Solving $-(x+1)(x-3)(x+2) > 0$ A Step-by-Step Guide

This article delves into a detailed explanation of how to solve the inequality $-(x+1)(x-3)(x+2) > 0$. Understanding inequalities is a crucial aspect of mathematics, particularly in algebra and calculus. This guide will walk you through each step, ensuring you grasp the underlying concepts and can apply them to similar problems. We will break down the problem into manageable parts, discuss the theory behind each step, and provide clear examples. Whether you're a student tackling homework or a math enthusiast looking to sharpen your skills, this comprehensive guide offers valuable insights and practical techniques. Let’s explore the methods to dissect this inequality and arrive at the correct solution.

1. Understanding the Inequality

To begin with, let's understand the inequality $-(x+1)(x-3)(x+2) > 0$. Polynomial inequalities such as this one involve finding the range of values for $x$ that satisfy the given condition. The negative sign in front of the expression complicates things slightly, so we’ll address it early on. Our main aim is to find the intervals on the number line where the expression $-(x+1)(x-3)(x+2)$ is greater than zero. In simpler terms, we need to determine the $x$ values for which the expression results in a positive number. This type of problem often involves analyzing the roots of the polynomial and testing intervals between these roots. It’s crucial to first identify these roots and then consider how the sign of the expression changes across these critical points. The approach involves transforming the inequality into a more manageable form, identifying key points, and then testing intervals to find the solution set. Let's start by addressing the negative sign, which will simplify our analysis and make the subsequent steps clearer.

2. Simplifying the Inequality

The initial step in solving the inequality $-(x+1)(x-3)(x+2) > 0$ is to simplify it by removing the negative sign. To do this, we can multiply both sides of the inequality by -1. Remember that when multiplying or dividing an inequality by a negative number, you must reverse the inequality sign. Thus, multiplying both sides by -1 gives us $(x+1)(x-3)(x+2) < 0$. This transformation is crucial because it allows us to work with a more standard form of a polynomial inequality. Now, we have a product of three factors less than zero, making it easier to analyze the sign changes across the roots. By removing the negative sign, we've essentially flipped the problem from finding where the expression is positive to where it is negative. This adjustment streamlines the subsequent steps, especially when we start analyzing intervals on the number line. The simplified form makes it clearer to identify the intervals where the product of the factors yields a negative result.

3. Finding the Roots

Next, we need to find the roots of the polynomial $(x+1)(x-3)(x+2) = 0$. The roots are the values of $x$ that make the polynomial equal to zero. To find these roots, we set each factor equal to zero and solve for $x$. This gives us three roots: $x = -1$, $x = 3$, and $x = -2$. These roots are critical because they divide the number line into intervals where the sign of the polynomial may change. The roots represent the points where the expression transitions from positive to negative or vice versa. Placing these roots on a number line helps visualize these intervals and understand how the sign of each factor behaves within them. Knowing the roots allows us to systematically test intervals and determine where the inequality $(x+1)(x-3)(x+2) < 0$ holds true. The roots are the key to unlocking the solution set for the inequality.

4. Creating a Sign Chart

To effectively solve the inequality $(x+1)(x-3)(x+2) < 0$, creating a sign chart is an invaluable technique. A sign chart helps us visualize how the sign of each factor, and thus the overall expression, changes across different intervals. We start by placing the roots we found, $x = -2$, $x = -1$, and $x = 3$, on a number line. These roots divide the number line into four intervals: $(-\infty, -2)$, $(-2, -1)$, $(-1, 3)$, and $(3, \infty)$. For each interval, we choose a test value and determine the sign of each factor $(x+1)$, $(x-3)$, and $(x+2)$. The product of these signs will give us the sign of the entire expression $(x+1)(x-3)(x+2)$ in that interval. For instance, in the interval $(-\infty, -2)$, we might choose $x = -3$. Substituting this value into each factor gives us $(-3+1) = -2$ (negative), $(-3-3) = -6$ (negative), and $(-3+2) = -1$ (negative). The product of three negative numbers is negative, so the expression is negative in this interval. By repeating this process for each interval, we can build a complete sign chart, which makes it straightforward to identify the intervals where the expression is less than zero.

5. Testing Intervals

The crucial step of testing intervals is where we use the sign chart to determine the solution set for the inequality $(x+1)(x-3)(x+2) < 0$. Once we have divided the number line into intervals using the roots $-2$, $-1$, and $3$, we need to test a value within each interval to see if it satisfies the inequality. Let's revisit the intervals: $(-\infty, -2)$, $(-2, -1)$, $(-1, 3)$, and $(3, \infty)$.

1. For the interval $(-\infty, -2)$, let’s choose a test value of $x = -3$. Plugging this into the expression:
   • $(x+1) = -3+1 = -2$ (negative)
   • $(x-3) = -3-3 = -6$ (negative)
   • $(x+2) = -3+2 = -1$ (negative) The product $(-2) \times (-6) \times (-1) = -12$ is negative, so this interval satisfies the inequality.
2. For the interval $(-2, -1)$, let’s choose a test value of $x = -1.5$. Plugging this into the expression:
   • $(x+1) = -1.5+1 = -0.5$ (negative)
   • $(x-3) = -1.5-3 = -4.5$ (negative)
   • $(x+2) = -1.5+2 = 0.5$ (positive) The product $(-0.5) \times (-4.5) \times (0.5) = 1.125$ is positive, so this interval does not satisfy the inequality.
3. For the interval $(-1, 3)$, let’s choose a test value of $x = 0$. Plugging this into the expression:
   • $(x+1) = 0+1 = 1$ (positive)
   • $(x-3) = 0-3 = -3$ (negative)
   • $(x+2) = 0+2 = 2$ (positive) The product $(1) \times (-3) \times (2) = -6$ is negative, so this interval satisfies the inequality.
4. For the interval $(3, \infty)$, let’s choose a test value of $x = 4$. Plugging this into the expression:
   • $(x+1) = 4+1 = 5$ (positive)
   • $(x-3) = 4-3 = 1$ (positive)
   • $(x+2) = 4+2 = 6$ (positive) The product $(5) \times (1) \times (6) = 30$ is positive, so this interval does not satisfy the inequality.

By testing these intervals, we can clearly see which ranges of $x$ values make the inequality $(x+1)(x-3)(x+2) < 0$ true. This methodical approach ensures we account for all possible sign changes and accurately determine the solution set.

6. Determining the Solution Set

Based on our interval testing, we can now determine the solution set for the inequality $(x+1)(x-3)(x+2) < 0$. From our analysis, we found that the inequality holds true for the intervals $(-\infty, -2)$ and $(-1, 3)$. This means that the values of $x$ in these intervals make the expression negative. Therefore, the solution set consists of all $x$ values that fall within these ranges. We express this solution set in interval notation as $(-\infty, -2) \cup (-1, 3)$. The union symbol $\cup$ indicates that the solution set includes both intervals. It’s important to note that since the inequality is strictly less than zero (and not less than or equal to), we use open intervals (parentheses) to exclude the roots $-2$, $-1$, and $3$ themselves. If the inequality had been $(x+1)(x-3)(x+2) \leq 0$, we would have included these roots in the solution set using closed intervals (brackets). The solution set $(-\infty, -2) \cup (-1, 3)$ precisely captures all $x$ values that satisfy the original inequality, providing a clear and concise answer.

7. Writing the Final Solution

Finally, let’s write the final solution to the inequality $-(x+1)(x-3)(x+2) > 0$. After simplifying and testing intervals, we determined that the solution set is $(-\infty, -2) \cup (-1, 3)$. This means that the inequality is satisfied when $x$ is less than -2 or when $x$ is between -1 and 3. It is essential to present the solution clearly and accurately, using correct notation. The interval notation provides a concise way to express the range of $x$ values that fulfill the condition. To recap, we started by simplifying the inequality by multiplying by -1, which reversed the inequality sign. We then found the roots of the polynomial, created a sign chart, and tested intervals to identify where the expression was less than zero. By combining these steps, we arrived at the solution set. Understanding and correctly presenting the solution set is the final step in mastering this type of inequality problem. The solution $(-\infty, -2) \cup (-1, 3)$ is the definitive answer to the posed inequality.

In summary, solving the inequality $-(x+1)(x-3)(x+2) > 0$ involves a methodical approach. We began by simplifying the inequality, found the roots, created a sign chart, tested intervals, and finally determined the solution set. Each step is crucial in arriving at the correct answer. The solution set, expressed in interval notation, is $(-\infty, -2) \cup (-1, 3)$. This solution represents all the $x$ values that satisfy the original inequality. By mastering these steps, you can confidently tackle similar polynomial inequalities. Remember, the key is to break down the problem into manageable parts, understand the underlying theory, and apply the techniques systematically. This comprehensive guide aims to equip you with the skills and knowledge necessary to solve such problems effectively. Practice and repetition will further solidify your understanding and improve your problem-solving abilities.