Solving Polynomial Inequality -3(x-4)(x+12)(x-11) < 0 In Interval Notation

Understanding Polynomial Inequalities

Polynomial inequalities are mathematical expressions that compare a polynomial to zero, using inequality symbols such as <, >, ≤, or ≥. Solving these inequalities involves finding the range of values for the variable (typically 'x') that make the inequality true. In this article, we will delve into the process of solving the polynomial inequality -3(x-4)(x+12)(x-11) < 0. This comprehensive guide will provide a step-by-step approach, ensuring clarity and understanding for readers of all backgrounds. Mastering polynomial inequalities is a crucial skill in algebra and calculus, as it forms the basis for more advanced mathematical concepts. This article aims to not only provide the solution to the given inequality but also to equip you with the knowledge and skills to tackle similar problems with confidence. By understanding the underlying principles and techniques, you will be able to approach polynomial inequalities with a systematic and logical approach, leading to accurate solutions and a deeper understanding of algebraic concepts. Furthermore, we will explore the practical applications of polynomial inequalities, highlighting their relevance in various fields such as engineering, economics, and computer science. This will help you appreciate the significance of this topic and its real-world implications. So, let's embark on this journey of solving polynomial inequalities and unlock the power of algebraic problem-solving.

The Importance of Interval Notation

Interval notation is a concise way to represent a set of numbers that form an interval on the number line. It's crucial for expressing solutions to inequalities, as it clearly indicates the range of values that satisfy the given condition. For instance, the interval (a, b) represents all numbers between a and b, excluding a and b themselves. On the other hand, [a, b] includes both a and b. Understanding interval notation is essential for accurately expressing the solution to the inequality. Using parentheses or square brackets correctly can significantly impact the interpretation of the solution set. For example, a seemingly small change like using a parenthesis instead of a square bracket can mean the difference between including or excluding a critical value. This seemingly minor detail can have significant consequences in various applications, particularly in fields that rely on precise mathematical modeling. Therefore, mastering interval notation is not merely a matter of mathematical formalism; it's a critical skill that ensures clear and unambiguous communication of mathematical results. Furthermore, familiarity with interval notation is essential for understanding more advanced mathematical concepts, such as limits, continuity, and calculus. These concepts often rely on the precise definition of intervals and their properties. So, by investing the time to understand and practice interval notation, you are laying a solid foundation for future mathematical endeavors. In this article, we will consistently use interval notation to express the solutions to polynomial inequalities, reinforcing its importance and providing ample opportunities for practice.

Step-by-Step Solution

1. Simplify the Inequality

Our first step in solving the polynomial inequality -3(x-4)(x+12)(x-11) < 0 is to simplify it. We can begin by dividing both sides of the inequality by -3. Remember, when we divide or multiply an inequality by a negative number, we must reverse the inequality sign. This is a crucial rule to remember, as it directly affects the accuracy of the final solution. Forgetting to flip the inequality sign is a common mistake that can lead to incorrect results. The reason for this rule lies in the fundamental properties of inequalities. Multiplying or dividing by a negative number essentially reflects the number line, which reverses the order of the numbers. Therefore, to maintain the truth of the inequality, we must also reverse the direction of the inequality sign. After dividing by -3, our inequality becomes: (x-4)(x+12)(x-11) > 0. Now we have a simpler form to work with. This step is essential as it eliminates the negative coefficient, making the subsequent steps easier to manage. By simplifying the inequality, we reduce the chances of making errors and gain a clearer understanding of the problem's structure. Furthermore, this simplification allows us to focus on the key factors that determine the solution set, namely the roots of the polynomial. This simplified form is now ready for us to identify the critical points and proceed with the analysis. This initial simplification is a crucial step towards efficiently and accurately solving the polynomial inequality.

2. Find the Critical Points

The critical points of a polynomial inequality are the values of x that make the polynomial equal to zero. These points are crucial because they divide the number line into intervals where the polynomial's sign remains constant. To find these points, we set each factor in our simplified inequality, (x-4)(x+12)(x-11) > 0, equal to zero and solve for x. This gives us: x - 4 = 0, which yields x = 4; x + 12 = 0, which yields x = -12; and x - 11 = 0, which yields x = 11. These values, -12, 4, and 11, are our critical points. They are the points where the polynomial expression changes its sign, transitioning from positive to negative or vice versa. This is because at these points, one or more of the factors in the polynomial expression becomes zero, causing the entire product to become zero. Understanding the concept of critical points is fundamental to solving polynomial inequalities. These points act as boundaries, defining the intervals where the polynomial expression maintains a consistent sign. By identifying these critical points, we can effectively analyze the behavior of the polynomial across different intervals and determine the solution set for the inequality. These critical points are the cornerstones of our solution process, guiding us to the intervals where the inequality holds true. They provide a clear demarcation, allowing us to systematically test each interval and determine its contribution to the final solution. Without accurately identifying the critical points, it would be impossible to determine the correct solution set for the polynomial inequality.

3. Create a Sign Chart

Now, we create a sign chart using the critical points we found: -12, 4, and 11. A sign chart is a visual tool that helps us determine the sign of the polynomial in each interval created by the critical points. We draw a number line and mark these points on it. This divides the number line into four intervals: (-∞, -12), (-12, 4), (4, 11), and (11, ∞). For each interval, we choose a test value within that interval and plug it into the factored polynomial (x-4)(x+12)(x-11). The sign of the result will tell us the sign of the polynomial in that entire interval. For example, in the interval (-∞, -12), we can choose x = -13. Plugging this into our polynomial, we get (-13-4)(-13+12)(-13-11) = (-17)(-1)(-24), which is negative. This means the polynomial is negative in the interval (-∞, -12). We repeat this process for each interval. This methodical approach ensures that we accurately determine the sign of the polynomial in every region of the number line. The sign chart provides a clear and concise visual representation of the polynomial's behavior, making it easier to identify the intervals that satisfy the inequality. By carefully analyzing the sign chart, we can quickly pinpoint the regions where the polynomial is positive or negative, allowing us to determine the solution set with confidence. This visual tool is invaluable for solving polynomial inequalities, as it simplifies the process and reduces the risk of errors. The sign chart acts as a roadmap, guiding us through the solution process and ensuring that we arrive at the correct answer.

4. Determine the Sign in Each Interval

Let's determine the sign of the polynomial (x-4)(x+12)(x-11) in each interval using test values. This is a critical step in solving the inequality, as it allows us to identify the regions where the polynomial is either positive or negative. This information is essential for determining the solution set that satisfies the given inequality. First, for the interval (-∞, -12), we chose x = -13, and we found the polynomial to be negative. This indicates that the polynomial expression is less than zero for all values of x in this interval. Next, in the interval (-12, 4), we can choose x = 0. Plugging this in, we get (0-4)(0+12)(0-11) = (-4)(12)(-11), which is positive. This tells us that the polynomial expression is greater than zero for all values of x in the interval (-12, 4). Then, for the interval (4, 11), let's choose x = 5. We get (5-4)(5+12)(5-11) = (1)(17)(-6), which is negative. Therefore, the polynomial is negative in this interval. Finally, in the interval (11, ∞), we can choose x = 12. We get (12-4)(12+12)(12-11) = (8)(24)(1), which is positive. This means the polynomial is positive in the interval (11, ∞). By systematically testing each interval, we have created a clear picture of the polynomial's behavior across the entire number line. This detailed analysis is crucial for identifying the intervals that satisfy the original inequality. The signs we have determined for each interval form the basis for our solution, allowing us to confidently identify the range of values that make the inequality true.

5. Identify the Intervals that Satisfy the Inequality

We are looking for the intervals where (x-4)(x+12)(x-11) > 0, meaning the polynomial is positive. From our sign chart analysis, we found that the polynomial is positive in the intervals (-12, 4) and (11, ∞). Therefore, these are the intervals that satisfy our inequality. It's important to note that because the inequality is strictly greater than zero (>), we do not include the critical points themselves in our solution. This is because at the critical points, the polynomial is equal to zero, not greater than zero. The choice of parentheses in our interval notation reflects this exclusion. If the inequality had been greater than or equal to zero (≥), we would have included the critical points by using square brackets in our interval notation. Understanding the distinction between strict and non-strict inequalities is crucial for accurately representing the solution set. Failing to account for this distinction can lead to incorrect solutions. The intervals (-12, 4) and (11, ∞) represent the range of values for x that make the polynomial expression positive, thus satisfying the inequality. These intervals provide a complete and accurate solution to the problem. The careful analysis we conducted using the sign chart has allowed us to confidently identify these intervals and express them in the appropriate notation.

6. Write the Solution in Interval Notation

Finally, we write the solution in interval notation. The intervals where the inequality (x-4)(x+12)(x-11) > 0 is satisfied are (-12, 4) and (11, ∞). We combine these intervals using the union symbol (∪) to express the complete solution. Therefore, the solution to the polynomial inequality is (-12, 4) ∪ (11, ∞). This interval notation provides a concise and accurate representation of all the values of x that satisfy the given inequality. The use of parentheses indicates that the endpoints -12, 4, and 11 are not included in the solution set, which is consistent with the strict inequality (>). The union symbol (∪) signifies that the solution set consists of all values within both intervals. This complete representation ensures that all possible solutions are captured and clearly communicated. Interval notation is the standard way to express solutions to inequalities in mathematics, providing a clear and unambiguous way to represent a range of values. By using interval notation, we can effectively communicate the solution to the polynomial inequality in a way that is easily understood by others. This final step completes the solution process, providing a clear and concise answer to the problem.

Conclusion

In this article, we have demonstrated a step-by-step method for solving the polynomial inequality -3(x-4)(x+12)(x-11) < 0. By simplifying the inequality, finding critical points, creating a sign chart, and identifying the intervals that satisfy the inequality, we arrived at the solution (-12, 4) ∪ (11, ∞). Polynomial inequalities are a fundamental topic in algebra, and mastering their solution is crucial for further studies in mathematics. The techniques discussed here can be applied to a wide range of polynomial inequalities, making this a valuable skill to acquire. Understanding the concepts behind each step, such as the importance of critical points and the use of sign charts, is key to solving these problems effectively. Furthermore, the ability to accurately represent the solution set using interval notation is essential for clear communication of mathematical results. By practicing these techniques and applying them to various problems, you can develop a strong understanding of polynomial inequalities and their solutions. This knowledge will serve as a solid foundation for more advanced mathematical concepts and applications. Remember, the key to success in mathematics is consistent practice and a thorough understanding of the underlying principles. So, continue to explore polynomial inequalities and challenge yourself with increasingly complex problems to further enhance your skills and knowledge.