Solving (x-3)^2(x+2)<0 A Step-by-Step Guide

In the realm of mathematics, particularly in algebra, solving inequalities is a fundamental skill. Inequalities, unlike equations, deal with relationships that are not strictly equal. They involve comparing expressions using symbols like < (less than), > (greater than), ≤ (less than or equal to), and ≥ (greater than or equal to). This article delves into the process of solving polynomial inequalities, focusing on a specific example: (x-3)^2(x+2) < 0. We will explore the steps involved, the underlying concepts, and how to express the solution in interval notation.

Understanding Polynomial Inequalities

Polynomial inequalities are inequalities that involve polynomial expressions. A polynomial expression is a mathematical expression consisting of variables and coefficients, involving only the operations of addition, subtraction, multiplication, and non-negative integer exponents. Solving these inequalities requires finding the values of the variable that make the inequality true. This often involves identifying critical points, testing intervals, and expressing the solution in a concise and accurate manner.

Critical points play a crucial role in solving polynomial inequalities. These are the values of the variable that make the polynomial expression equal to zero or undefined. In the context of polynomial inequalities, critical points are typically the roots of the polynomial. They divide the number line into intervals, within each of which the polynomial's sign remains constant. By identifying these critical points and testing values within each interval, we can determine the solution set of the inequality.

To successfully solve polynomial inequalities, a systematic approach is essential. This approach generally involves the following steps:

1. Transform the inequality: If necessary, rearrange the inequality so that one side is zero. This allows us to focus on the sign of the polynomial expression on the other side.
2. Factor the polynomial: Factor the polynomial expression as much as possible. This helps identify the critical points more easily.
3. Find the critical points: Determine the values of the variable that make each factor equal to zero. These are the critical points.
4. Create a sign chart: Draw a number line and mark the critical points on it. These points divide the number line into intervals. Create a sign chart to analyze the sign of each factor and the overall polynomial expression within each interval.
5. Test each interval: Choose a test value within each interval and substitute it into the original inequality. Determine whether the inequality is true or false for that test value. This will tell you the sign of the polynomial within that interval.
6. Identify the solution set: Based on the sign chart and the inequality, identify the intervals where the inequality is satisfied. The solution set consists of these intervals.
7. Express the solution in interval notation: Write the solution set using interval notation. This notation uses parentheses and brackets to indicate whether the endpoints are included or excluded from the solution.

Step-by-Step Solution for (x-3)^2(x+2) < 0

Now, let's apply these steps to solve the inequality (x-3)^2(x+2) < 0. This inequality involves a polynomial expression that is already factored, making our task slightly easier. We will go through each step in detail to ensure clarity and understanding.

1. Identify the Critical Points

The critical points are the values of x that make the expression (x-3)^2(x+2) equal to zero. To find these points, we set each factor equal to zero and solve for x:

• (x-3)^2 = 0 => x-3 = 0 => x = 3
• x+2 = 0 => x = -2

Thus, the critical points are x = 3 and x = -2. These points divide the number line into three intervals: (-∞, -2), (-2, 3), and (3, ∞).

2. Create a Sign Chart

A sign chart is a visual tool that helps us analyze the sign of the polynomial expression in each interval. We create a table with the critical points as dividers and the factors of the polynomial as rows. This allows us to systematically determine the sign of each factor and the overall expression in each interval.

3. Test Each Interval

To determine the sign of the expression in each interval, we choose a test value within each interval and substitute it into the expression (x-3)^2(x+2). The sign of the result will tell us the sign of the expression in that interval.

• Interval (-∞, -2): Choose x = -3
  • (x-3)^2(x+2) = (-3-3)^2(-3+2) = (36)(-1) = -36 < 0
  • The expression is negative in this interval.
• Interval (-2, 3): Choose x = 0
  • (x-3)^2(x+2) = (0-3)^2(0+2) = (9)(2) = 18 > 0
  • The expression is positive in this interval.
• Interval (3, ∞): Choose x = 4
  • (x-3)^2(x+2) = (4-3)^2(4+2) = (1)(6) = 6 > 0
  • The expression is positive in this interval.

4. Identify the Solution Set

We are looking for the intervals where (x-3)^2(x+2) < 0, which means the expression is negative. From the sign chart and our tests, we see that the expression is negative only in the interval (-∞, -2).

Note that x = 3 is a critical point where (x-3)^2 = 0, but since the inequality is strictly less than zero (< 0), we do not include x = 3 in the solution set. The same logic applies to x = -2, where (x+2) = 0.

5. Express the Solution in Interval Notation

The solution set is the interval (-∞, -2). In interval notation, this is written as (-∞, -2). This notation indicates that the solution includes all values of x less than -2, but not including -2 itself.

Conclusion

In summary, the solution to the inequality (x-3)^2(x+2) < 0 is (-∞, -2). This solution represents all real numbers less than -2. Solving polynomial inequalities involves a systematic approach that includes identifying critical points, creating a sign chart, testing intervals, and expressing the solution in interval notation. Understanding these steps and the underlying concepts is crucial for mastering algebraic problem-solving.

By carefully analyzing the factors and their signs in different intervals, we can accurately determine the solution set. This process not only provides the answer but also enhances our understanding of polynomial behavior and inequalities.

The method outlined in this article can be applied to a wide range of polynomial inequalities, making it a valuable tool for students and anyone working with mathematical expressions. Practice and familiarity with these techniques will lead to greater confidence and proficiency in solving inequalities.

Key Takeaways

• Solving polynomial inequalities involves finding the values of the variable that make the inequality true.
• Critical points are the roots of the polynomial and divide the number line into intervals.
• A sign chart helps analyze the sign of the polynomial expression in each interval.
• Interval notation is used to express the solution set concisely.
• The solution to (x-3)^2(x+2) < 0 is (-∞, -2).

Final Answer: The solution to the inequality $(x-3)^2(x+2)<0$ is written in interval notation as $(-\infty, -2)$.