Samuel's Race: Finding The Right Inequality

Hey guys! Let's break down a cool math problem about Samuel and his 3-mile race. He's got a goal, a time constraint, and we need to figure out which inequality represents his situation. This is a super common type of problem, and understanding it will help you with all sorts of word problems. So, let's dive in! We will use the proper math language for this. Ready to ace this race with me?

Understanding the Problem: Samuel's Race

Alright, so here's the deal: Samuel is running a 3-mile race, and he's aiming to finish it in under 33 minutes. Think about that for a sec – under 33 minutes means he wants to be faster than that time. He's already been running, so he's not starting from scratch. He's already clocked 10.5 minutes. Our mission is to find the inequality that correctly models this scenario. This means we'll need to figure out how much more time he can spend running to meet his goal and select the correct inequality to represent the situation. This kind of problem is all about translating words into math symbols. It's like learning a new language, but instead of words, we use numbers and symbols like less than, greater than, and equal to. This is really useful in real life, not just in math class. When you're planning a trip, managing your budget, or even just trying to get somewhere on time, you're essentially using these same skills. The better you get at this, the easier life will become, trust me. So, let's figure out the proper way to solve Samuel’s problem!

First, we know the race has a time limit of 33 minutes. This is a crucial piece of information, setting the boundary for Samuel's run. Then, we see that he has already run for 10.5 minutes. He still has some time left, and this time, we do not know. To find the remaining time, we use the variable x. The goal is to determine the inequality that accurately represents the situation, not just to solve for x. The goal of this task is to ensure that you completely understand the mathematical concept. This problem is similar to problems you might face in the real world when dealing with time management or any scenario where you have a goal with a limit. Now let’s move on to the next part and analyze the answer!

Decoding the Inequalities: A Closer Look

Okay, now that we've got the problem down, let's look at the possible answers. Remember, we need an inequality, which means we'll be using symbols like < (less than), > (greater than), ≤ (less than or equal to), or ≥ (greater than or equal to). The goal is to represent the situation where Samuel wants to complete his race in under 33 minutes. Keep in mind that “under” implies that he must finish in a time that is strictly less than 33 minutes. The time he already spent running (10.5 minutes) plus the remaining time (x) must be less than 33 minutes. He cannot exceed the 33-minute mark, meaning the total time must be less than 33. Understanding the difference between these symbols is key, guys. Let’s not be confused, and get it right! Each of these inequalities represents a different relationship between two values. The correct answer needs to accurately reflect Samuel's goal of finishing under 33 minutes.

Let’s look closely at all of the possible answers. The goal is to represent the relationship between the time Samuel has already spent running, the remaining time, and the target time. The correct inequality will accurately model how these elements are related. The correct inequality must show that the sum of the time already spent running and the remaining time must be less than 33 minutes. Let’s dive deep!

• A. 10.5 + x ≤ 33: This inequality states that the sum of 10.5 minutes (already run) and the remaining time (x) is less than or equal to 33 minutes. This would mean he could finish in exactly 33 minutes, but he wants to finish under 33 minutes. This one is incorrect because it includes the possibility of him finishing in exactly 33 minutes, which goes against the instructions of the problem.
• B. 10.5 + x < 33: This inequality correctly states that the sum of 10.5 minutes and the remaining time (x) is less than 33 minutes. This matches Samuel's goal of finishing the race in under 33 minutes. This is the correct answer! This option perfectly reflects the original problem statement.

Now, let's see how our inequality represents the correct answer. The total time Samuel spends running (10.5 minutes plus the remaining time, x) must be less than 33 minutes to finish in under 33 minutes. The correct inequality, 10.5 + x < 33, reflects this relationship. So, the correct answer is B. Easy, right?

Why the Correct Answer Matters

Understanding inequalities isn't just about getting the right answer on a test. It's a fundamental skill that helps you analyze and solve real-world problems. When you can translate a scenario into a mathematical expression, you're able to make informed decisions and predictions. For example, if you were trying to budget for a project with a set deadline, you'd use inequalities to ensure you stay within your time and financial constraints. These concepts also appear in science, engineering, and even computer programming. It's used in lots of practical ways. It's a core skill that empowers you to think critically and solve problems effectively.

Learning inequalities helps you in many aspects of your life. When you get better at these types of problems, you also get better at critical thinking. So, keep practicing, and you'll find it gets easier every time. Plus, it is very satisfying to get the right answer, right?

Tips for Tackling Inequality Problems

Want to get better at solving these types of problems? Here are a few tips to help you out:

• Read Carefully: Take your time to read the problem. Make sure you understand what's being asked. Highlight key information, like the target time and the given information.
• Translate Words to Symbols: Practice turning phrases like