What Values Of X Satisfy The Inequality 5(x-2)(x+4) > 0?
Navigating the realm of inequalities can sometimes feel like traversing a complex maze, but with a systematic approach and a clear understanding of the underlying principles, we can unravel even the most intricate problems. In this comprehensive exploration, we will embark on a journey to decipher the solution set to the inequality 5(x-2)(x+4) > 0. We will dissect the inequality, identify its critical points, and employ a strategic methodology to determine the intervals where the inequality holds true. By the end of this detailed analysis, you will possess a profound understanding of how to solve similar inequalities and confidently navigate the world of mathematical problem-solving.
Understanding the Inequality: A Foundation for Success
Before we delve into the intricacies of solving the inequality, it is crucial to establish a solid foundation by understanding the fundamental concepts at play. The inequality 5(x-2)(x+4) > 0 presents a scenario where the product of three factors, namely 5, (x-2), and (x+4), must be greater than zero. This implies that the product must be positive. To satisfy this condition, either all three factors must be positive, or one factor must be positive while the other two are negative. This forms the cornerstone of our approach to solving the inequality.
To gain a clearer perspective, let's identify the critical points of the inequality. These are the values of x that make any of the factors equal to zero. Setting each factor to zero, we find:
- x - 2 = 0 => x = 2
- x + 4 = 0 => x = -4
These critical points, x = 2 and x = -4, serve as dividers on the number line, splitting it into three distinct intervals: (-∞, -4), (-4, 2), and (2, ∞). Our next task is to examine the behavior of the inequality within each of these intervals.
Interval Analysis: A Strategic Approach
To determine the solution set, we will employ a systematic approach known as interval analysis. This method involves selecting a test value within each interval and evaluating the inequality at that value. The sign of the resulting product will indicate whether the inequality holds true within that interval.
Let's consider the first interval, (-∞, -4). We can choose a test value, say x = -5, and substitute it into the inequality:
5((-5)-2)((-5)+4) = 5(-7)(-1) = 35 > 0
Since the result is positive, the inequality holds true in the interval (-∞, -4).
Next, let's examine the interval (-4, 2). We can choose a test value, say x = 0, and substitute it into the inequality:
5(0-2)(0+4) = 5(-2)(4) = -40 < 0
In this case, the result is negative, indicating that the inequality does not hold true in the interval (-4, 2).
Finally, let's analyze the interval (2, ∞). We can choose a test value, say x = 3, and substitute it into the inequality:
5(3-2)(3+4) = 5(1)(7) = 35 > 0
The result is positive, implying that the inequality holds true in the interval (2, ∞).
The Solution Set: Unveiling the Answer
Based on our interval analysis, we have identified that the inequality 5(x-2)(x+4) > 0 holds true in the intervals (-∞, -4) and (2, ∞). To express the solution set in a concise and mathematically rigorous manner, we can use set notation.
The solution set is the union of the two intervals where the inequality is satisfied. Therefore, the solution set can be written as:
{x | x < -4 or x > 2}
This notation signifies the set of all x values that are either less than -4 or greater than 2. This precisely captures the solution to the inequality.
Visualizing the Solution: A Graphical Representation
To further solidify our understanding, let's visualize the solution set on a number line. We can represent the critical points, -4 and 2, as open circles, as the inequality is strict (greater than, not greater than or equal to). The intervals where the inequality holds true, (-∞, -4) and (2, ∞), can be highlighted with arrows extending outwards from the critical points.
The graphical representation provides a clear visual confirmation of our algebraic solution, reinforcing the concept that the inequality is satisfied for all x values less than -4 or greater than 2.
Conclusion: Mastering Inequalities
In this comprehensive exploration, we have successfully unraveled the solution set to the inequality 5(x-2)(x+4) > 0. We began by establishing a solid foundation, understanding the inequality's core principles and identifying its critical points. We then employed interval analysis, a strategic method that allowed us to determine the intervals where the inequality holds true. Finally, we expressed the solution set in set notation and visualized it on a number line, providing a clear and concise representation of our findings.
By mastering the techniques and principles discussed in this article, you are well-equipped to tackle a wide range of inequalities with confidence. Remember, the key to success lies in a systematic approach, a thorough understanding of the underlying concepts, and a commitment to practice. So, embrace the challenge, delve into the world of inequalities, and unlock your mathematical potential.
What values of x satisfy the inequality 5(x-2)(x+4) > 0?
Solving Inequalities: Find the Solution Set for 5(x-2)(x+4) > 0