Solving And Graphing The Inequality (x-1)(x-4)(x-5) ≤ 0

Understanding Polynomial Inequalities

In the realm of mathematics, polynomial inequalities play a crucial role in understanding the behavior of polynomial functions. Solving inequalities, especially those involving higher-degree polynomials, requires a systematic approach that combines algebraic techniques with graphical interpretations. This article delves into the process of solving the inequality (x-1)(x-4)(x-5) ≤ 0, a classic example that showcases the principles involved in dealing with such problems. This particular inequality is a cubic inequality, meaning it involves a polynomial of degree three. Solving it entails finding the values of x that make the expression less than or equal to zero. To effectively tackle this, we need to understand the relationship between the factors of the polynomial and the intervals where the inequality holds true. Key to this process is identifying the critical points, which are the values of x that make the polynomial equal to zero. These points divide the number line into intervals, and by testing values within each interval, we can determine the solution set of the inequality. Let's explore the step-by-step method to solve this inequality and graphically represent its solution on a number line.

Step 1: Find the Critical Points

The first step in solving the inequality $(x-1)(x-4)(x-5) ext{ ≤ } 0$ is to identify the 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, and the sign of the expression remains constant within each interval. To find the critical points, we set each factor equal to zero and solve for x:

• $x - 1 = 0 ext{ => } x = 1$
• $x - 4 = 0 ext{ => } x = 4$
• $x - 5 = 0 ext{ => } x = 5$

Thus, the critical points are 1, 4, and 5. These points are the roots of the polynomial and are the values of x where the graph of the polynomial intersects the x-axis. They act as boundaries, separating the number line into intervals where the polynomial is either positive or negative. The critical points themselves are part of the solution set because the inequality includes the 'equal to' condition (≤ 0). This means that the values of x that make the expression equal to zero are also solutions. In the context of the graph of the polynomial, these critical points correspond to the x-intercepts. Understanding the nature and significance of these critical points is essential for correctly determining the intervals that satisfy the inequality.

Step 2: Create a Number Line and Intervals

Now that we have the critical points, we can create a number line and divide it into intervals. Mark the critical points (1, 4, and 5) on the number line. These points partition the number line into four intervals:

• $(-\infty, 1)$
• $(1, 4)$
• $(4, 5)$
• $(5, \infty)$

Each of these intervals represents a range of x values, and within each interval, the expression $(x-1)(x-4)(x-5)$ will have a consistent sign (either positive or negative). The critical points themselves are included in the solution set because the inequality is non-strict (≤ 0). When visualizing this on the number line, it's helpful to use closed circles or brackets to indicate that the critical points are included in the solution. The number line serves as a visual aid, allowing us to clearly see how the critical points segment the possible x values. This segmentation is fundamental to the next step, where we'll test each interval to determine whether it satisfies the given inequality. The careful construction of this number line is a crucial step in solving polynomial inequalities, as it provides a clear framework for analyzing the sign of the polynomial expression.

Step 3: Test Each Interval

To determine the sign of the expression $(x-1)(x-4)(x-5)$ in each interval, we select a test value within each interval and substitute it into the expression. This process will reveal whether the expression is positive or negative in that interval. Here’s how we proceed:

1. Interval $(-\infty, 1)$: Choose $x = 0$. Substituting into the expression gives $(0-1)(0-4)(0-5) = (-1)(-4)(-5) = -20$, which is negative.
2. Interval $(1, 4)$: Choose $x = 2$. Substituting into the expression gives $(2-1)(2-4)(2-5) = (1)(-2)(-3) = 6$, which is positive.
3. Interval $(4, 5)$: Choose $x = 4.5$. Substituting into the expression gives $(4.5-1)(4.5-4)(4.5-5) = (3.5)(0.5)(-0.5) = -0.875$, which is negative.
4. Interval $(5, \infty)$: Choose $x = 6$. Substituting into the expression gives $(6-1)(6-4)(6-5) = (5)(2)(1) = 10$, which is positive.

By testing these values, we can create a sign chart or simply note the sign of the expression in each interval. This step is vital because it links the algebraic expression to the number line intervals, showing us exactly where the expression is less than or equal to zero. The sign of the expression is determined by the product of the signs of its factors. Therefore, understanding the sign changes at the critical points is crucial for accurately solving the inequality. This process of testing intervals is a cornerstone of solving polynomial and rational inequalities, providing a clear and reliable method for finding the solution set.

Step 4: Identify the Solution Set

The inequality we are solving is $(x-1)(x-4)(x-5) ext{ ≤ } 0$. This means we are looking for the intervals where the expression is either negative or zero. From the previous step, we found that the expression is negative in the intervals $(-\infty, 1)$ and $(4, 5)$. Additionally, the expression is equal to zero at the critical points $x = 1$, $x = 4$, and $x = 5$. Therefore, the solution set includes these critical points and the intervals where the expression is negative.

Combining these findings, the solution set for the inequality is:

$$(-\infty, 1] \cup [4, 5]$$

This notation represents the union of two intervals: all real numbers less than or equal to 1, and all real numbers between 4 and 5, inclusive. The square brackets indicate that the endpoints (1, 4, and 5) are included in the solution set, as the inequality includes the “equal to” condition. The union symbol ($\cup$) signifies that the solution set is the combination of these intervals. Identifying the solution set accurately depends on correctly interpreting the results of the interval testing and understanding the inequality's conditions. This step is the culmination of the algebraic analysis, translating the sign information into a concise representation of all x values that satisfy the original inequality. Understanding how to express the solution set using interval notation is an essential skill in mathematics, especially when dealing with inequalities.

Step 5: Graph the Solution Set on a Number Line

The final step is to graph the solution set on a number line. This visual representation provides a clear picture of the values of x that satisfy the inequality $(x-1)(x-4)(x-5) ext{ ≤ } 0$. To graph the solution set, we mark the critical points 1, 4, and 5 on the number line. Since the inequality includes the “equal to” condition (≤), we use closed circles (or brackets) at these points to indicate that they are included in the solution. Then, we shade the intervals $(-\infty, 1]$ and $[4, 5]$ to represent all the values of x in these intervals that satisfy the inequality.

The graph will have a shaded region extending from negative infinity up to and including 1, and another shaded region extending from 4 to 5, including both 4 and 5. The unshaded regions represent the intervals where the expression $(x-1)(x-4)(x-5)$ is greater than zero, and thus do not satisfy the inequality. This graphical representation is a powerful tool for understanding the solution set, as it visually highlights the range of x values that meet the condition. Graphing the solution set on a number line not only completes the problem-solving process but also enhances comprehension and reinforces the connection between algebraic solutions and their geometric interpretations. This visualization is especially helpful in communicating the solution clearly and effectively.

Conclusion

In summary, solving the inequality $(x-1)(x-4)(x-5) ext{ ≤ } 0$ involves finding the critical points, creating intervals on a number line, testing each interval to determine the sign of the expression, identifying the solution set, and graphing it on a number line. This process demonstrates the interplay between algebra and graphical representation in solving polynomial inequalities. By following these steps, you can effectively tackle similar problems and gain a deeper understanding of the behavior of polynomial functions. The solution set $(-\infty, 1] \cup [4, 5]$ represents all the values of x that make the expression less than or equal to zero. This article has provided a comprehensive guide to solving this specific inequality, but the methods and principles discussed are widely applicable to a variety of polynomial inequalities. Mastering these techniques is essential for success in algebra and calculus, where understanding the solutions to inequalities is fundamental to many concepts and applications. The combination of algebraic manipulation and graphical interpretation provides a robust approach to solving inequalities and understanding their solutions.