Solving The Polynomial Inequality X^3 - X^2 > 9(x - 1)

Understanding inequalities is a fundamental concept in mathematics, crucial for solving a wide array of problems. This article delves into the solution of the inequality $x^3 - x^2 > 9(x - 1)$, providing a step-by-step approach to understanding and resolving it. We will explore the algebraic manipulations involved, the importance of factoring, and how to interpret the solution set. By the end of this discussion, you will have a clear understanding of how to tackle such inequalities and the reasoning behind each step.

1. Transforming the Inequality

To begin solving the inequality, we need to manipulate it into a more manageable form. The initial inequality is $x^3 - x^2 > 9(x - 1)$. The first step is to expand the right side of the inequality, which gives us $x^3 - x^2 > 9x - 9$. This expansion simplifies the expression, allowing us to consolidate all terms on one side. Our goal is to set one side of the inequality to zero, making it easier to identify potential roots and intervals for testing. This process is similar to solving equations, but with inequalities, we need to be mindful of how operations affect the direction of the inequality sign.

Next, we subtract $9x$ and add $9$ to both sides of the inequality. This action results in the inequality $x^3 - x^2 - 9x + 9 > 0$. By bringing all terms to one side, we now have a polynomial expression that we can analyze. This is a crucial step, as it transforms the problem into one of finding the values of $x$ for which the polynomial is greater than zero. The next step will involve factoring this polynomial, which will help us identify the critical points where the expression changes sign.

This transformation is essential because it allows us to use techniques like factoring and sign analysis to determine the solution set. Factoring is a powerful tool in algebra, and in this case, it will help us break down the cubic polynomial into simpler expressions. By identifying the roots of the polynomial, we can divide the number line into intervals and test each interval to see if it satisfies the original inequality. This systematic approach ensures that we find all possible solutions.

2. Factoring the Polynomial

Now that we have the inequality in the form $x^3 - x^2 - 9x + 9 > 0$, the next crucial step is to factor the cubic polynomial. Factoring helps us break down the complex expression into simpler components, making it easier to identify the roots and the intervals where the polynomial is positive or negative. We will use factoring by grouping, a technique that involves pairing terms and extracting common factors.

To begin, we group the first two terms and the last two terms: $(x^3 - x^2) + (-9x + 9) > 0$. From the first group, we can factor out $x^2$, giving us $x^2(x - 1)$. From the second group, we can factor out $-9$, resulting in $-9(x - 1)$. Now, the inequality looks like this: $x^2(x - 1) - 9(x - 1) > 0$. Notice that $(x - 1)$ is a common factor in both terms. This is a key observation that allows us to further simplify the expression.

We can now factor out the common term $(x - 1)$ from the entire expression. This gives us $(x - 1)(x^2 - 9) > 0$. The expression $x^2 - 9$ is a difference of squares, which can be factored further. Recall that the difference of squares $a^2 - b^2$ can be factored as $(a - b)(a + b)$. Applying this to $x^2 - 9$, we get $(x - 3)(x + 3)$. Therefore, the fully factored inequality is $(x - 1)(x - 3)(x + 3) > 0$. This factored form is much easier to work with, as it reveals the roots of the polynomial and allows us to determine the intervals where the inequality holds true.

The roots of the polynomial are the values of $x$ that make the expression equal to zero. In this case, the roots are $x = 1$, $x = 3$, and $x = -3$. These roots are critical points that divide the number line into intervals. By analyzing the sign of the polynomial in each interval, we can determine where the polynomial is greater than zero and thus find the solution to the inequality. Factoring the polynomial is a pivotal step in this process, as it simplifies the inequality and allows us to systematically identify the solution set.

3. Identifying Critical Points

The critical points are the values of $x$ that make the polynomial equal to zero. These points are crucial because they divide the number line into intervals, and within each interval, the sign of the polynomial remains constant. From the factored inequality $(x - 1)(x - 3)(x + 3) > 0$, we can identify the critical points by setting each factor equal to zero and solving for $x$. This gives us the critical points $x = -3$, $x = 1$, and $x = 3$.

To find these critical points, we set each factor to zero: $x - 1 = 0$, $x - 3 = 0$, and $x + 3 = 0$. Solving these equations gives us $x = 1$, $x = 3$, and $x = -3$, respectively. These values are where the polynomial changes its sign, moving from positive to negative or vice versa. Therefore, they are the boundaries of the intervals that we need to test to find the solution to the inequality. The critical points are the cornerstone of our solution strategy.

The critical points divide the number line into four intervals: $(- ext{∞}, -3)$, $(-3, 1)$, $(1, 3)$, and $(3, ext{∞})$. Each interval represents a range of $x$ values, and within each interval, the polynomial $(x - 1)(x - 3)(x + 3)$ will either be consistently positive or consistently negative. This is because the sign of the polynomial can only change at the critical points where one or more of the factors become zero. Therefore, to solve the inequality, we need to determine the sign of the polynomial in each of these intervals. Understanding the role of critical points is fundamental to solving inequalities of this nature.

4. Testing Intervals

With the critical points identified, the next step is to test each interval to determine where the inequality $(x - 1)(x - 3)(x + 3) > 0$ holds true. We will choose a test value within each interval and substitute it into the factored inequality. The sign of the result will indicate whether the polynomial is positive or negative in that interval. This process will help us identify the intervals that satisfy the inequality and form the solution set.

Let's start with the first interval, $(- ext{∞}, -3)$. We can choose a test value, such as $x = -4$. Substituting this into the factored inequality, we get $(-4 - 1)(-4 - 3)(-4 + 3) = (-5)(-7)(-1) = -35$. Since the result is negative, the polynomial is negative in this interval, and this interval does not satisfy the inequality.

Next, we consider the interval $(-3, 1)$. Let's choose a test value, such as $x = 0$. Substituting this into the factored inequality, we get $(0 - 1)(0 - 3)(0 + 3) = (-1)(-3)(3) = 9$. Since the result is positive, the polynomial is positive in this interval, and this interval satisfies the inequality.

For the interval $(1, 3)$, let's choose a test value, such as $x = 2$. Substituting this into the factored inequality, we get $(2 - 1)(2 - 3)(2 + 3) = (1)(-1)(5) = -5$. Since the result is negative, the polynomial is negative in this interval, and this interval does not satisfy the inequality.

Finally, we consider the interval $(3, ext{∞})$. Let's choose a test value, such as $x = 4$. Substituting this into the factored inequality, we get $(4 - 1)(4 - 3)(4 + 3) = (3)(1)(7) = 21$. Since the result is positive, the polynomial is positive in this interval, and this interval satisfies the inequality. By testing each interval, we have determined where the polynomial is greater than zero. This information is crucial for stating the solution to the inequality. Testing intervals is a systematic way to analyze the behavior of the polynomial between the critical points.

5. Stating the Solution

After testing the intervals, we have identified the ranges of $x$ where the inequality $(x - 1)(x - 3)(x + 3) > 0$ holds true. We found that the polynomial is positive in the intervals $(-3, 1)$ and $(3, ext{∞})$. Therefore, the solution to the inequality is the union of these intervals. This means that any value of $x$ within these intervals will satisfy the original inequality $x^3 - x^2 > 9(x - 1)$.

To express the solution set, we use interval notation. The solution set is $(-3, 1) ext{ ∪ } (3, ext{∞})$. This notation indicates that the solution includes all values of $x$ between $-3$ and $1$ (excluding $-3$ and $1$) and all values of $x$ greater than $3$. The union symbol $ ext{ ∪ }$ combines these two intervals into a single solution set. It is important to note that the parentheses indicate that the endpoints are not included in the solution, as the inequality is strict (greater than, not greater than or equal to).

In summary, the solution set represents all the values of $x$ that make the inequality $x^3 - x^2 > 9(x - 1)$ true. By transforming the inequality, factoring the polynomial, identifying critical points, and testing intervals, we have systematically determined the solution set. The interval notation $(-3, 1) ext{ ∪ } (3, ext{∞})$ concisely and accurately expresses the solution. This step-by-step approach demonstrates the logical progression required to solve polynomial inequalities. Stating the solution in the correct notation is the final step in solving the inequality, providing a clear and concise answer.

By following these steps, you can solve similar polynomial inequalities. Understanding the principles of factoring, critical points, and interval testing is essential for mastering these types of problems. The solution set we obtained, $(-3, 1) ext{ ∪ } (3, ext{∞})$, represents the values of $x$ that satisfy the original inequality, and this detailed explanation should provide a solid foundation for tackling future mathematical challenges.