Inequality For The Difference Between Two Times A Number And Eighteen

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In mathematics, inequalities are used to express relationships where one quantity is not necessarily equal to another. They are essential tools for modeling and solving real-world problems involving constraints and limitations. Understanding how to translate verbal statements into mathematical inequalities is a fundamental skill in algebra. In this article, we will focus on dissecting the statement: "the difference between two times a number and eighteen is at most fourteen" and accurately represent it as an inequality. Mastering this skill allows us to solve a wide range of problems, from simple algebraic equations to complex optimization challenges. So, let's delve into the world of inequalities and learn how to convert words into mathematical expressions.

Understanding the Statement

To effectively translate the given statement into a mathematical inequality, we must first break it down into its core components. The statement, "the difference between two times a number and eighteen is at most fourteen," contains several key phrases that need careful consideration. Let's dissect each part to ensure we capture its precise meaning:

  1. "A number": This phrase indicates that we are dealing with an unknown quantity, which we will represent using a variable. Let's use the variable x to denote this number. The choice of x is arbitrary; we could use any letter, but x is a common convention in algebra.

  2. "Two times a number": This part means we need to multiply our variable x by 2. So, "two times a number" translates to 2x or simply 2x. This operation is straightforward and represents a scaling of the unknown number.

  3. "The difference between… and eighteen": This indicates a subtraction operation. We are finding the difference between "two times a number" (2x) and eighteen. The order of subtraction is crucial. The phrase implies that we are subtracting eighteen from two times the number, so it will be 2x - 18.

  4. "Is at most fourteen": This is the most critical part of the statement for determining the type of inequality. "At most" means that the quantity can be equal to fourteen or less than fourteen. It cannot exceed fourteen. In mathematical terms, this is represented by the less than or equal to symbol (≤). Therefore, "is at most fourteen" translates to ≤ 14. This phrase sets the upper limit of the expression 2x - 18.

By carefully dissecting each component of the statement, we can begin to see how the inequality will take shape. The combination of these elements will give us a clear mathematical representation of the original verbal statement. Understanding each phrase's role is crucial for accurate translation and problem-solving.

Translating the Statement into an Inequality

Now that we have broken down the statement into its individual components, we can piece them together to form a mathematical inequality. This process involves translating each phrase into its corresponding mathematical symbol or operation and then combining them in the correct order. This step is crucial for accurately representing the problem in a way that can be solved algebraically. Let's walk through the translation step by step:

  1. We identified "a number" as the variable x.
  2. "Two times a number" was translated to 2x.
  3. "The difference between two times a number and eighteen" became 2x - 18. Remember, the order of subtraction is important here.
  4. "Is at most fourteen" was interpreted as ≤ 14.

Putting it all together, we combine the expression 2x - 18 with the inequality symbol ≤ and the number 14. This gives us the inequality:

2x - 18 ≤ 14

This inequality mathematically represents the original statement. It states that the result of subtracting 18 from two times the number x is less than or equal to 14. This translation is a precise representation of the verbal statement, capturing all the nuances and conditions it implies. Understanding how to perform this translation is essential for solving a wide range of mathematical problems involving inequalities. By accurately converting verbal statements into mathematical expressions, we can apply algebraic techniques to find solutions and gain insights into the relationships between quantities.

Analyzing the Options

Now that we have derived the correct inequality, 2x - 18 ≤ 14, let's analyze the given options to identify the one that matches our result. This step involves comparing our derived inequality with each of the provided options and understanding why some options are correct while others are not. Evaluating the options helps reinforce our understanding of the inequality and the nuances of the mathematical symbols used.

The options are:

A. 2x - 18 ≥ 14 B. 2x - 18 ≤ 14 C. 2(x - 18) ≤ 14 D. 2(x - 18) ≥ 14

Let's examine each option:

  • Option A: 2x - 18 ≥ 14

    This inequality states that the difference between two times a number and eighteen is greater than or equal to fourteen. This contradicts the original statement, which specified that the difference is "at most" fourteen, meaning it should be less than or equal to fourteen. Therefore, option A is incorrect.

  • Option B: 2x - 18 ≤ 14

    This inequality matches our derived inequality exactly. It correctly states that the difference between two times a number and eighteen is less than or equal to fourteen. This aligns perfectly with the original statement, making option B the correct choice.

  • Option C: 2(x - 18) ≤ 14

    This inequality represents a different mathematical statement. It suggests that two times the difference between the number and eighteen is less than or equal to fourteen. This is not the same as the original statement, which specified the difference between "two times a number" and eighteen. Therefore, option C is incorrect.

  • Option D: 2(x - 18) ≥ 14

    Similar to option C, this inequality also represents a different mathematical statement. It suggests that two times the difference between the number and eighteen is greater than or equal to fourteen. This contradicts the original statement and is therefore incorrect.

By carefully comparing each option with our derived inequality, we can confidently identify option B as the correct answer. This process of elimination and comparison reinforces our understanding of the problem and the importance of accurate translation.

The Correct Answer

Based on our analysis, the correct inequality that represents the statement "the difference between two times a number and eighteen is at most fourteen" is:

B. 2x - 18 ≤ 14

This inequality accurately captures the relationship described in the original statement. The expression 2x - 18 represents the difference between two times a number (x) and eighteen, and the symbol ≤ indicates that this difference is less than or equal to fourteen. This option precisely matches our derived inequality and reflects the correct interpretation of the given statement. Choosing the correct answer reinforces our understanding of how to translate verbal statements into mathematical inequalities.

Importance of Accurate Translation

The process of translating verbal statements into mathematical expressions, especially inequalities, is a critical skill in mathematics. Accurate translation is essential for correctly modeling real-world problems and finding appropriate solutions. Inaccurate translation can lead to incorrect solutions and misunderstandings of the underlying concepts. This skill is not only valuable in algebra but also in various fields such as physics, engineering, economics, and computer science, where mathematical models are used to describe and analyze complex systems.

The importance of accurate translation stems from several key factors:

  1. Correct Problem Representation: A precise translation ensures that the mathematical expression accurately reflects the relationships and conditions described in the original statement. This is crucial for setting up the problem correctly and applying the appropriate solution techniques. An inaccurate translation can lead to a misrepresentation of the problem, making it difficult or impossible to find a valid solution.

  2. Effective Problem Solving: Once a statement is accurately translated into a mathematical expression, we can apply various algebraic techniques to solve it. This might involve solving equations, graphing inequalities, or using other mathematical tools. However, the effectiveness of these techniques depends on the accuracy of the initial translation. A correct translation provides a solid foundation for problem-solving.

  3. Clear Communication: Mathematical expressions are a precise and unambiguous way to communicate ideas. An accurate translation ensures that the intended meaning of the original statement is conveyed clearly and without confusion. This is particularly important when working with others on mathematical problems, as it facilitates effective collaboration and understanding.

  4. Real-World Applications: Many real-world problems can be modeled and solved using mathematical inequalities. For example, constraints on resources, budget limitations, and performance requirements can often be expressed as inequalities. Accurate translation allows us to apply mathematical tools to analyze these problems and make informed decisions. Whether it's optimizing a production process, managing a budget, or designing a system, accurate translation is essential for practical applications.

  5. Logical Reasoning: The process of translation requires careful logical reasoning and attention to detail. We must understand the meaning of each phrase and how it relates to the overall statement. This process helps develop critical thinking skills and enhances our ability to analyze and interpret information. It's not just about memorizing rules; it's about understanding the logic behind the mathematical concepts.

In summary, accurate translation is a cornerstone of mathematical problem-solving. It enables us to correctly represent problems, apply appropriate techniques, communicate effectively, address real-world challenges, and develop critical thinking skills. Mastering this skill is essential for success in mathematics and various other fields.

Conclusion

In this article, we have explored the process of translating a verbal statement into a mathematical inequality. We focused on the statement "the difference between two times a number and eighteen is at most fourteen" and demonstrated how to break it down into its components, translate each part into mathematical symbols, and combine them to form the inequality 2x - 18 ≤ 14. We also analyzed the given options, identified the correct answer, and discussed the importance of accurate translation in mathematical problem-solving. This skill is essential for various applications and helps in developing logical reasoning and critical thinking abilities. Mastering the translation of verbal statements into mathematical expressions empowers us to tackle a wide range of problems and communicate mathematical ideas effectively.