Hamburger & Cheeseburger Sales: Inequality Math

Hey guys, let's dive into a fun math problem that's super relevant if you've ever thought about running a food stand, or even just managing resources! We've got Dwayne here, who's in the business of slinging delicious hamburgers and cheeseburgers. He's got a crucial limitation: 100 burger buns. Each hamburger, let's call the number of these h, goes for $3.00. And each cheeseburger, which we'll denote as c, is a bit pricier at $3.50. The big question is: Which system of inequalities best represents the number of hamburgers (h) and cheeseburgers (c) he can make, given his limited buns? This isn't just about making money; it's about understanding constraints and how they shape our options. When we talk about systems of inequalities, we're essentially setting up boundaries for our variables. Think of it like this: you can't just make an infinite number of burgers because you'll eventually run out of buns, right? So, Dwayne's situation gives us a classic example of how math helps us model real-world limitations. We need to figure out not just one condition, but a set of conditions that Dwayne's burger sales must satisfy. This means we'll be looking at the number of buns, the types of burgers, and potentially other factors that might come into play. It's a great way to get a grip on algebra and see how it applies to everyday scenarios. We'll break down exactly why certain inequalities are formed and what they mean in terms of Dwayne's burger business. So, stick around as we unravel this tasty mathematical puzzle!

Understanding the Constraints: The Buns are Key!

Alright, let's get down to brass tacks, folks. The most critical constraint Dwayne faces is his supply of burger buns. He has exactly 100 burger buns. Now, whether he's making a plain hamburger or a fancy cheeseburger, each one requires one bun. This is the fundamental piece of information we need to start building our mathematical model. So, if h represents the number of hamburgers and c represents the number of cheeseburgers, the total number of buns used will be the sum of buns for hamburgers and buns for cheeseburgers. Since each burger uses one bun, the total buns used is simply h + c. Because Dwayne only has 100 buns, the total number of buns he uses cannot exceed this amount. It can be exactly 100, or it can be less than 100 if he decides not to use all his buns for some reason (though typically, we assume he wants to maximize production). This leads us directly to our first inequality: h + c ≤ 100. This inequality is super important because it directly translates the physical limitation of the buns into a mathematical statement. It tells us that any combination of hamburgers (h) and cheeseburgers (c) that Dwayne chooses to make must have a total that is less than or equal to 100. For instance, he could make 50 hamburgers and 50 cheeseburgers (50 + 50 = 100), which perfectly uses up all the buns. He could also make 30 hamburgers and 40 cheeseburgers (30 + 40 = 70), which leaves him with 30 buns leftover. However, he cannot make 70 hamburgers and 50 cheeseburgers because that would require 120 buns (70 + 50 = 120), and he simply doesn't have that many. This inequality sets a clear upper limit on his production based on the bun availability. It’s the cornerstone of our system, guys, because without enough buns, no burgers can be made, regardless of how much money he could potentially earn. This constraint is the first layer of our mathematical representation.

Beyond Buns: The Non-Negative Reality

Now, let's think about the nature of the things we're counting: hamburgers (h) and cheeseburgers (c). Can Dwayne make a negative number of hamburgers? Absolutely not! You can't produce or sell -5 hamburgers, can you? The same goes for cheeseburgers. This might seem like a super obvious point, but in mathematics, we need to explicitly state these kinds of basic assumptions. When we talk about quantities of physical items like burgers, they must be non-negative. This means the number of hamburgers (h) must be greater than or equal to zero, and the number of cheeseburgers (c) must also be greater than or equal to zero. Mathematically, we write these conditions as two separate inequalities: h ≥ 0 and c ≥ 0. These are often called non-negativity constraints, and they are fundamental in many real-world optimization problems, especially those involving quantities of goods or resources. Without these constraints, our mathematical model might suggest solutions that don't make any practical sense. For example, if we only had the h + c ≤ 100 inequality, a hypothetical solution like h = -10 and c = 50 would technically satisfy the inequality (-10 + 50 = 40, which is ≤ 100), but it's a nonsensical answer in the context of selling burgers. Therefore, including h ≥ 0 and c ≥ 0 ensures that our solutions are realistic and grounded in the physical world. These two simple inequalities, combined with the bun constraint, form the core of the system that governs Dwayne's burger production. They ensure that we're only considering valid, achievable numbers of burgers.

Putting It All Together: The System of Inequalities

So, we've identified the key limitations and realities facing Dwayne's burger business. We have the constraint on the total number of buns, and we have the fundamental requirement that the number of burgers cannot be negative. To represent the entire situation mathematically, we need to combine all these conditions into a single system of inequalities. A system of inequalities is simply a collection of two or more inequalities that must all be true simultaneously. In Dwayne's case, all the conditions we discussed must hold true for any valid production plan he comes up with. The conditions are:

1. The total number of burgers (h + c) cannot exceed the number of buns (100). This translates to: h + c ≤ 100
2. The number of hamburgers (h) must be zero or a positive number. This translates to: h ≥ 0
3. The number of cheeseburgers (c) must be zero or a positive number. This translates to: c ≥ 0

When we write these three inequalities together, we get the complete system that represents Dwayne's situation:

h + c ≤ 100
h ≥ 0
c ≥ 0

This set of inequalities is what defines the feasible region for Dwayne's burger production. Any pair of values for h and c that satisfies all three inequalities is a possible combination of hamburgers and cheeseburgers he can make. For example, making 60 hamburgers and 30 cheeseburgers is valid because 60 + 30 = 90 (which is ≤ 100), and both 60 and 30 are ≥ 0. Making 100 hamburgers and 0 cheeseburgers is also valid (100 + 0 = 100 ≤ 100, and 100 ≥ 0, 0 ≥ 0). However, making 70 hamburgers and 40 cheeseburgers is not valid because 70 + 40 = 110, which is greater than 100. This system is the mathematical blueprint for Dwayne's operational possibilities, guys. It’s the language of constraints!

What About the Prices? Do They Matter for This Question?

This is a super important point to clarify, and it often trips people up! The problem asks for the system of inequalities that represents the number of hamburgers (h) and cheeseburgers (c) Dwayne must make, given his limited buns. The prices of the burgers ($3 for a hamburger, $3.50 for a cheeseburger) are given, but they are not used in forming the inequalities for this specific question. Why? Because the question is only about the physical limitations imposed by the buns. The prices would become relevant if the question were different, for example, if it asked: "What is the maximum amount of money Dwayne can make?" or "If Dwayne wants to make at least $200, what combinations of burgers are possible?"

In those scenarios, we would introduce another inequality (or perhaps an objective function to maximize) that incorporates the prices. For instance, to represent the total revenue, we'd use an expression like 3h + 3.50c. If Dwayne wanted to make at least $200, the inequality might be 3h + 3.50c ≥ 200. However, for the question as stated – focusing solely on the number of burgers based on the bun constraint – the prices are extra information that we don't need for this particular set of inequalities. It's crucial to identify what the question is actually asking for. Sometimes, word problems throw in extra details to test your understanding of what information is relevant to the specific mathematical concept being applied. So, while the prices are interesting from a business perspective, for the system of inequalities based on bun count, they are not part of the equations. We stick to the constraints that directly limit the quantity of h and c based on the available buns and the non-negative nature of items.

Visualizing the Solution: The Feasible Region

Let's take this a step further, guys, because seeing is believing! We can actually graph this system of inequalities to visualize all the possible combinations of hamburgers (h) and cheeseburgers (c) that Dwayne can make. Remember our system?

h + c ≤ 100
h ≥ 0
c ≥ 0

When we graph inequalities, we usually treat h as our horizontal axis (the x-axis) and c as our vertical axis (the y-axis). The inequalities h ≥ 0 and c ≥ 0 tell us that we are only interested in the first quadrant of the coordinate plane. This makes perfect sense because we can't have negative burgers! So, our entire possible region is above the h-axis and to the right of the c-axis.

Now, let's look at the main inequality: h + c ≤ 100. To graph this, we first consider the line h + c = 100. If we plot this line, it intercepts the h-axis at (100, 0) (when c=0, h=100) and the c-axis at (0, 100) (when h=0, c=100). We can draw a straight line connecting these two points.

Since the inequality is ≤ (less than or equal to), we need to shade the region that satisfies this condition. We can test a point, like the origin (0,0). Is 0 + 0 ≤ 100? Yes, it is! So, we shade the area below the line h + c = 100.

Combining all our conditions, the region that satisfies h ≥ 0, c ≥ 0, and h + c ≤ 100 is a triangle. The vertices (corners) of this triangle are at:

• (0, 0): 0 hamburgers, 0 cheeseburgers (no burgers made)
• (100, 0): 100 hamburgers, 0 cheeseburgers (all buns for hamburgers)
• (0, 100): 0 hamburgers, 100 cheeseburgers (all buns for cheeseburgers)

Any point with integer coordinates (since you can't make fractions of burgers) within or on the boundary of this triangle represents a valid combination of hamburgers and cheeseburgers that Dwayne can produce given his 100 buns. This visual representation, the feasible region, is incredibly powerful. It shows us the entire scope of possibilities. If Dwayne wanted to maximize profit, he'd evaluate the profit at each of these vertices (and potentially other points if the profit function wasn't linear, but that's another story!). For this problem, understanding that the system of inequalities defines this triangular region is the key takeaway. It’s a neat way to see math come alive, right?

Conclusion: Mastering the Math of Limitations

So there you have it, gang! We've successfully broken down Dwayne's burger-selling scenario into a clear and concise system of inequalities. The core challenge was understanding the constraints: the finite number of burger buns and the non-negative nature of producing items. By translating these real-world limitations into mathematical statements, we arrived at the system:

h + c ≤ 100
h ≥ 0
c ≥ 0

This system perfectly captures the boundaries within which Dwayne must operate. It tells us that the total number of hamburgers (h) and cheeseburgers (c) cannot exceed 100, and that neither h nor c can be negative. We also clarified that the prices, while important for profit calculations, are not part of this specific system of inequalities because the question was focused purely on the quantity constraints. Finally, we touched upon how this system defines a visualizable feasible region – a triangle on a graph – that represents all possible valid combinations of burgers.

Understanding systems of inequalities like this is a fundamental skill in mathematics, especially when you start moving into areas like linear programming. It's all about setting up the rules of the game mathematically so you can analyze the possibilities, make informed decisions, or find optimal solutions. Whether you're managing inventory, planning a project, or even just figuring out how much pizza you can order for a party based on a budget, these principles apply. Keep practicing, keep questioning, and you'll find that math is an incredibly powerful tool for navigating the real world. Happy problem-solving, everyone!