Two Positive Integers Are 3 Units Apart On A Number Line, And Their Product Is 108. Which Equation Can Be Used To Find *m*, The Greater Integer?

In this article, we delve into a classic mathematical problem involving two positive integers that are 3 units apart on a number line and have a product of 108. Our main goal is to determine the correct equation that can be used to solve for m, the greater of these two integers. This problem is a wonderful illustration of how algebraic principles can be applied to solve real-world scenarios. Understanding how to set up equations from word problems is a critical skill in algebra and beyond. This article aims to not only provide the solution but also to explain the underlying concepts in a clear and comprehensive manner, ensuring that readers of all backgrounds can grasp the logic and methodology involved. We will dissect the problem step by step, clarifying the relationships between the integers and translating them into algebraic expressions. By the end of this exploration, you will have a solid understanding of how to approach similar problems and a deeper appreciation for the power of algebra in problem-solving.

Problem Statement

Let's revisit the core of the problem. We are given two key pieces of information about our integers: their distance on the number line and their product. The problem states that the two positive integers are 3 units apart on a number line. This tells us that the difference between the greater integer (m) and the smaller integer is 3. In mathematical terms, if m is the greater integer, the smaller integer can be represented as m - 3. The second crucial piece of information is that the product of these two integers is 108. This means that when we multiply the greater integer (m) by the smaller integer (m - 3), the result is 108. This can be written as an equation, which is the foundation for solving the problem. By carefully translating these relationships into algebraic expressions, we set the stage for identifying the correct equation from the given options. This initial setup is the most critical part of the problem-solving process, as it lays the groundwork for all subsequent steps. A clear understanding of these relationships is essential for success.

Analyzing the Options

Now, let's carefully examine the options provided to us. Each option presents a different equation, and our task is to determine which one accurately represents the problem's conditions. Option A states m( m - 3) = 108. This equation suggests that the product of the greater integer (m) and an integer 3 units less than m is 108. This aligns perfectly with our understanding of the problem, where the two integers are m and m - 3, and their product is 108. Option B presents m( m + 3) = 108. This equation implies that the product of the greater integer (m) and an integer 3 units greater than m is 108. However, this contradicts the problem statement, which specifies that the integers are 3 units apart, not that one is 3 more than the other in the context of the product. Option C gives us (m + 3)(m - 3) = 108. This equation represents the product of two integers that are 6 units apart (since the difference between m + 3 and m - 3 is 6), not 3 units as stated in the problem. Therefore, this equation does not accurately reflect the problem's conditions. By methodically analyzing each option and comparing it to the problem statement, we can narrow down the possibilities and identify the correct equation.

Detailed Explanation of the Correct Equation

Let's take a closer look at why option A, m( m - 3) = 108, is the correct equation. This equation precisely captures the relationship between the two integers and their product. As we established earlier, m represents the greater integer, and m - 3 represents the smaller integer, since they are 3 units apart on the number line. The equation m( m - 3) = 108 states that when we multiply these two integers together, the result is 108, which is exactly what the problem statement tells us. To further illustrate this, let's consider a scenario. If m were 12, then the smaller integer would be 12 - 3 = 9. The product of 12 and 9 is indeed 108, which validates the equation's structure. This equation is a quadratic equation, which means it can be solved to find the possible values of m. Solving this equation would involve expanding the expression, rearranging the terms, and then either factoring or using the quadratic formula. However, for the purpose of this problem, we are only asked to identify the correct equation, not to solve it. The key takeaway here is the ability to translate word problems into algebraic equations, a fundamental skill in mathematics. This equation serves as a mathematical model of the problem, allowing us to use algebraic techniques to find the solution.

Why Other Options are Incorrect

To reinforce our understanding, let's delve deeper into why the other options are incorrect. Option B, m( m + 3) = 108, is incorrect because it represents the product of the greater integer (m) and an integer that is 3 units greater than m. This contradicts the problem statement, which specifies that the integers are 3 units apart, meaning one integer is 3 less than the other, not 3 more. In this equation, m + 3 would represent a number larger than m, and their product being 108 doesn't align with the given conditions. Option C, (m + 3)(m - 3) = 108, is also incorrect. This equation represents the product of two integers that are 6 units apart, not 3. The expression (m + 3) represents an integer 3 units greater than m, and (m - 3) represents an integer 3 units less than m. The difference between these two integers is ( m + 3) - (m - 3) = 6. This equation is actually a difference of squares, which simplifies to m^2 - 9 = 108. While this equation has its own mathematical validity, it doesn't accurately represent the problem we are trying to solve. By understanding why these options are incorrect, we solidify our grasp of the problem's conditions and the correct algebraic representation.

Solving the Equation (Optional)

Although the problem only asks us to identify the correct equation, let's briefly explore how we would solve it. The correct equation is m( m - 3) = 108. To solve for m, we first need to expand the equation: m^2 - 3m = 108. Next, we rearrange the equation to form a quadratic equation in standard form: m^2 - 3m - 108 = 0. Now, we can try to factor the quadratic. We are looking for two numbers that multiply to -108 and add up to -3. These numbers are -12 and 9. So, we can factor the equation as (m - 12)(m + 9) = 0. Setting each factor equal to zero gives us two possible solutions for m: m - 12 = 0, which gives m = 12, and m + 9 = 0, which gives m = -9. Since the problem specifies that the integers are positive, we discard the solution m = -9. Therefore, the greater integer m is 12. The smaller integer would then be m - 3 = 12 - 3 = 9. And indeed, 12 * 9 = 108, which confirms our solution. This step demonstrates the practical application of the equation we identified and how it leads to the solution of the problem. Understanding the solution process further enhances our comprehension of the problem and the underlying mathematical principles.

Key Concepts and Takeaways

In this exploration, we've covered several key concepts that are fundamental to algebra and problem-solving. First and foremost, we've emphasized the importance of translating word problems into algebraic equations. This is a crucial skill that allows us to apply mathematical techniques to real-world scenarios. We learned how to identify the relationships between the given quantities and represent them using variables and equations. Secondly, we've highlighted the significance of analyzing and understanding the problem statement. Misinterpreting the conditions can lead to incorrect equations and solutions. We carefully examined the information provided, such as the distance between the integers and their product, and used it to construct the correct equation. Thirdly, we've demonstrated the process of evaluating different options and justifying the correct answer. By systematically analyzing each equation and comparing it to the problem statement, we were able to eliminate incorrect options and identify the accurate one. Finally, we briefly touched upon the process of solving quadratic equations, which is a common technique in algebra. While it wasn't the primary focus of this problem, it's an important skill to have for solving similar problems in the future. The key takeaway is that a combination of careful analysis, algebraic manipulation, and problem-solving strategies is essential for success in mathematics.

Conclusion

In conclusion, the correct equation that can be used to solve for m, the greater integer, is A. m( m - 3) = 108. This equation accurately represents the relationship between the two positive integers that are 3 units apart and have a product of 108. We arrived at this solution by carefully analyzing the problem statement, translating the given information into algebraic expressions, and evaluating the provided options. This problem serves as a valuable exercise in algebraic thinking and problem-solving skills. By understanding the underlying concepts and methodologies, we can confidently tackle similar challenges in mathematics and beyond. The ability to translate word problems into mathematical equations is a fundamental skill that opens the door to solving a wide range of problems in various fields, from science and engineering to finance and economics. We hope this detailed explanation has provided you with a clear understanding of the problem and the solution process, and that it empowers you to approach future mathematical challenges with confidence.