What Is The Smallest Element Of The Set A = {x In N | 1 < 15 - 2x < 11}?

by ADMIN 73 views

In the realm of mathematics, problem-solving is an art that intertwines logic, precision, and a dash of creative thinking. This article delves into a specific mathematical problem, exploring the steps involved in identifying the smallest element within a defined set. We will dissect the given set, apply the necessary inequalities, and arrive at the solution, all while emphasizing the underlying principles of mathematical reasoning. So, let's embark on this mathematical journey and uncover the smallest element within the set A.

Understanding the Problem Statement

The crux of the problem lies in deciphering the set A. We are given that A is a set of natural numbers (denoted by N) that satisfy a particular condition. This condition is expressed as a double inequality: 1 < 15 - 2x < 11. In essence, we are seeking all natural numbers 'x' that, when plugged into the expression 15 - 2x, yield a result that is strictly greater than 1 and strictly less than 11. The challenge is to find the smallest natural number that fits this criterion. This type of problem is foundational in understanding sets, inequalities, and the properties of natural numbers, making it a valuable exercise for students and anyone interested in sharpening their mathematical acumen.

Decoding the Set Definition

Before we jump into solving the inequality, let's break down the components of the set definition. The notation A = {x in N | 1 < 15 - 2x < 11} tells us several key things:

  • A is a set: This means that A is a collection of distinct elements.
  • x in N: This signifies that the elements 'x' belong to the set of natural numbers. Natural numbers are the positive whole numbers (1, 2, 3, ...).
  • | (the "such that" symbol): This indicates that the elements 'x' must satisfy a specific condition.
  • 1 < 15 - 2x < 11: This is the condition that the elements 'x' must fulfill. It's a double inequality, meaning that the expression 15 - 2x must be simultaneously greater than 1 and less than 11.

Understanding these components is crucial for translating the mathematical notation into a concrete problem that we can solve. Now, we are ready to tackle the inequality and determine the values of 'x' that belong to the set A.

Solving the Inequality: A Step-by-Step Approach

To find the elements of set A, we need to isolate 'x' in the inequality 1 < 15 - 2x < 11. We can do this by performing algebraic operations on all parts of the inequality, ensuring that we maintain the validity of the relationships. Here's a step-by-step breakdown:

  1. Subtract 15 from all parts:

    • 1 - 15 < 15 - 2x - 15 < 11 - 15
    • -14 < -2x < -4

  2. Divide all parts by -2 (and reverse the inequality signs): Remember that dividing by a negative number reverses the direction of the inequality signs.

    • -14 / -2 > -2x / -2 > -4 / -2
    • 7 > x > 2

  3. Rewrite the inequality in a more conventional order:

    • 2 < x < 7

This final inequality, 2 < x < 7, tells us that 'x' must be greater than 2 and less than 7. Since 'x' belongs to the set of natural numbers, the possible values for 'x' are 3, 4, 5, and 6. Therefore, the set A can be explicitly written as A = {3, 4, 5, 6}.

Identifying the Smallest Element

Now that we have determined the elements of set A, the final step is straightforward: we simply need to identify the smallest number within the set. By inspection, it's clear that the smallest element in the set A = {3, 4, 5, 6} is 3. Therefore, the answer to the problem is 3.

Alternative Approaches to Solving the Inequality

While the step-by-step method outlined above is a standard approach, there are alternative ways to tackle the inequality. One such method involves splitting the double inequality into two separate inequalities and solving them individually:

  1. Split the inequality:

    • 1 < 15 - 2x and 15 - 2x < 11

  2. Solve the first inequality (1 < 15 - 2x):

    • Subtract 15 from both sides: -14 < -2x
    • Divide by -2 (and reverse the sign): 7 > x

  3. Solve the second inequality (15 - 2x < 11):

    • Subtract 15 from both sides: -2x < -4
    • Divide by -2 (and reverse the sign): x > 2

  4. Combine the solutions:

    • We have x > 2 and x < 7, which is equivalent to 2 < x < 7.

This alternative approach yields the same result, reinforcing the importance of understanding the underlying principles of inequalities and their manipulation.

Key Takeaways and General Problem-Solving Strategies

This problem, while seemingly simple, highlights several crucial aspects of mathematical problem-solving. Here are some key takeaways and strategies that can be applied to a wide range of mathematical challenges:

  • Understanding Definitions: The foundation of any mathematical problem lies in understanding the definitions of the terms and concepts involved. In this case, understanding the notation for sets, natural numbers, and inequalities was paramount.
  • Breaking Down Complex Problems: Complex problems can often be simplified by breaking them down into smaller, more manageable steps. Solving the double inequality involved a series of algebraic manipulations, each of which was relatively straightforward.
  • Attention to Detail: Mathematical precision is essential. Paying close attention to the signs of inequalities, the order of operations, and the properties of numbers is crucial for avoiding errors.
  • Alternative Approaches: Often, there are multiple ways to solve a problem. Exploring different approaches can deepen understanding and provide alternative perspectives.
  • Verification: Once a solution is obtained, it's always a good practice to verify it. In this case, we can plug the smallest element (3) back into the original inequality to confirm that it satisfies the condition.

Connecting to Broader Mathematical Concepts

This problem serves as a building block for more advanced mathematical concepts. The principles of working with inequalities are fundamental in calculus, optimization problems, and various other areas of mathematics. Understanding sets and their properties is essential in discrete mathematics, logic, and computer science. Moreover, the problem-solving strategies employed here – breaking down complex problems, paying attention to detail, and exploring alternative approaches – are universally applicable across mathematical disciplines.

Conclusion: The Power of Mathematical Reasoning

In conclusion, finding the smallest element of the set A involved a systematic application of mathematical principles. We deciphered the set definition, solved the inequality, and identified the smallest natural number that satisfied the given condition. This exercise underscores the power of mathematical reasoning in problem-solving and highlights the importance of understanding fundamental concepts. By mastering these skills, we equip ourselves to tackle more complex mathematical challenges and appreciate the beauty and elegance of mathematical thought. So, keep practicing, keep exploring, and keep sharpening your mathematical acumen! The world of mathematics awaits your discoveries.

Ultimately, the solution to the problem is 3, which corresponds to option (d). This problem serves as an excellent illustration of how a clear understanding of mathematical definitions and techniques can lead to a straightforward solution.

Repair Input Keyword

What is the smallest element of the set A = {x in N | 1 < 15 - 2x < 11}?

Title

Smallest Element of Set A Where 1 < 15 - 2x < 11 Math Problem Solved