Framing Art: Inequalities For Sam's Frame Purchase
Hey guys! Let's dive into a fun, practical problem involving a talented artist named Sam. Sam's got a vision: he wants to showcase his amazing artwork right in his own home. To do this, he needs frames—lots of them! But, like many of us, Sam's working with a budget and a specific number of pieces he wants to display. So, how can we help Sam figure out the best way to get those frames without breaking the bank? This is where the magic of inequalities comes into play. Inequalities, in math terms, help us define the limits or constraints of a situation. In Sam's case, we need to consider how many frames he wants to buy, how much he can spend, and the different costs of the frames available. Let's break it down step by step so we can set up the perfect system of inequalities for Sam.
Setting Up the Scenario
So, Sam is our artist, and he's got two main constraints: the number of frames and his budget. First, he wants to frame no fewer than 10 of his artworks. This means he needs at least 10 frames. Second, he can't spend more than $225 in total. Now, he's got two types of frames to choose from: large frames and medium frames. The large frames cost $24 each, while the medium frames cost $18 each. Our mission, should we choose to accept it, is to figure out a system of inequalities that represents these conditions. This system will help Sam (and us) understand the possible combinations of large and medium frames he can buy while staying within his constraints. Think of it like creating a roadmap that guides Sam to the perfect framing solution. We want to ensure he gets to display his art beautifully without overspending or falling short on the number of frames he needs. So, let's get started and define our variables to make this problem crystal clear.
Defining Variables
Before we can create our inequalities, we need to define our variables. Variables are like the placeholders in our mathematical expressions, representing the unknown quantities we're trying to figure out. In this case, we have two unknowns: the number of large frames and the number of medium frames Sam should buy. Let's assign variables to these: Let x represent the number of large frames Sam purchases. And, let y represent the number of medium frames Sam purchases. Now that we have our variables defined, we can start translating the constraints from our problem into mathematical inequalities. Remember, inequalities are used because we're dealing with ranges of possible values rather than exact numbers. Sam can buy at least 10 frames, and he can spend no more than $225. These phrases indicate that we'll be using inequalities like "greater than or equal to" (≥) and "less than or equal to" (≤). With our variables clearly defined, we're ready to build the inequalities that will guide Sam's frame-buying decision. This is where the real mathematical fun begins!
Inequality for the Number of Frames
Sam wants to frame no fewer than 10 pieces of his artwork. This means the total number of frames he buys must be greater than or equal to 10. Since x represents the number of large frames and y represents the number of medium frames, we can express this constraint as an inequality: x + y ≥ 10 This inequality tells us that the sum of the number of large frames and the number of medium frames must be at least 10. It's a simple but crucial piece of the puzzle. For example, Sam could buy 5 large frames and 5 medium frames (5 + 5 = 10), which satisfies this inequality. Or, he could buy 3 large frames and 8 medium frames (3 + 8 = 11), which also works. The key is that the total must be 10 or more. This inequality ensures that Sam meets his minimum requirement for displaying his artwork. Now, let's move on to the next constraint: Sam's budget. This will involve the costs of the frames and the total amount he can spend. Remember, large frames cost $24 each, and medium frames cost $18 each. We need to incorporate these costs into our next inequality to make sure Sam doesn't overspend.
Inequality for the Budget
Now, let's tackle Sam's budget. He can spend a maximum of $225. The cost of the large frames is $24 each, so the total cost for x large frames is 24x. Similarly, the cost of the medium frames is $18 each, so the total cost for y medium frames is 18y. The sum of these costs must be less than or equal to $225. We can write this as an inequality: 24x + 18y ≤ 225 This inequality ensures that Sam's total spending on frames stays within his budget. For example, if Sam buys 2 large frames and 5 medium frames, the total cost would be (24 * 2) + (18 * 5) = 48 + 90 = $138, which is within his budget. However, if he buys 5 large frames and 5 medium frames, the total cost would be (24 * 5) + (18 * 5) = 120 + 90 = $210, which is also within his budget. But what if he bought only large frames? He could afford 225 / 24 = 9.375, so at most 9 large frames, and 0 medium frames. The combination of this inequality and the previous one (x + y ≥ 10) will help Sam find the right balance between the number of frames and the cost. So, with both inequalities in place, we have a system that represents Sam's constraints. Let's put it all together and see what it looks like.
The System of Inequalities
Alright, guys, we've built our two inequalities, and now it's time to combine them into a system. A system of inequalities is simply a set of two or more inequalities that are considered together. In Sam's case, the system represents all the constraints he needs to satisfy simultaneously. Here's the system of inequalities we've created:
- x + y ≥ 10
- 24x + 18y ≤ 225
This system tells us that: The sum of the number of large frames (x) and the number of medium frames (y) must be greater than or equal to 10. The total cost of the frames (24x for large frames and 18y for medium frames) must be less than or equal to $225. This system of inequalities provides a mathematical model for Sam's framing problem. By solving this system, Sam can find the possible combinations of large and medium frames that meet his needs and stay within his budget. Graphing these inequalities can also give a visual representation of the feasible region, showing all possible solutions. So, with this system in hand, Sam is well-equipped to make an informed decision about his frame purchases. Let's recap the entire process to make sure we've covered everything.
Recapping the Process
Let's do a quick recap of what we've done to help Sam figure out his frame situation. We started by understanding Sam's constraints: he wants to frame at least 10 pieces of art and can't spend more than $225. We then defined our variables, with x representing the number of large frames and y representing the number of medium frames. Next, we translated Sam's constraints into mathematical inequalities: x + y ≥ 10 (the number of frames) and 24x + 18y ≤ 225 (the budget). Finally, we combined these inequalities into a system, which represents all the conditions Sam needs to meet simultaneously. This system of inequalities provides a clear, mathematical way for Sam to explore his options and find the best combination of large and medium frames for his artwork. By understanding and using these inequalities, Sam can confidently make his frame purchases, knowing he's staying within his budget and meeting his display goals. This problem demonstrates how math can be applied to real-life situations, helping us make informed decisions and solve practical challenges. Whether it's framing artwork, planning a budget, or anything in between, inequalities can be a powerful tool. Keep practicing, and you'll become a pro at using them to solve all sorts of problems!