Tire Cost Inequality: Teresa's Budget Solution

Hey guys! Let's break down this math problem about Teresa buying new tires for her mountain bike. She's got a budget, but things might cost a little more or less than she planned. We need to figure out how to write an inequality that shows the possible cost of each tire. It sounds tricky, but we'll get through it together. So, buckle up and let's dive into the world of inequalities and tire costs!

Understanding the Problem

In this section, we're going to break down the problem step by step to make sure we fully understand what's going on.

First, let's talk about Teresa's tire budget. Teresa has set aside $120 for two new tires for her mountain bike. That's the base amount she's planning to spend. But here's the twist: she knows the actual cost might vary. After doing some research, Teresa realizes she might spend up to $25 more or less than her budgeted amount. This variation is super important because it means we can't just say the tires will cost exactly $120. We need to consider a range of possibilities. This is where inequalities come into play. Inequalities, unlike equations, allow us to express a range of values rather than a single, exact number. They use symbols like < (less than), > (greater than), ≤ (less than or equal to), and ≥ (greater than or equal to). These symbols will help us define the boundaries of Teresa's spending. Now, let's think about what we're trying to find. The core of the problem is to determine the possible cost of each tire. Teresa is buying two tires, and the $120 budget (plus or minus the $25) covers the total cost for both. So, we'll need to figure out how to represent the cost of a single tire in our inequality. This involves dividing the total possible cost range by two, since she's buying two tires. By understanding this, we can better construct our inequality, which will show the minimum and maximum amount Teresa might spend on a single tire.

Setting up the Inequality

Okay, so how do we actually write this thing? Let's break it down. The key here is to represent the unknown – the cost of each tire – with a variable. Let's use 'x' to stand for the cost of one tire. Since Teresa is buying two tires, the total cost for both tires would be 2 * x, which we write as 2x. Now, let's think about the range of possible spending. Teresa's budget is $120, but she might spend $25 more or less. This gives us two important numbers to work with. The lowest amount Teresa might spend is $120 minus $25, which is $95. The highest amount she might spend is $120 plus $25, which is $145. So, the total cost for the two tires (2x) will fall somewhere between $95 and $145. This is where the inequality comes in. We can express this range using two inequalities combined into one. We know that 2x must be greater than or equal to $95 (the lowest possible cost) and less than or equal to $145 (the highest possible cost). We can write this as a compound inequality: 95 ≤ 2x ≤ 145. This inequality tells us that the total cost of the two tires, represented by 2x, is somewhere between $95 and $145, including those amounts. This is a super important step, guys, because it sets the stage for figuring out the cost of a single tire. Now that we have the total cost range, we can move on to isolating 'x' and finding the range for the cost of each individual tire.

Solving the Inequality

Alright, we've got our inequality: 95 ≤ 2x ≤ 145. Now, let's get down to solving it. Remember, our goal is to figure out the possible cost range for one tire, so we need to isolate 'x' in the middle of the inequality. To do this, we need to get rid of the '2' that's multiplying 'x'. And how do we do that? By dividing! But here's the important thing: because we're dealing with an inequality, whatever we do to one part, we have to do to all parts. So, we're going to divide every single part of the inequality by 2. That means we'll divide 95 by 2, 2x by 2, and 145 by 2. Let's do the math. 95 divided by 2 is 47.5. 2x divided by 2 is simply x (that's what we wanted!). And 145 divided by 2 is 72.5. So, after dividing, our inequality looks like this: 47.5 ≤ x ≤ 72.5. What does this mean? It tells us that the cost of one tire (x) must be greater than or equal to $47.50 and less than or equal to $72.50. In other words, each tire could cost anywhere between $47.50 and $72.50 for Teresa to stay within her budget plus the $25 variation. See? We're making progress! Now we have a clear range for the cost of each tire. In the next section, we'll put it all together and talk about what our solution really means in the context of Teresa's tire-buying adventure.

Interpreting the Solution

Woohoo! We've solved the inequality, and we know that 47.5 ≤ x ≤ 72.5. But what does this actually mean for Teresa and her tire shopping? Well, let's break it down. This inequality tells us the possible range of prices for each tire. The 'x' represents the cost of one tire, and the inequality states that this cost must be between $47.50 and $72.50, including those amounts. So, the cheapest each tire could be is $47.50, and the most expensive it could be is $72.50, if Teresa wants to stay within her budget, give or take that $25 wiggle room. This is super helpful information for Teresa! She can now go to the bike shop or browse online knowing what price range to look for. If she finds tires that cost less than $47.50 each, that's great – she'll be under budget! But if she finds tires that cost more than $72.50 each, she'll know she needs to either adjust her budget or look for a different set of tires. Interpreting the solution is just as important as solving the inequality itself. It's about understanding what the math actually means in the real world. In this case, it gives Teresa a practical guide for making her purchase. We’ve taken a mathematical problem and turned it into something useful for Teresa. That's the power of math, guys! We can use these skills in all sorts of everyday situations. Now, let’s wrap up with a quick recap of everything we’ve done.

Conclusion

Alright, we did it! We took Teresa's tire budget dilemma and turned it into a math problem we could solve. Let's quickly recap the steps we took to get here. First, we carefully understood the problem. We figured out that Teresa had a budget of $120 for two tires, but she might spend up to $25 more or less. This meant we needed to think in terms of a range of possible costs, not just one fixed number. Next, we set up the inequality. We used 'x' to represent the cost of one tire, and we wrote the compound inequality 95 ≤ 2x ≤ 145 to show the total possible cost range for both tires. Then, we solved the inequality. We divided all parts of the inequality by 2 to isolate 'x' and find the possible cost range for a single tire, which gave us 47.5 ≤ x ≤ 72.5. Finally, we interpreted the solution. We figured out that each tire could cost between $47.50 and $72.50 for Teresa to stay within her budget. Awesome! By breaking down the problem step by step, we were able to turn a potentially confusing situation into a clear and understandable solution. This is a great example of how math can be used to solve real-world problems. So next time you're faced with a budgeting question or any situation involving a range of possibilities, remember the power of inequalities! You've got this!