Solving Cubic Inequality Algebraically Step-by-Step Guide
Introduction
In this article, we will delve into the process of solving a cubic inequality algebraically. Specifically, we will address the inequality x^3 - 3x^2 - 28x < 0. This type of problem often appears in algebra and calculus courses, and mastering the technique to solve it is crucial for further mathematical studies. We will provide a detailed, step-by-step explanation to ensure a clear understanding of each stage involved. By following this guide, you'll gain the ability to tackle similar inequalities with confidence. This comprehensive approach is designed to clarify the methodology, making it accessible even if you're new to the topic. Let's break down each step, ensuring you grasp the underlying concepts and techniques necessary for solving such algebraic challenges. Understanding inequalities is fundamental in various fields, including engineering, economics, and computer science. Thus, mastering this skill equips you with a valuable tool for problem-solving in a wide range of contexts. Our focus will be on developing a robust understanding, so you can apply these methods effectively in various scenarios. This skill is also essential for higher-level mathematics, such as calculus and analysis, where inequalities frequently appear. Therefore, dedicating time to mastering this concept will pay off in your future academic endeavors. We aim to make the process as straightforward as possible, by presenting each step with clarity and detailed explanations.
Step 1: Factor the Polynomial
The first critical step in solving the inequality x^3 - 3x^2 - 28x < 0 is to factor the polynomial expression. Factoring simplifies the inequality and allows us to identify the critical points, which are essential for determining the solution intervals. Begin by looking for a common factor in all terms. In this case, we can see that x is a common factor. Factoring out x from the polynomial gives us: x(x^2 - 3x - 28) < 0. Now, we need to factor the quadratic expression x^2 - 3x - 28. To do this, we look for two numbers that multiply to -28 and add to -3. These numbers are -7 and 4. Therefore, we can factor the quadratic as (x - 7)(x + 4). Putting it all together, the factored form of the inequality is x(x - 7)(x + 4) < 0. This factored form is crucial because it allows us to identify the roots of the polynomial, which are the values of x that make the expression equal to zero. The roots are the points where the graph of the polynomial crosses the x-axis, and they divide the number line into intervals that we will analyze in the subsequent steps. Factoring is a fundamental algebraic skill, and proficiency in this area is essential for solving various types of equations and inequalities. It's a technique that's used extensively in mathematics, so mastering it will be beneficial in many contexts. This initial step of factoring sets the stage for the rest of the solution process. Without it, identifying the critical points would be significantly more challenging. Therefore, ensuring you have a solid grasp of factoring techniques is vital for your mathematical journey.
Step 2: Identify the Critical Points
After factoring the inequality x(x - 7)(x + 4) < 0, the next step 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, within which the polynomial's sign remains consistent. To find the critical points, we set each factor equal to zero and solve for x. From the factor x, we get x = 0. From the factor (x - 7), we get x - 7 = 0, which implies x = 7. Lastly, from the factor (x + 4), we get x + 4 = 0, which implies x = -4. Therefore, the critical points are x = -4, 0, and 7. These critical points are the boundaries where the polynomial can change its sign. Think of them as the pivotal values that determine whether the expression is positive, negative, or zero. The identification of critical points is a fundamental step in solving inequalities, as it allows us to create a sign chart or test intervals. Without these critical points, it would be impossible to determine the intervals where the inequality holds true. This process is not only applicable to cubic inequalities but also to inequalities involving other polynomials. Thus, understanding this step is crucial for a wide range of mathematical problems. By pinpointing these critical values, we can systematically analyze the behavior of the polynomial in each interval, leading us to the final solution. Remember, these points are not necessarily part of the solution, as the inequality might be strict (i.e., < or >). They simply serve as dividing lines for our analysis.
Step 3: Create a Sign Chart
Once we have identified the critical points, which are x = -4, 0, and 7, the next crucial step is to create a sign chart. A sign chart helps us visualize how the sign of the polynomial x(x - 7)(x + 4) changes across different intervals defined by the critical points. Start by drawing a number line and marking the critical points -4, 0, and 7 on it. These points divide the number line into four intervals: (-∞, -4), (-4, 0), (0, 7), and (7, ∞). Now, we need to determine the sign of the polynomial in each interval. To do this, we pick a test value within each interval and substitute it into the factored form of the polynomial, x(x - 7)(x + 4). For the interval (-∞, -4), let's choose x = -5. Substituting this value, we get (-5)(-5 - 7)(-5 + 4) = (-5)(-12)(-1) = -60, which is negative. So, the polynomial is negative in this interval. For the interval (-4, 0), let's choose x = -1. Substituting this value, we get (-1)(-1 - 7)(-1 + 4) = (-1)(-8)(3) = 24, which is positive. So, the polynomial is positive in this interval. For the interval (0, 7), let's choose x = 1. Substituting this value, we get (1)(1 - 7)(1 + 4) = (1)(-6)(5) = -30, which is negative. So, the polynomial is negative in this interval. For the interval (7, ∞), let's choose x = 8. Substituting this value, we get (8)(8 - 7)(8 + 4) = (8)(1)(12) = 96, which is positive. So, the polynomial is positive in this interval. We can now create a sign chart by noting the sign of the polynomial in each interval: Interval (-∞, -4): Negative Interval (-4, 0): Positive Interval (0, 7): Negative Interval (7, ∞): Positive. The sign chart is a powerful tool because it visually represents the behavior of the polynomial across the entire number line. This visual representation makes it easy to identify the intervals where the inequality holds true. Without the sign chart, it would be much harder to determine the solution set. This method is applicable to any polynomial inequality, making it a versatile technique to have in your mathematical toolkit.
Step 4: Determine the Solution Set
After creating the sign chart, the final step is to determine the solution set for the inequality x^3 - 3x^2 - 28x < 0, which we have factored as x(x - 7)(x + 4) < 0. Our sign chart shows the intervals where the polynomial is negative, positive, or zero. We are looking for the intervals where the polynomial is strictly less than zero, meaning we want the intervals where the polynomial is negative. From the sign chart, we identified that the polynomial is negative in the intervals (-∞, -4) and (0, 7). Since the inequality is strict (i.e., < and not ≤), we do not include the critical points in the solution set. The critical points are where the polynomial equals zero, and we only want the values where it is strictly less than zero. Therefore, the solution set consists of all x values in the intervals (-∞, -4) and (0, 7). We can write the solution set in interval notation as (-∞, -4) ∪ (0, 7). The symbol ∪ represents the union of the two intervals, meaning the solution includes all values in either interval. This solution set represents all real numbers x that satisfy the original inequality. Any value within these intervals, when substituted into the original inequality, will result in a negative value. Determining the solution set is the culmination of all the previous steps. The factoring, identifying critical points, and creating a sign chart were all necessary to arrive at this final answer. Understanding how to interpret the sign chart and translate it into the solution set is a crucial skill in solving inequalities. This process provides a clear and systematic way to solve polynomial inequalities, ensuring accuracy and understanding.
Conclusion
In conclusion, solving the inequality x^3 - 3x^2 - 28x < 0 involves several key steps: factoring the polynomial, identifying the critical points, creating a sign chart, and determining the solution set. We began by factoring the given polynomial into x(x - 7)(x + 4) < 0. This factorization allowed us to identify the critical points, which are the values of x that make the polynomial equal to zero. These critical points were found to be x = -4, 0, and 7. Next, we created a sign chart to analyze the sign of the polynomial in the intervals defined by these critical points. This involved choosing test values in each interval and substituting them into the factored polynomial to determine whether the result was positive or negative. The sign chart visually represented the behavior of the polynomial across the number line, making it easier to identify the intervals where the inequality held true. Finally, we determined the solution set based on the sign chart. Since we were looking for where the polynomial was strictly less than zero, we identified the intervals where the polynomial was negative. The solution set was found to be (-∞, -4) ∪ (0, 7). This means that any value of x within these intervals will satisfy the original inequality. Mastering this process is essential for solving polynomial inequalities and is a fundamental skill in algebra and calculus. By following these steps, you can systematically approach and solve similar problems, building confidence in your mathematical abilities. Understanding inequalities is not only crucial for academic success but also has practical applications in various fields, making this a valuable skill to develop.
