At An End-of-the-year Sale, Gabriela Bought More Than 12 Bottles Of Hand Soaps And Lotions. If $x$ Represents The Number Of Hand Soaps And $y$ Represents The Number Of Lotions She Bought, Which Inequality Best Represents Her
At an End-of-the-Year Sale: Modeling Gabriela's Purchase with Inequalities
As the year comes to a close, many retailers offer end-of-the-year sales to clear out inventory and make room for new products. For Gabriela, this means scoring great deals on hand soaps and lotions. In this article, we'll explore how to model Gabriela's purchase using inequalities, a fundamental concept in mathematics.
Understanding the Problem
Gabriela bought more than 12 bottles of hand soaps and lotions. Let's denote the number of hand soaps as $x$ and the number of lotions as $y$. We want to find the inequality that best represents her purchase.
The Inequality
To model Gabriela's purchase, we need to consider the total number of bottles she bought. Since she bought more than 12 bottles, we can write an inequality to represent this situation. The inequality should be in the form of $x + y > 12$, where $x$ is the number of hand soaps and $y$ is the number of lotions.
Why This Inequality?
The inequality $x + y > 12$ represents the situation where the total number of bottles (hand soaps and lotions) is greater than 12. This is because the sum of the number of hand soaps ($x$) and the number of lotions ($y$) is always greater than 12, since Gabriela bought more than 12 bottles in total.
Visualizing the Inequality
To better understand the inequality, let's visualize it on a coordinate plane. We can plot the line $x + y = 12$, which represents the boundary between the region where Gabriela bought 12 or fewer bottles and the region where she bought more than 12 bottles.
# Inequality Representation
x + y > 12
Visualizing the Inequality
- Plot the line x + y = 12 on a coordinate plane
- Shade the region above the line to represent the inequality x + y > 12
Solving the Inequality
To solve the inequality $x + y > 12$, we can isolate one variable in terms of the other. Let's solve for $y$:
This inequality represents the region above the line $y = 12 - x$.
Graphing the Inequality
To graph the inequality, we can plot the line $y = 12 - x$ and shade the region above it. This will give us a visual representation of the inequality.
# Graphing the Inequality
y = 12 - x
Graphing the Inequality
- Plot the line y = 12 - x on a coordinate plane
- Shade the region above the line to represent the inequality y > 12 - x
Conclusion
In this article, we explored how to model Gabriela's purchase using inequalities. We found that the inequality $x + y > 12$ best represents her purchase, where $x$ is the number of hand soaps and $y$ is the number of lotions. We also visualized the inequality on a coordinate plane and solved for one variable in terms of the other. By understanding and working with inequalities, we can better model real-world situations and make informed decisions.
Key Takeaways
- Inequalities are used to model real-world situations where one quantity is greater than, less than, or equal to another quantity.
- The inequality $x + y > 12$ represents the situation where the total number of bottles (hand soaps and lotions) is greater than 12.
- To solve the inequality, we can isolate one variable in terms of the other.
- Graphing the inequality involves plotting the line and shading the region above it.
Further Exploration
- Explore other inequalities that model real-world situations, such as $x - y > 5$ or $2x + 3y > 15$.
- Practice solving and graphing inequalities to become more comfortable with this concept.
- Apply inequalities to real-world problems, such as budgeting, scheduling, or optimizing resources.
At an End-of-the-Year Sale: Modeling Gabriela's Purchase with Inequalities - Q&A
In our previous article, we explored how to model Gabriela's purchase using inequalities. We found that the inequality $x + y > 12$ best represents her purchase, where $x$ is the number of hand soaps and $y$ is the number of lotions. In this article, we'll answer some frequently asked questions about inequalities and modeling real-world situations.
Q: What is an inequality?
A: An inequality is a mathematical statement that compares two quantities, indicating that one quantity is greater than, less than, or equal to another quantity. Inequalities are used to model real-world situations where one quantity is not equal to another.
Q: How do I write an inequality?
A: To write an inequality, you need to compare two quantities. For example, if you want to represent the situation where the number of hand soaps ($x$) is greater than 5, you can write the inequality $x > 5$.
Q: What is the difference between an inequality and an equation?
A: An equation is a mathematical statement that states that two quantities are equal. For example, $x + 2 = 7$ is an equation. An inequality, on the other hand, states that one quantity is greater than, less than, or equal to another quantity. For example, $x + 2 > 7$ is an inequality.
Q: How do I solve an inequality?
A: To solve an inequality, you need to isolate one variable in terms of the other. For example, if you have the inequality $x + 2 > 7$, you can subtract 2 from both sides to get $x > 5$.
Q: Can I graph an inequality?
A: Yes, you can graph an inequality. To graph an inequality, you need to plot the line that represents the boundary between the region where the inequality is true and the region where it is false. For example, if you have the inequality $x + y > 12$, you can plot the line $x + y = 12$ and shade the region above it to represent the inequality.
Q: What are some real-world applications of inequalities?
A: Inequalities have many real-world applications, such as:
- Budgeting: Inequalities can be used to model budgeting situations, such as "I have $100 to spend on groceries, and I want to buy at least 5 pounds of chicken."
- Scheduling: Inequalities can be used to model scheduling situations, such as "I have 3 hours to complete a project, and I want to spend at least 2 hours on the most important task."
- Optimizing resources: Inequalities can be used to model optimizing resource situations, such as "I have 10 employees to assign to 3 tasks, and I want to assign at least 2 employees to each task."
Q: How can I practice solving and graphing inequalities?
A: You can practice solving and graphing inequalities by:
- Working through example problems in your textbook or online resources.
- Creating your own example problems and solving them.
- Graphing inequalities on a coordinate plane and shading the region above or below the line.
Q: What are some common mistakes to avoid when working with inequalities?
A: Some common mistakes to avoid when working with inequalities include:
- Forgetting to flip the inequality sign when multiplying or dividing both sides by a negative number.
- Not isolating one variable in terms of the other.
- Not plotting the line and shading the region correctly when graphing an inequality.
Conclusion
In this article, we answered some frequently asked questions about inequalities and modeling real-world situations. We hope that this article has helped you to better understand inequalities and how to apply them to real-world problems. Remember to practice solving and graphing inequalities to become more comfortable with this concept.
Key Takeaways
- Inequalities are used to model real-world situations where one quantity is greater than, less than, or equal to another quantity.
- To write an inequality, you need to compare two quantities.
- To solve an inequality, you need to isolate one variable in terms of the other.
- To graph an inequality, you need to plot the line and shade the region above or below it.
Further Exploration
- Explore other inequalities that model real-world situations, such as $x - y > 5$ or $2x + 3y > 15$.
- Practice solving and graphing inequalities to become more comfortable with this concept.
- Apply inequalities to real-world problems, such as budgeting, scheduling, or optimizing resources.