the possible combinations of apples and grapes in a grocery bag. The graph represents the possible numbers of apples and grapes in the bag, and it can be used to make informed decisions about the contents of the bag.

## References

*   [Graphing Inequalities](https://www.mathsisfun.com/inequalities/graphing-inequalities.html)
*   [Algebraic Equations](https://www.mathsisfun.com/algebra/equations.html)

## Further Reading

*   [Graphing Quadratic Equations](https://www.mathsisfun.com/algebra/graphing-quadratic-equations.html)
*   [Solving Systems of Linear Equations](https://www.mathsisfun.com/algebra/systems-linear-equations.html)

## Code

```python
import matplotlib.pyplot as plt
import numpy as np

# Define the x and y values
x = np.linspace(0, 15, 100)
y = 5 - (1/3)*x

# Create the plot
plt.plot(x, y)
plt.xlabel('x (weight of apples)')
plt.ylabel('y (weight of grapes)')
plt.title('Graph of the Inequality')
plt.fill_between(x, y, color='blue', alpha=0.3)
plt.grid(True)
plt.show()