Inequality For Daisy's Earnings: Hours To Earn $280

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Hey guys! Let's break down this problem about Daisy's earnings and figure out which inequality helps us determine how many hours she needs to work to make at least $280. This is a classic example of how math concepts, like inequalities, can be applied to real-life situations. So, grab your thinking caps, and let's dive in!

Understanding the Problem

The core of this problem lies in understanding the relationship between Daisy's hourly wage, the number of hours she works, and her desired earnings. Daisy earns $10 for every hour she works, and she wants to earn at least $280. The phrase "at least" is super important here because it tells us we're dealing with an inequality, not just a simple equation. We need to find an inequality that represents this situation accurately.

To really get this, let's think step-by-step. If Daisy works 1 hour, she earns $10. If she works 2 hours, she earns $20, and so on. We can see that her total earnings are calculated by multiplying her hourly wage ($10) by the number of hours she works (which we'll call 'h'). The question asks us to find the number of hours Daisy needs to work to earn at least $280. This means her earnings should be equal to or greater than $280. This "equal to or greater than" is the key to choosing the correct inequality symbol.

Let's consider a few scenarios. If Daisy wants to earn exactly $280, we could set up an equation: 10h = 280. This would tell us the exact number of hours she needs to work. But the question asks for at least $280, meaning she could earn more. This is where inequalities come into play. Inequalities allow us to represent a range of possible values, not just a single value.

So, how do we translate "at least $280" into a mathematical symbol? "At least" means the value can be equal to $280 or greater than $280. In mathematical notation, this is represented by the "greater than or equal to" symbol, which looks like this: ≥. This symbol is crucial for setting up the correct inequality.

Now, let's consider why the other options might be incorrect. An inequality using the "less than or equal to" symbol (≤) would represent Daisy earning no more than $280, which isn't what the problem asks. An inequality using the "greater than" symbol (>) would mean Daisy needs to earn more than $280, but not exactly $280, which also isn't quite right. We need to include the possibility of her earning exactly $280.

By carefully dissecting the problem and understanding the meaning of "at least," we can confidently choose the correct inequality. It's all about translating the words into mathematical symbols and understanding what each symbol represents.

Analyzing the Options

Okay, let's look at the answer choices provided and see which one fits our understanding of the problem. We need an inequality that shows Daisy's total earnings (which is $10 multiplied by the number of hours, 'h') being greater than or equal to $280.

Here are the options:

A. $10h

B. 10h>28010h > 280

C. 10h≥28010h ≥ 280

Now, let’s break down each option and see why some are right and others are wrong.

Option A, 10h≤28010h ≤ 280, uses the "less than or equal to" symbol (≤). This inequality would mean that ten times the number of hours Daisy works is less than or equal to $280. In simpler terms, it represents the scenario where Daisy earns at most $280. This is not what we want because the problem specifically states Daisy needs to earn at least $280. This option limits her earnings to $280 or less, which doesn't match the problem's condition. So, we can eliminate option A.

Option B, 10h>28010h > 280, uses the "greater than" symbol (>). This inequality means that ten times the number of hours Daisy works is greater than $280. While this option does represent Daisy earning more than $280, it doesn't include the possibility of her earning exactly $280. Remember, the problem says "at least $280," which means $280 is also a valid earning amount. This option is close, but it misses a crucial part of the condition, so it's not the correct answer.

Option C, 10h≥28010h ≥ 280, uses the "greater than or equal to" symbol (≥). This inequality means that ten times the number of hours Daisy works is greater than or equal to $280. This is exactly what we're looking for! It represents Daisy earning $280 or more, fulfilling the condition of earning at least $280. This option includes both the possibility of earning more than $280 and the possibility of earning exactly $280. Therefore, option C is the correct answer.

By carefully analyzing each option and comparing it to our understanding of the problem, we can confidently identify the correct inequality. It's all about understanding the nuances of mathematical symbols and how they translate to real-world scenarios.

The Correct Inequality

So, based on our analysis, the correct inequality is:

C. $10h ≥ 280

This inequality accurately represents the situation because it states that Daisy's earnings ($10 multiplied by the number of hours she works, 'h') must be greater than or equal to $280. The "greater than or equal to" symbol (≥) is key here because it includes the possibility of Daisy earning exactly $280, as well as earning more than $280, which aligns perfectly with the "at least" condition in the problem.

To further solidify our understanding, let's think about what this inequality tells us in practical terms. If we were to solve this inequality for 'h', we would divide both sides by 10:

h ≥ 28

This result means that Daisy needs to work 28 hours or more to earn at least $280. If she works exactly 28 hours, she will earn $280 (10 * 28 = 280). If she works more than 28 hours, she will earn more than $280. This perfectly matches the problem's requirements.

Now, let's briefly revisit why the other options are incorrect to reinforce our understanding. Option A, 10h≤28010h ≤ 280, would imply that Daisy earns $280 or less, which is the opposite of what we want. Option B, 10h>28010h > 280, would mean Daisy earns more than $280, but it excludes the possibility of her earning exactly $280. Only option C captures the full meaning of "at least $280."

Understanding why the correct answer is correct and why the incorrect answers are incorrect is a crucial step in mastering problem-solving in mathematics. It's not just about finding the right answer; it's about understanding the underlying concepts and reasoning.

Therefore, we can confidently say that option C, $10h ≥ 280, is the correct inequality to determine how many hours Daisy needs to work each week to earn at least $280.

Solving the Inequality (Bonus)

Just for fun, and to make sure we fully grasp this concept, let's actually solve the inequality we've identified as correct:

10h≥28010h ≥ 280

To solve for 'h' (the number of hours), we need to isolate 'h' on one side of the inequality. We can do this by dividing both sides of the inequality by 10. Remember, when we divide or multiply both sides of an inequality by a positive number, the direction of the inequality symbol remains the same.

So, dividing both sides by 10, we get:

h ≥ 28

This solution tells us that Daisy needs to work 28 hours or more to earn at least $280. This makes sense, right? If she works 28 hours, she earns $10 * 28 = $280. If she works more than 28 hours, she'll earn even more.

Let's think about this in a real-world context. Suppose Daisy wants to buy a new gadget that costs $350. How many hours would she need to work? We can set up a similar inequality:

10h≥35010h ≥ 350

Dividing both sides by 10, we get:

h ≥ 35

So, Daisy would need to work at least 35 hours to afford the gadget. This shows how inequalities can be used to plan and make financial decisions.

Solving inequalities is a powerful skill, and it builds upon the foundational understanding of setting them up correctly. By understanding the relationship between the words in a problem and the mathematical symbols, you can tackle a wide range of real-world scenarios.

Key Takeaways

Alright, guys, let's recap the key things we've learned from this problem. Understanding these takeaways will help you tackle similar problems with confidence!

  1. Keywords are Crucial: Pay close attention to keywords like "at least," "no more than," "greater than," and "less than." These words are your clues to identifying the correct inequality symbol.
  2. Translate Words to Symbols: The ability to translate real-world scenarios into mathematical expressions is essential. Practice breaking down problems into smaller parts and representing each part with the appropriate symbol or operation.
  3. Understand Inequality Symbols: Make sure you know what each inequality symbol means:
    • ≤ means "less than or equal to"
    • ≥ means "greater than or equal to"
    • < means "less than"
  4. Check Your Answer: After you've chosen an inequality, take a moment to think about whether it makes sense in the context of the problem. Does it accurately represent the situation described?
  5. Solve (If Needed): Sometimes, you might need to solve the inequality to fully answer the question. Remember the rules for solving inequalities, especially when multiplying or dividing by negative numbers (which we didn't need to do in this problem, but it's good to keep in mind!).

By keeping these takeaways in mind, you'll be well-equipped to handle inequality problems and apply them to real-life situations. Remember, math is a tool for understanding the world around us, so keep practicing and exploring!

So, the next time you encounter a problem involving "at least" or "no more than," you'll be ready to tackle it like a pro! Keep up the great work, and remember, math can be fun when you break it down step by step. You've got this!