Solving (x-7)(x+2) > 0 A Comprehensive Guide To Polynomial Inequalities

Polynomial inequalities, such as (x-7)(x+2) > 0, are a fundamental topic in algebra and calculus. Mastering the techniques to solve these inequalities is crucial for understanding various mathematical concepts and their applications in real-world problems. This article provides a comprehensive guide on how to solve the polynomial inequality (x-7)(x+2) > 0, graph the solution set on a real number line, and express the solution set in interval notation. We will explore the underlying principles, step-by-step methods, and practical examples to ensure a clear understanding of the topic.

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 a mathematical statement that compares a polynomial expression to another value, often zero, using inequality symbols such as >, <, ≥, or ≤. These inequalities are used to describe intervals on the number line where the polynomial satisfies the given condition. In our case, we are looking for the values of x that make the polynomial (x-7)(x+2) greater than zero.

Polynomial inequalities are not just abstract mathematical constructs; they have significant applications in various fields. For instance, in optimization problems, we often need to find the range of values for a variable that satisfies certain constraints, which can be expressed as polynomial inequalities. In calculus, understanding inequalities is crucial for determining intervals where a function is increasing or decreasing, and for finding maximum and minimum values. Moreover, in economics and engineering, polynomial inequalities can model real-world scenarios such as cost analysis, resource allocation, and system stability.

The inequality (x-7)(x+2) > 0 is a quadratic inequality because the polynomial is of degree 2. Solving it involves finding the intervals where the product of the factors (x-7) and (x+2) is positive. This requires understanding how the sign of each factor changes as x varies and how these signs combine to determine the sign of the product. We will explore these concepts in detail in the following sections.

Step-by-Step Solution to (x-7)(x+2) > 0

To solve the polynomial inequality (x-7)(x+2) > 0, we follow a systematic approach that involves finding the critical points, creating a sign chart, and determining the intervals where the inequality holds true. Here’s a detailed step-by-step guide:

Step 1: Find the 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 where the sign of the polynomial remains constant. To find the critical points, we set each factor equal to zero and solve for x:

• x - 7 = 0 implies x = 7
• x + 2 = 0 implies x = -2

Thus, the critical points are x = 7 and x = -2. These points will serve as the boundaries of our intervals on the number line.

Step 2: Create a Sign Chart

A sign chart is a visual tool that helps us determine the sign of each factor and the overall polynomial in the intervals defined by the critical points. We create a number line and mark the critical points -2 and 7. This divides the number line into three intervals: (-∞, -2), (-2, 7), and (7, ∞). We then choose a test value within each interval and determine the sign of each factor (x-7) and (x+2).

In the interval (-∞, -2), we chose the test value x = -3. Plugging this value into the factors, we get:

• x - 7 = -3 - 7 = -10 (negative)
• x + 2 = -3 + 2 = -1 (negative)

The product (x-7)(x+2) is positive because the product of two negative numbers is positive.

In the interval (-2, 7), we chose the test value x = 0:

• x - 7 = 0 - 7 = -7 (negative)
• x + 2 = 0 + 2 = 2 (positive)

The product (x-7)(x+2) is negative because the product of a negative and a positive number is negative.

In the interval (7, ∞), we chose the test value x = 8:

• x - 7 = 8 - 7 = 1 (positive)
• x + 2 = 8 + 2 = 10 (positive)

The product (x-7)(x+2) is positive because the product of two positive numbers is positive.

Step 3: Determine the Solution Set

We are looking for the intervals where (x-7)(x+2) > 0, meaning the product is positive. From the sign chart, we see that the polynomial is positive in the intervals (-∞, -2) and (7, ∞). Therefore, the solution set consists of these intervals.

Graphing the Solution Set on a Real Number Line

Visualizing the solution set on a real number line helps to reinforce the concept and provides a clear understanding of the solution. To graph the solution set, we draw a number line and mark the critical points -2 and 7. Since the inequality is strict (> 0), we use open circles at these points to indicate that they are not included in the solution. We then shade the intervals where the polynomial is positive, which are (-∞, -2) and (7, ∞).

<----------------|----------------|---------------->
      (-∞)       -2               7        (∞)
     o-------      o-------     

The shaded regions represent the solution set of the inequality.

Expressing the Solution Set in Interval Notation

Interval notation is a concise way to represent the solution set of an inequality. It uses intervals to describe the range of values that satisfy the inequality. For the inequality (x-7)(x+2) > 0, the solution set consists of two intervals: (-∞, -2) and (7, ∞). We use parentheses to indicate that the endpoints are not included in the solution because the inequality is strict. To combine these intervals into a single expression, we use the union symbol ∪.

Therefore, the solution set in interval notation is:

(-∞, -2) ∪ (7, ∞)

This notation clearly conveys that the solution includes all real numbers less than -2 and all real numbers greater than 7.

Common Mistakes to Avoid

Solving polynomial inequalities can be tricky, and it's easy to make mistakes if you're not careful. Here are some common mistakes to avoid:

1. Forgetting to Find Critical Points: Critical points are the foundation of solving polynomial inequalities. Without them, you can't determine the intervals where the inequality holds true. Always set each factor equal to zero and solve for x.
2. Using Closed Intervals for Strict Inequalities: When the inequality is strict (> or <), the critical points are not included in the solution. Use open intervals (parentheses) to indicate this. Conversely, for non-strict inequalities (≥ or ≤), use closed intervals (brackets) to include the critical points.
3. Incorrectly Interpreting the Sign Chart: The sign chart is a crucial tool, but it's only effective if interpreted correctly. Make sure you understand how the signs of the factors combine to determine the sign of the polynomial. A single mistake in the sign chart can lead to an incorrect solution.
4. Dividing by a Variable Expression: When dealing with inequalities, you cannot divide both sides by a variable expression without knowing its sign. This is because dividing by a negative number reverses the inequality sign. Instead, bring all terms to one side and factor.
5. Not Checking the Solution: After finding the solution set, it's always a good idea to check it by plugging in test values from the intervals. This can help you catch errors and ensure that your solution is correct.

Advanced Techniques and Applications

While the basic method of finding critical points and using a sign chart works for most polynomial inequalities, there are more advanced techniques and applications to be aware of. For example, when dealing with higher-degree polynomials, the same principles apply, but the sign chart may have more intervals to consider. In such cases, it's essential to be organized and systematic in constructing the sign chart.

Another important concept is the Intermediate Value Theorem, which states that if a continuous function (like a polynomial) changes sign between two points, it must have a root (a critical point) in that interval. This theorem provides a theoretical basis for the sign chart method and helps us understand why the sign of the polynomial remains constant within each interval.

Polynomial inequalities also have practical applications in optimization problems, where we seek to maximize or minimize a function subject to certain constraints. These constraints can often be expressed as inequalities, and solving them is a crucial step in finding the optimal solution. For instance, in business, companies may use polynomial inequalities to model cost functions and determine the production levels that minimize costs while meeting demand.

Conclusion

Solving the polynomial inequality (x-7)(x+2) > 0 involves a systematic approach that includes finding critical points, creating a sign chart, and expressing the solution set in interval notation. This process is not only a fundamental skill in algebra but also a valuable tool for solving real-world problems in various fields. By understanding the underlying principles and avoiding common mistakes, you can confidently tackle polynomial inequalities of any degree. Remember to practice regularly and apply these techniques to different types of problems to solidify your understanding. Mastering polynomial inequalities is a significant step towards more advanced mathematical concepts and applications. Through this comprehensive guide, you should now have a solid grasp on how to solve polynomial inequalities and apply them in various contexts.