Inequality For Word Problem: 3 Times 2 Less Than A Number

by Andrew McMorgan 58 views

Hey guys! Ever find yourself staring blankly at a word problem, especially when it involves inequalities? It's a common struggle, but don't sweat it! We're here to break down a classic example and show you how to translate those tricky words into mathematical expressions. Let's dive into a problem that might seem daunting at first, but we'll conquer it together: "Three times two less than a number is greater than or equal to five times the number." We'll use 'n' to represent our mystery number and walk through the process of choosing the correct inequality. So, grab your thinking caps, and let's get started!

Understanding the Problem

Before we jump into the options, let's really understand what the problem is saying. The core of translating word problems lies in carefully dissecting each phrase and recognizing the mathematical operations they imply. When dealing with inequalities, it's crucial to pay close attention to keywords like "greater than," "less than," "at least," and, in our case, "greater than or equal to." These phrases dictate the symbols we'll use in our inequality. The problem states: "Three times two less than a number is greater than or equal to five times the number." Let's break this down bit by bit.

First, we identify our variable. The problem clearly states, "Let n = a number." This is our foundation. Now, we need to decipher the phrase "two less than a number." In mathematical terms, this translates to n - 2. It's vital to get the order right here. We're subtracting 2 from the number, not the other way around. Next, we encounter "three times two less than a number." This means we're multiplying the entire expression (n - 2) by 3. Using parentheses ensures we maintain the correct order of operations: 3(n - 2). This part of the problem emphasizes the importance of understanding order of operations in mathematical expressions. Without it, we might incorrectly translate the phrase and arrive at the wrong inequality. For instance, writing 3n - 2 would imply that we are only multiplying 3 by n, not by the entire quantity of "two less than a number." Then we have "is greater than or equal to," which directly translates to the inequality symbol ≥. This symbol is a key component in expressing the relationship described in the problem. It tells us that the expression on the left side of the inequality is either larger than or the same as the expression on the right side. Finally, we have "five times the number," which is simply 5n. This part is relatively straightforward, but it's crucial to connect it correctly to the rest of the inequality. We're comparing "three times two less than a number" to "five times the number," and the phrase "is greater than or equal to" dictates the direction of the comparison. Now that we've dissected each part, we can start building our inequality. We know that 3(n - 2) is on one side, ≥ is our comparison, and 5n is on the other side. This careful breakdown is essential for accurately representing the word problem in a mathematical format. It’s not just about plugging in numbers; it’s about understanding the relationships between them as described in the words. By systematically approaching the problem in this way, we minimize the chances of error and ensure we’re on the right track to solving it.

Evaluating the Options

Okay, now that we've dissected the problem and know what we're looking for, let's evaluate the given options. This is where our understanding of the problem truly shines. We'll compare each option against our translation to see which one accurately represents the relationship described. Let's walk through each choice:

A. 3(2) - n ≥ 5n

This option immediately stands out as incorrect. Why? Because it completely misinterprets the phrase "two less than a number." Instead of subtracting 2 from n, it subtracts n from 3(2), which is 6. This is a fundamental error in translating the word problem. It’s crucial to recognize the importance of order in mathematical operations and how it’s reflected in word problems. The phrase "two less than a number" clearly indicates that the subtraction should involve n being the primary value from which 2 is subtracted. Option A flips this relationship, indicating a misunderstanding of basic algebraic principles. Furthermore, this option doesn't capture the essence of multiplying "two less than a number" by 3. It only multiplies 3 by 2 and then subtracts n. This further reinforces the error in interpreting the original problem statement. So, we can confidently eliminate option A as it does not align with our dissected understanding of the problem.

B. 3n - 2 ≥ 5n

Option B is closer, but it still misses a crucial element. It correctly identifies 5n on one side of the inequality and uses the "greater than or equal to" symbol (≥). However, it falters in translating "three times two less than a number." While it includes 3n and subtracts 2, it doesn't multiply the entire expression (n - 2) by 3. It only subtracts 2 from 3n, which is a different mathematical relationship. This distinction is key to understanding the correct translation. The phrase “three times two less than a number” implies a quantity, the quantity being “two less than a number”, that is then multiplied by three. Therefore, the entire expression “n – 2” should be enclosed in parentheses and multiplied by 3: 3(n - 2). Option B neglects this crucial use of parentheses, which results in a misrepresentation of the original problem statement. This option highlights the importance of paying close attention to the structure of mathematical expressions and how they represent relationships described in word problems. Therefore, option B, while closer than option A, is still incorrect and can be eliminated.

C. 3(n - 2) ≥ 5n

This option looks promising! Let’s break it down. On the right side, we have 5n, which accurately represents “five times the number.” The “≥” symbol correctly translates “is greater than or equal to.” And on the left side, we have 3(n - 2). Does this accurately represent “three times two less than a number”? Yes, it does! The (n - 2) correctly translates “two less than a number,” and the 3 outside the parentheses signifies that we’re multiplying the entire quantity (n - 2) by 3. This option demonstrates a clear understanding of how to translate the word problem into a mathematical inequality. It correctly captures the order of operations and the relationships between the different components of the problem. The inclusion of parentheses is critical here, as it ensures that we are multiplying the entire expression “two less than a number” by 3, rather than just multiplying 3 by n and then subtracting 2. This subtle but significant detail makes option C a strong contender for the correct answer. So far, option C aligns perfectly with our understanding of the problem.

D. 3(2 - n) ≥ 5n

Option D is tricky! It includes the parentheses and 5n which indicates multiplying by 3, but it changes the order of subtraction within the parentheses. It has 3(2 - n) instead of 3(n - 2). This seemingly small difference drastically alters the meaning. Remember, we're looking for “two less than a number,” which means we subtract 2 from n, not the other way around. This option demonstrates the importance of being precise in translating mathematical language. The phrase “two less than a number” has a very specific meaning, and reversing the order of subtraction changes the entire relationship. This highlights the need for meticulous attention to detail when working with word problems and inequalities. Option D, despite having some correct components, ultimately misrepresents the core relationship described in the problem. This subtle error serves as a valuable reminder to always double-check the order of operations and the direction of relationships in mathematical translations. Therefore, option D is incorrect and can be eliminated.

The Solution

Based on our evaluation, option C, 3(n - 2) ≥ 5n, is the correct inequality that represents the given relationship. We meticulously dissected the problem, translated each phrase into mathematical terms, and compared each option against our translation. Option C stood out because it accurately captured the order of operations, the relationship between the components, and the meaning of the phrase "two less than a number." It showed that we understood how to represent mathematical relationships in a concise and precise way.

Key Takeaways for Mastering Inequality Translations

So, what did we learn from this exercise? Here are some key takeaways for mastering the art of translating word problems into inequalities:

  1. Break it Down: The first takeaway is the importance of breaking down the problem into smaller, manageable parts. Don't try to swallow the whole thing at once. Identify the key phrases and translate them individually. This makes the process less overwhelming and helps you avoid errors.

  2. Order Matters: Pay close attention to the order of operations and the order of words. Phrases like "two less than a number" have a specific meaning, and changing the order changes the entire relationship. This is a crucial aspect of mathematical translation that can easily trip up beginners.

  3. Keywords are Your Friends: Recognize those inequality keywords! "Greater than," "less than," "at least," "at most," and "greater than or equal to" are all clues that tell you which inequality symbol to use. Make a mental note of these keywords and their corresponding symbols.

  4. Use Parentheses Wisely: Parentheses are your allies! They help you group terms and ensure the correct order of operations. If you're multiplying a quantity, make sure you enclose that quantity in parentheses. This is where many students make mistakes, so pay extra attention to this point.

  5. Double-Check Your Work: Always double-check your translation! Does the inequality you wrote accurately represent the relationships described in the word problem? Read the problem again and make sure everything aligns. This final step can save you from making careless errors.

By following these tips, you'll be well on your way to conquering any word problem that comes your way. Remember, practice makes perfect! The more you translate word problems, the easier it will become. Keep up the great work, and you'll be an inequality pro in no time! We hope this step-by-step guide has been helpful. Now go out there and tackle those inequalities with confidence! You've got this!