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

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Polynomial inequalities, such as (x-1)(x-4)(x-5) ≤ 0, might appear daunting at first. However, with a systematic approach, these inequalities can be solved effectively. This article provides a step-by-step guide on how to solve polynomial inequalities and graph their solution sets on a number line. Understanding these methods is crucial for various mathematical applications, making it an essential skill for students and professionals alike. Mastering the art of solving polynomial inequalities involves a combination of algebraic manipulation, graphical representation, and logical deduction. Each step in the process builds upon the previous one, ensuring a clear and accurate solution. This comprehensive guide aims to break down the complexity of polynomial inequalities into manageable steps, empowering you to tackle these problems with confidence.

Understanding Polynomial Inequalities

Before diving into the solution, it's essential to understand what polynomial inequalities are and why they matter. A polynomial inequality is an inequality that involves a polynomial expression. These inequalities can take various forms, such as less than (<<), greater than (>), less than or equal to (≤), or greater than or equal to (≥). The solutions to these inequalities are the values of the variable that make the inequality true. Understanding polynomial inequalities is not just an academic exercise; it has practical applications in various fields such as engineering, economics, and computer science. For instance, engineers might use polynomial inequalities to determine the range of values for a physical parameter within which a system remains stable. Economists might use them to model market behavior and predict price fluctuations. In computer science, polynomial inequalities can be used in optimization algorithms and complexity analysis.

Polynomial inequalities differ from polynomial equations in a crucial way: while equations have specific solutions (roots), inequalities have solution sets, which are intervals or unions of intervals. This difference necessitates a different approach to solving them. The solution set of a polynomial inequality represents the range of values for the variable that satisfy the given condition. For example, consider the inequality x^2 - 4 > 0. The solution set includes all values of x that make the expression greater than zero. This set consists of two intervals: x < -2 and x > 2. Graphically, this means that all points on the number line to the left of -2 and to the right of 2 satisfy the inequality. This concept of solution sets is fundamental to understanding and solving polynomial inequalities.

Step-by-Step Solution for (x-1)(x-4)(x-5) ≤ 0

Now, let's tackle the inequality (x-1)(x-4)(x-5) ≤ 0 step by step.

Step 1 Find the Zeros

The first step is to find the zeros of the polynomial. These are the values of x that make the polynomial equal to zero. To find these, set each factor equal to zero and solve:

  • x - 1 = 0 => x = 1
  • x - 4 = 0 => x = 4
  • x - 5 = 0 => x = 5

Finding the zeros is a critical first step because these values divide the number line into intervals, within which the polynomial's sign remains constant. These zeros are the points where the polynomial crosses the x-axis, and they serve as boundaries for the intervals where the polynomial is either positive or negative. In the given inequality, the zeros are 1, 4, and 5. These points will be used to create a sign chart, which is a visual tool to determine the intervals where the polynomial is less than or equal to zero. The accuracy of this step is paramount, as any error in finding the zeros will propagate through the rest of the solution. Each zero corresponds to a root of the polynomial, and these roots play a crucial role in determining the solution set of the inequality.

Step 2 Create a Sign Chart

A sign chart helps visualize the sign of the polynomial in different intervals. Place the zeros on a number line and consider the intervals they create:

---(-∞)---(1)---(4)---(5)---(+∞)---

The sign chart is a powerful tool for solving inequalities because it provides a clear visual representation of where the polynomial is positive, negative, or zero. To create the sign chart, draw a number line and mark the zeros that were found in the previous step. These zeros divide the number line into intervals. For each interval, we will determine the sign of the polynomial by testing a value within that interval. The sign chart allows us to easily identify the intervals that satisfy the inequality. It is a systematic way to organize the information and avoid errors. The sign chart method is applicable to a wide range of polynomial inequalities, making it a versatile technique in algebra.

Step 3 Test Intervals

Choose a test value in each interval and plug it into the polynomial (x-1)(x-4)(x-5) to determine the sign:

  • Interval (-∞, 1): Test x = 0 => (0-1)(0-4)(0-5) = (-1)(-4)(-5) = -20 (Negative)
  • Interval (1, 4): Test x = 2 => (2-1)(2-4)(2-5) = (1)(-2)(-3) = 6 (Positive)
  • Interval (4, 5): Test x = 4.5 => (4.5-1)(4.5-4)(4.5-5) = (3.5)(0.5)(-0.5) = -0.875 (Negative)
  • Interval (5, ∞): Test x = 6 => (6-1)(6-4)(6-5) = (5)(2)(1) = 10 (Positive)

Testing intervals is a crucial step in determining the solution set of the inequality. By choosing a test value within each interval, we can evaluate the sign of the polynomial in that interval. The sign of the polynomial remains consistent within each interval because the polynomial can only change signs at its zeros. Therefore, the sign of the test value will be representative of the sign of the polynomial throughout the entire interval. This method simplifies the process of finding the solution set, as we only need to perform a few calculations to determine the sign in each interval. The choice of test values is arbitrary, but it is best to choose values that are easy to work with and not too close to the zeros to avoid calculation errors.

Step 4 Determine the Solution Set

Since we are looking for (x-1)(x-4)(x-5) ≤ 0, we want the intervals where the polynomial is negative or zero. From our tests, these are:

  • (-∞, 1]
  • [4, 5]

Therefore, the solution set is (-∞, 1] ∪ [4, 5].

The solution set is determined by identifying the intervals where the polynomial satisfies the inequality. In this case, we are looking for intervals where (x-1)(x-4)(x-5) is less than or equal to zero. From the sign chart, we identified two intervals: (-∞, 1) and (4, 5). Since the inequality includes