Inequality For Savings: Martina's Cell Phone Goal

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Hey guys! Let's break down this math problem about Martina and her quest to buy a new cell phone. It's all about setting up the right inequality to figure out how many weeks she needs to save. We'll go through it step by step, making sure it's super clear. So, grab your thinking caps, and let's dive in!

Understanding the Problem

First, let's recap the situation. Martina has her eyes on a cell phone that costs $800,000. Right now, she's got $200,000 saved up. The awesome news is that she's planning to save an additional $50,000 every single week. The question we need to answer is: which inequality will help us figure out the number of weeks (w) Martina needs to save to reach her goal?

To solve this, we need to translate the word problem into a mathematical expression. Think of it like this: her current savings plus the money she saves each week needs to be greater than or equal to the cost of the phone. That's the key idea here. We're not just looking for when she has exactly $800,000; she needs to have at least that amount.

Now, let's start building our inequality. We know her current savings is a constant, and the amount she saves per week will depend on the number of weeks. This is where our variable w comes in. We need to combine these pieces in a way that reflects the total savings Martina will have. We'll see how to do this in the next section.

Building the Inequality

Okay, so let's put the pieces together to create our inequality. We know Martina starts with $200,000. This is our base amount. Then, she saves $50,000 each week, which we can represent as $50,000 w, where w is the number of weeks. So, the total amount Martina has saved after w weeks is $200,000 + $50,000w.

Now, here's the crucial part: Martina needs to have at least $800,000 to buy the phone. In mathematical terms, "at least" means greater than or equal to. So, the total amount she saves must be greater than or equal to $800,000. This gives us the inequality:

$200,000 + $50,000w ≥ $800,000

This inequality is the heart of the problem. It tells us that Martina's initial savings plus her weekly savings multiplied by the number of weeks must be greater than or equal to the cost of the phone. The symbol ≥ is super important here; it makes sure we're including the possibility that she saves exactly $800,000, which is enough to buy the phone.

In the next part, we'll look at why this inequality is the right representation and how it helps us understand the problem better. We'll also think about what the other options might look like and why they wouldn't be correct.

Why This Inequality Works

Let's really make sure we understand why the inequality $200,000 + $50,000w ≥ $800,000 is the correct way to represent Martina's savings goal. Think of it as a story: Martina starts with $200,000, and every week, her savings grow by $50,000. We need to find the point where her savings are enough to buy the phone.

The inequality captures this perfectly. The left side, $200,000 + $50,000w, represents Martina's total savings after w weeks. The right side, $800,000, is the target she needs to reach. The "greater than or equal to" symbol (≥) is key because it means Martina can buy the phone when her savings are exactly $800,000 or more. If we used just a "greater than" symbol (>), we'd be saying she needs to have more than $800,000, which isn't quite right.

Now, let's think about what other inequalities might look like and why they wouldn't work. For example, an inequality like $200,000 + $50,000w ≤ $800,000 would mean Martina's savings need to be less than or equal to the phone's price, which is the opposite of what we want. An inequality with a subtraction, like $200,000 - $50,000w ≥ $800,000, wouldn't make sense because Martina is saving money, not losing it.

So, our inequality correctly shows that Martina's starting money plus her weekly savings must be at least the price of the cell phone. In the final section, we'll recap everything and highlight the most important takeaways from this problem.

Key Takeaways

Alright, let's wrap things up and make sure we've got the main points down. This problem was all about translating a real-world scenario—Martina saving for a phone—into a mathematical inequality. We saw how each part of the inequality represents a piece of the story:

  • $200,000: Martina's initial savings.
  • $50,000w: The amount she saves each week, multiplied by the number of weeks.
  • $800,000: The cost of the cell phone.
  • ≥: The "greater than or equal to" symbol, meaning Martina needs to save at least this much.

The final inequality, $200,000 + $50,000w ≥ $800,000, perfectly captures the relationship between these amounts. Remember, the key to these problems is breaking them down step by step. Identify the knowns (like Martina's starting savings and the phone's price), figure out the unknown (the number of weeks), and then use the right mathematical symbols to connect them.

Inequalities are super useful in real life, whether you're saving for a phone, planning a budget, or figuring out how much time you need to study for a test. They help us set goals and understand the limits we need to work within.

So, next time you see a word problem, don't sweat it! Just take it one step at a time, and you'll be able to turn it into math in no time. Keep practicing, guys, and you'll become inequality pros!