Hey guys! Let's break down this math problem step by step so we can figure out the correct inequality. We're dealing with Serena, who's making some money cutting lawns, and we want to represent her earnings on Monday using an inequality. So, let's dive right in!
Understanding the Problem
First, let's make sure we totally get what the problem is asking. Our main goal here is to figure out the inequality that shows how many hours Serena worked on Monday. We know a few key things:
- Serena earns $9 for each hour she spends cutting lawns.
- She also gets about $15 in tips every day.
- On Monday, she made no less than $110. This is super important – "no less than" tells us we're dealing with a minimum amount.
So, we need to use this info to create a mathematical inequality. Think of it like building a sentence using math symbols!
Building the Inequality: A Step-by-Step Approach
Let's break down how to build this inequality piece by piece. We'll start by thinking about how each part of Serena's earnings contributes to the total.
Hourly Earnings
Serena makes $9 per hour, and we're using h to represent the number of hours she worked. So, her earnings from cutting lawns can be expressed as 9 * h*, or simply 9h. This means that for every hour Serena works, she earns $9. If she works 2 hours, she earns 9 * 2 = $18. If she works 5 hours, she earns 9 * 5 = $45, and so on. The hourly earnings is a crucial part of her total income, and it directly depends on the number of hours she puts in.
Daily Tips
She also gets $15 in tips each day. This is a fixed amount, meaning it doesn't change no matter how many hours she works. Whether she works for one hour or eight hours, she still gets those $15 in tips. This fixed amount is added to her hourly earnings to calculate her total income for the day. Tips are a great bonus, and in this case, they help Serena reach her goal of earning at least $110 on Monday.
Total Earnings
Now, let's combine these two parts to represent her total earnings. Her total earnings are the sum of her hourly earnings and her daily tips. So, we can write this as: 9h + 15. This expression, 9h + 15, represents the total amount of money Serena makes on any given day, depending on the number of hours she works. It's the foundation of our inequality, showing how her hourly wage and tips come together.
The "No Less Than" Condition
Here's where the inequality part comes in. We know that Serena made "no less than" $110 on Monday. What does this mean in math terms? It means she made $110 or more. The phrase "no less than" is a key indicator that we're dealing with a "greater than or equal to" (≥) inequality. It sets a minimum threshold for her earnings. Think of it as a floor – Serena's earnings couldn't fall below $110.
Putting It All Together
So, we can now put all the pieces together to form our inequality. We know that her total earnings (9h + 15) must be greater than or equal to $110. This translates directly into the inequality: 9h + 15 ≥ 110. This is the complete inequality that represents the situation. It tells us that Serena's hourly earnings plus her tips must be at least $110. This inequality is the key to solving the problem and finding the possible values for h, the number of hours she worked.
The Correct Inequality
Based on our breakdown, the inequality that represents h, the number of hours Serena worked on Monday, is:
9h + 15 ≥ 110
This inequality tells us that Serena's total earnings, which include her hourly pay (9h) and her tips ($15), must be greater than or equal to $110. The "greater than or equal to" sign (≥) is crucial because it reflects the "no less than" condition in the problem. It means Serena could have earned exactly $110, or she could have earned more.
Why Other Options Are Incorrect
It's just as important to understand why the other options are wrong. This helps us solidify our understanding of inequalities and how they represent real-world situations. Let's take a quick look at why the other options don't fit.
Option A: 15h + 9 ≤ 110
This option is incorrect because it flips the hourly wage and the tips. It suggests that Serena earns $15 per hour and gets $9 in tips, which isn't what the problem states. Also, the "less than or equal to" sign (≤) implies that Serena earned $110 or less, contradicting the "no less than" condition. This option misrepresents both the amounts and the relationship between Serena's earnings and the $110 threshold.
Option B: 15h + 9 ≥ 110
Similar to option A, this one incorrectly assigns $15 as the hourly wage and $9 as the tips. While it does use the correct "greater than or equal to" sign (≥), the incorrect amounts make the entire inequality wrong. It's important to match the correct values to the corresponding variables and constants in the problem. This option gets the inequality direction right but the values wrong.
Option C: 9h + 15 ≤ 110
This option has the correct hourly wage ($9) and tip amount ($15), but it uses the "less than or equal to" sign (≤). This means it's saying Serena earned $110 or less, which goes against the problem's statement that she earned "no less than" $110. The inequality needs to reflect the minimum earnings requirement, and this option fails to do so. The correct values are present, but the inequality symbol is incorrect.
Key Takeaways
- Read carefully: Always make sure you fully understand the problem and what it's asking.
- Identify keywords: Phrases like "no less than" are clues that tell you which inequality symbol to use.
- Break it down: Deconstruct the problem into smaller parts (hourly earnings, tips, total earnings) to make it easier to build the inequality.
- Check your answer: Make sure your inequality accurately reflects the situation described in the problem.
By following these steps, you can confidently tackle inequality problems and apply them to real-world scenarios.
Real-World Applications of Inequalities
Understanding inequalities isn't just about solving math problems; it's a skill that's super useful in everyday life. Inequalities help us make decisions and understand limits in various situations. Let's explore some real-world applications to see how these mathematical concepts play out in practical scenarios.
Budgeting and Spending
One common application of inequalities is in budgeting. Imagine you have a certain amount of money to spend each month. You can use an inequality to represent your spending limit. For example, if your budget is $500, you can write an inequality like this: spending ≤ $500. This means your total spending should be less than or equal to $500. You can then break down your spending into different categories, like rent, food, and entertainment, and create inequalities for each category to ensure you stay within your overall budget. This helps you manage your finances effectively and avoid overspending.
Time Management
Inequalities can also be used to manage time. Suppose you have a project due in a week, and you need to allocate your time wisely. You can use an inequality to represent the amount of time you spend on the project each day. For instance, if you want to spend at least 2 hours each day on the project, you can write: time spent ≥ 2 hours. This helps you ensure you're dedicating enough time to the project to meet the deadline. Time management is crucial in both academic and professional settings, and inequalities can be a valuable tool for planning and prioritizing tasks.
Health and Fitness
In the realm of health and fitness, inequalities can help you set goals and track progress. For example, if you want to walk at least 10,000 steps a day, you can represent this goal with the inequality: steps taken ≥ 10,000. Similarly, if you want to limit your calorie intake to 2,000 calories per day, you can write: calories consumed ≤ 2,000. These inequalities help you monitor your activity levels and dietary habits, ensuring you stay on track with your health goals. Inequalities provide a clear, quantifiable way to set and achieve fitness targets.
Comparing Prices and Deals
When shopping, inequalities can help you compare prices and determine the best deals. For instance, if you're comparing two products with different prices and discounts, you can use inequalities to figure out which option is cheaper. Suppose Product A costs $50 with a 20% discount, and Product B costs $40 with a 10% discount. You can calculate the final price for each product and use an inequality to compare them. This helps you make informed purchasing decisions and get the most for your money. Inequalities empower you to be a savvy shopper and find the best value.
Setting Goals and Limits
In many areas of life, inequalities help us set goals and limits. Whether it's saving money, limiting screen time, or achieving a certain grade in a class, inequalities provide a mathematical framework for defining boundaries and targets. For example, if you want to save at least $1,000 by the end of the year, you can represent this goal with the inequality: savings ≥ $1,000. By setting these kinds of goals and limits, you can use inequalities to track your progress and make adjustments as needed. This makes inequalities a versatile tool for personal and professional development.
Conclusion
So, there you have it! We've cracked the code on this problem by understanding the key information and translating it into a mathematical inequality. Remember, inequalities are super useful for representing situations where there's a range of possible values, not just one specific answer. Keep practicing, and you'll become a pro at solving these types of problems! This stuff isn't just about math class; it's about understanding the world around us in a more precise and logical way.