Graphing Inequalities A Grocery Bag Weight Problem

In the realm of mathematics, we often encounter real-world scenarios that can be elegantly represented and solved using algebraic concepts. Today, we're diving into a fascinating problem involving a grocery bag filled with apples and grapes, exploring the interplay between variables, inequalities, and graphical representations. This exploration will not only solidify your understanding of these mathematical tools but also demonstrate their practical applications in everyday situations. Our main focus is to decipher the constraints on the number of apples and the weight of grapes that can fit into a grocery bag without exceeding a weight limit. This is a perfect example of how inequalities can model real-world limitations, and how graphs can visually represent the range of possible solutions.

Imagine a grocery bag containing a mix of delicious fruits: apples and grapes. Let's say the bag holds $x$ apples, and each apple weighs $\frac{1}{3}$ of a pound. Additionally, there are $y$ pounds of grapes in the bag. The critical constraint we face is that the total weight of the grocery bag must be less than 5 pounds. Our mission is to determine which graph accurately represents the possible combinations of apples and grapes that can be in the bag while adhering to this weight restriction. This problem elegantly combines algebraic representation with graphical interpretation. We need to translate the word problem into a mathematical inequality, which will then allow us to visualize the solution set on a graph. The graph will act as a visual aid, showing us all the possible combinations of apples and grapes that satisfy the given condition. By analyzing the graph, we can gain a deeper understanding of the relationship between the number of apples and the weight of grapes, and how they both contribute to the overall weight of the bag. This exercise not only enhances our problem-solving skills but also highlights the power of mathematics in representing and analyzing real-world scenarios.

To begin, let's translate the given information into a mathematical inequality. The weight of the apples is calculated by multiplying the number of apples, $x$, by the weight of each apple, $\frac{1}{3}$ pound. This gives us $\frac{1}{3}x$ pounds for the total weight of the apples. We also have $y$ pounds of grapes in the bag. The combined weight of the apples and grapes is therefore $\frac{1}{3}x + y$. The problem states that the total weight must be less than 5 pounds. This translates directly into the inequality:

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

This inequality is the cornerstone of our problem. It mathematically expresses the constraint on the combined weight of the apples and grapes. Understanding this inequality is crucial, as it dictates the possible values of $x$ and $y$ that satisfy the condition. Any combination of apples and grapes that makes the left-hand side of the inequality less than 5 will be a valid solution. Conversely, any combination that results in a weight of 5 pounds or more is not permissible. This inequality is a linear inequality in two variables, which means its graph will be a region bounded by a straight line. To graph this inequality effectively, we will need to understand how to manipulate it into a more graph-friendly form, allowing us to easily identify the boundary line and the region that represents the solution set.

To make the inequality easier to graph, we can rearrange it to isolate $y$. This form, known as the slope-intercept form (similar to the equation of a line $y = mx + b$), will allow us to easily identify the slope and y-intercept, which are crucial for plotting the boundary line. Starting with the inequality:

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

We can subtract $\frac{1}{3}x$ from both sides to isolate $y$:

$$y < -\frac{1}{3}x + 5$$

Now, the inequality is in a form that is much easier to interpret graphically. We can see that the boundary line will have a slope of $-\frac{1}{3}$ and a y-intercept of 5. The negative slope indicates that the line will descend from left to right, and the y-intercept tells us where the line will cross the vertical axis. The "less than" sign (<) in the inequality tells us that the solution region will be below the line. This means that any point below the line on the graph will represent a combination of apples and grapes that satisfies the weight constraint. This transformation of the inequality is a key step in visualizing the solution set. By rearranging the inequality, we've made it much easier to identify the critical parameters needed to plot the boundary line and determine the region that represents all possible solutions. This demonstrates the power of algebraic manipulation in making mathematical concepts more accessible and understandable.

Now that we have the inequality in slope-intercept form ($y < -\frac{1}{3}x + 5$), we can proceed with graphing it. The boundary line is represented by the equation $y = -\frac{1}{3}x + 5$. To graph this line, we can use the slope and y-intercept we identified earlier. The y-intercept is 5, so we start by plotting a point at (0, 5) on the y-axis. The slope is $-\frac{1}{3}$, which means for every 3 units we move to the right on the x-axis, we move 1 unit down on the y-axis. Using this information, we can plot another point. For example, if we move 3 units to the right from (0, 5), we end up at x = 3, and we move 1 unit down to y = 4, giving us the point (3, 4). Now, we can draw a line through these two points. However, since our inequality uses a "less than" sign (<), the boundary line should be a dashed line rather than a solid line. This is because points on the line itself are not included in the solution set; the total weight must be strictly less than 5 pounds.

Next, we need to determine which side of the line represents the solution region. Since the inequality is $y < -\frac{1}{3}x + 5$, we are looking for all points where the y-value is less than the corresponding value on the line. This means we shade the region below the dashed line. The shaded region represents all possible combinations of apples ($x$) and grapes ($y$) that satisfy the condition that the total weight of the grocery bag is less than 5 pounds. It's important to remember that in the context of this problem, $x$ (the number of apples) and $y$ (the weight of grapes) cannot be negative. Therefore, we are only interested in the portion of the shaded region that lies in the first quadrant (where both x and y are positive). The graph visually represents the solution set, providing a clear and intuitive understanding of the relationship between the number of apples and the weight of grapes within the given constraint.

The graph we've created is a powerful tool for understanding the possible combinations of apples and grapes within the 5-pound weight limit. The shaded region, bounded by the dashed line and the axes, represents all the valid solutions. Any point within this region corresponds to a combination of $x$ apples and $y$ pounds of grapes that results in a total weight less than 5 pounds. For example, if we pick a point within the shaded region, say (3, 2), this corresponds to 3 apples and 2 pounds of grapes. We can verify that this combination satisfies our inequality:

$$\frac{1}{3}(3) + 2 = 1 + 2 = 3 < 5$$

So, 3 apples and 2 pounds of grapes is a valid combination.

On the other hand, any point outside the shaded region represents a combination that exceeds the 5-pound limit. For instance, let's consider the point (6, 4), which corresponds to 6 apples and 4 pounds of grapes. Plugging these values into our inequality, we get:

$$\frac{1}{3}(6) + 4 = 2 + 4 = 6 \nless 5$$

This combination results in a total weight of 6 pounds, which is greater than 5 pounds, so it's not a valid solution. The dashed line itself is crucial to our interpretation. It signifies that the points on the line are not included in the solution set. This is because the inequality is strictly "less than" 5 pounds. If the inequality were "less than or equal to" ($\leq$), then the boundary line would be solid, indicating that points on the line are also valid solutions. The graph provides a comprehensive visual representation of the solution space, allowing us to quickly assess whether a given combination of apples and grapes is feasible within the weight constraint. This demonstrates the power of graphical representation in making complex mathematical concepts more accessible and intuitive.

Now that we understand how to set up the inequality, transform it for graphing, and interpret the graph, we can tackle the original problem: identifying the correct graph from a set of options. The correct graph will have the following key features:

1. A dashed boundary line representing the equation $y = -\frac{1}{3}x + 5$.
2. A y-intercept at 5.
3. A negative slope (the line should descend from left to right).
4. The region below the dashed line should be shaded, representing the solution set where the total weight is less than 5 pounds.
5. The shaded region should be limited to the first quadrant, as the number of apples ($x$) and the weight of grapes ($y$) cannot be negative.

By carefully examining the provided graphs and comparing them to these criteria, we can systematically eliminate incorrect options and identify the graph that accurately represents the possible numbers of apples and pounds of grapes that can be in the grocery bag. This process reinforces our understanding of the relationship between the inequality and its graphical representation. It also highlights the importance of paying attention to details, such as the type of boundary line (dashed or solid) and the direction of shading, as these features convey crucial information about the solution set. By methodically analyzing each graph against the established criteria, we can confidently pinpoint the correct visual representation of the problem's constraints and solutions.

This problem, while seemingly simple, illustrates a powerful concept in mathematics: optimization under constraints. In the real world, we often face situations where we need to maximize or minimize a quantity (like the number of items we can carry) while adhering to certain limitations (like weight or budget). This grocery bag problem is a microcosm of these larger challenges. The inequality we derived represents a constraint, and the graph visually displays the feasible region, which is the set of all possible solutions that satisfy the constraint.

We can extend this problem in various ways to explore more complex scenarios. For instance, we could introduce a cost constraint, where each apple and each pound of grapes has a price, and we have a limited budget. This would add another inequality to the system, creating a system of linear inequalities. The solution would then be the intersection of the feasible regions of both inequalities. Another extension could involve considering different types of fruits with varying weights and prices, further complicating the problem but also making it more realistic.

These types of problems are foundational to fields like operations research and linear programming, which are used in logistics, supply chain management, finance, and many other industries. Understanding how to model constraints using inequalities and visualize solutions graphically is a valuable skill that extends far beyond the classroom. It allows us to analyze and optimize real-world situations, making informed decisions based on available resources and limitations. This grocery bag problem, therefore, serves as an excellent entry point into the fascinating world of mathematical modeling and its practical applications.

In this exploration, we've delved into the world of inequalities and graphing through the lens of a simple yet insightful grocery bag problem. We started by translating a real-world scenario into a mathematical inequality, representing the weight constraint on a bag of apples and grapes. We then learned how to transform this inequality into slope-intercept form, making it easier to graph. By plotting the boundary line and shading the appropriate region, we created a visual representation of the solution set, showcasing all possible combinations of apples and grapes that satisfy the weight limit. This graphical representation allowed us to interpret the solutions intuitively and identify valid combinations. We also discussed how to identify the correct graph from a set of options by focusing on key features like the type of boundary line, the y-intercept, the slope, and the shaded region.

Furthermore, we explored the real-world implications of this problem, highlighting its connection to optimization under constraints and its relevance to fields like operations research and linear programming. By extending the problem to include additional constraints, such as a budget limit, we saw how the complexity can increase while still relying on the same fundamental principles. This journey through inequalities and graphing demonstrates the power of mathematics in modeling and solving real-world problems. It also underscores the importance of visual representations in making abstract concepts more accessible and understandable. The grocery bag problem, therefore, serves as a valuable learning experience, reinforcing our understanding of mathematical tools and their practical applications.