Integer Solutions To (x^2-4)(x^2-10)<0 Inequality A Comprehensive Guide
Navigating the realm of inequalities often presents intriguing challenges, especially when dealing with polynomial expressions. In this comprehensive exploration, we will embark on a journey to decipher the integer solutions that satisfy the inequality (x2-4)(x2-10)<0. This seemingly simple expression holds within it a wealth of mathematical concepts, including quadratic equations, inequalities, and the crucial interplay between algebraic expressions and number theory. Our endeavor will not only uncover the specific integer solutions but also illuminate the underlying principles that govern such problems, equipping you with the skills to tackle similar mathematical puzzles with confidence. We will dissect the inequality, meticulously analyze its components, and unveil the elegant solutions that lie within. This exploration will serve as a beacon, guiding you through the often-intricate world of inequalities and fostering a deeper appreciation for the beauty and precision of mathematics.
Decoding the Inequality: A Step-by-Step Approach
To effectively unravel the integer solutions for the inequality (x2-4)(x2-10)<0, we must first embark on a journey of simplification and analysis. This involves breaking down the complex expression into manageable components, identifying critical points, and understanding the behavior of the inequality across different intervals. Let's embark on this step-by-step approach, armed with the tools of algebra and a keen eye for detail.
1. Factoring the Expressions
The first step in our quest is to factor the expressions within the inequality. This will allow us to identify the key values of x that influence the sign of the expression. The expression (x^2-4) is a difference of squares, which can be factored as (x-2)(x+2). Similarly, the expression (x^2-10) cannot be factored neatly using integers, but it's important to recognize that its roots are approximately ±√10, which lie between 3 and 4. By factoring, we transform the original inequality into a more revealing form: (x-2)(x+2)(x^2-10) < 0. This factored form illuminates the critical points where the expression might change its sign.
2. Identifying Critical Points
Critical points are the values of x that make the expression equal to zero. These points are crucial because they divide the number line into intervals where the expression maintains a consistent sign (either positive or negative). From the factored form (x-2)(x+2)(x^2-10) < 0, we can identify the following critical points:
- x = 2: This point comes from the factor (x-2).
- x = -2: This point comes from the factor (x+2).
- x = √10: This point comes from the factor (x^2-10).
- x = -√10: This point comes from the factor (x^2-10).
These four critical points, two integers and two irrational numbers, act as signposts, marking the boundaries of intervals where the expression's sign remains constant. They are the keys to unlocking the solution set of our inequality.
3. Constructing a Sign Table
A sign table is a powerful tool for visualizing the sign of the expression (x-2)(x+2)(x^2-10) across different intervals. We create a table with the critical points arranged in ascending order along the top row: -√10, -2, 2, √10. The left column represents the factors of the expression: (x-2), (x+2), and (x^2-10). Within the table, we fill in the sign of each factor in each interval. For example, (x-2) is negative for x < 2 and positive for x > 2. Similarly, (x+2) is negative for x < -2 and positive for x > -2. The factor (x^2-10) is negative between -√10 and √10, and positive outside this interval.
By multiplying the signs of the factors in each interval, we determine the sign of the entire expression (x-2)(x+2)(x^2-10). We are looking for intervals where the expression is negative, as dictated by the inequality (x2-4)(x2-10)<0. The sign table provides a clear roadmap to identifying these intervals, paving the way for us to pinpoint the integer solutions.
4. Identifying Intervals Where the Inequality Holds
After constructing the sign table, we can readily identify the intervals where the expression (x2-4)(x2-10) is negative. These are the intervals that satisfy the inequality. By carefully examining the sign table, we find that the expression is negative in the following intervals:
- (-√10, -2)
- (2, √10)
These intervals represent the solution set to the inequality. However, our ultimate goal is to find the integer solutions. Therefore, we must focus our attention on the integers that lie within these intervals.
Pinpointing Integer Solutions: A Quest for Whole Numbers
Now that we have identified the intervals where the inequality (x2-4)(x2-10)<0 holds, our next mission is to pinpoint the integer solutions within these intervals. This involves examining the intervals (-√10, -2) and (2, √10) and determining which whole numbers fall within their boundaries. Remember that √10 is approximately 3.16, a crucial piece of information for our integer search.
1. Integers in the Interval (-√10, -2)
The interval (-√10, -2) represents all real numbers strictly greater than -√10 and strictly less than -2. Since -√10 is approximately -3.16, we are looking for integers greater than -3.16 and less than -2. The only integer that satisfies this condition is -3. Therefore, -3 is one of the integer solutions to our inequality.
2. Integers in the Interval (2, √10)
Similarly, the interval (2, √10) represents all real numbers strictly greater than 2 and strictly less than √10. Again, knowing that √10 is approximately 3.16, we seek integers greater than 2 and less than 3.16. The only integer that fits this description is 3. Thus, 3 is another integer solution to the inequality.
3. The Complete Set of Integer Solutions
By carefully examining the intervals where the inequality holds and identifying the integers within those intervals, we have successfully unearthed the complete set of integer solutions. We found that the integers -3 and 3 are the only whole numbers that satisfy the inequality (x2-4)(x2-10)<0. This concludes our search, providing a definitive answer to the question at hand.
The Final Count: How Many Integer Solutions Exist?
After our meticulous exploration and identification of the integer solutions, we arrive at the final count. We discovered that there are two integers that satisfy the inequality (x2-4)(x2-10)<0: -3 and 3. This seemingly simple answer is the culmination of a journey through algebraic manipulation, inequality analysis, and careful consideration of number theory principles. The elegance of mathematics lies in its ability to provide precise answers to seemingly complex questions, and this problem serves as a testament to that beauty. By systematically dissecting the inequality, identifying critical points, and analyzing intervals, we have successfully unveiled the integer solutions, demonstrating the power of a structured approach to problem-solving.
Conclusion: A Triumph of Mathematical Reasoning
In conclusion, our journey to determine the integer solutions for the inequality (x2-4)(x2-10)<0 has been a rewarding exercise in mathematical reasoning. We began by simplifying the expression, factoring it into manageable components, and identifying critical points. We then constructed a sign table to visualize the behavior of the expression across different intervals, allowing us to pinpoint the intervals where the inequality holds. Finally, we focused on the integers within these intervals, successfully identifying -3 and 3 as the only integer solutions. This problem serves as a reminder of the importance of a systematic and methodical approach to mathematical challenges. By breaking down complex problems into smaller, more manageable steps, we can unlock the solutions that lie within. The skills and techniques we have employed in this exploration will undoubtedly prove valuable in tackling future mathematical endeavors, fostering a deeper understanding and appreciation for the power and elegance of mathematics.
