Solving (x-9)/(x-20) > 3 A Step-by-Step Guide
In this article, we will delve into the process of solving the inequality (x-9)/(x-20) > 3. Inequalities play a crucial role in various fields of mathematics and have practical applications in real-world scenarios. Mastering the techniques to solve inequalities is essential for students and professionals alike. This article aims to provide a comprehensive guide, breaking down each step with detailed explanations to ensure a clear understanding of the solution.
Understanding Inequalities
Before we dive into the solution, let's briefly discuss what inequalities are and why they are important. An inequality is a mathematical statement that compares two expressions using inequality symbols such as >, <, ≥, or ≤. Unlike equations, which state that two expressions are equal, inequalities indicate a range of values that satisfy the given condition. Inequalities are used extensively in calculus, optimization problems, and economic modeling, among other areas. They help in determining boundaries, constraints, and possible solutions in various scenarios.
Rewriting the Inequality
To solve the inequality (x-9)/(x-20) > 3, the first step is to manipulate it into a form that is easier to work with. The goal is to get all terms on one side, leaving zero on the other side. This is done by subtracting 3 from both sides of the inequality:
(x-9)/(x-20) - 3 > 0
This step is crucial because it allows us to combine the terms into a single fraction, making it simpler to analyze the sign changes. The importance of this step cannot be overstated; it sets the foundation for the rest of the solution. Subtracting 3 from both sides maintains the integrity of the inequality, ensuring that the solution set remains unchanged. The resulting inequality is now ready for the next step, which involves finding a common denominator.
Combining Terms
Next, we need to combine the terms on the left side of the inequality. To do this, we find a common denominator, which in this case is (x-20). We rewrite 3 as 3(x-20)/(x-20) and subtract it from the first term:
(x-9)/(x-20) - 3(x-20)/(x-20) > 0
Now we can combine the numerators:
(x-9 - 3(x-20))/(x-20) > 0
Expanding the terms in the numerator, we get:
(x-9 - 3x + 60)/(x-20) > 0
Simplifying the numerator gives us:
(-2x + 51)/(x-20) > 0
This simplified form is much easier to analyze. It allows us to identify the critical values, which are the points where the expression changes sign. The process of finding a common denominator and simplifying the expression is a fundamental technique in solving rational inequalities. It is essential to perform these steps accurately to arrive at the correct solution.
Finding Critical Values
Critical values are the points where the expression (-2x + 51)/(x-20) equals zero or is undefined. These points divide the number line into intervals, which we will test to determine the solution set. The critical values are found by setting the numerator and the denominator equal to zero.
First, we set the numerator equal to zero:
-2x + 51 = 0
Solving for x, we get:
x = 51/2 = 25.5
Next, we set the denominator equal to zero:
x - 20 = 0
Solving for x, we get:
x = 20
So, the critical values are x = 20 and x = 25.5. These values are crucial because they mark the points where the expression can change its sign. Understanding critical values is paramount in solving inequalities, as they define the intervals where the inequality holds true or false. The next step involves using these critical values to create a sign chart.
Creating a Sign Chart
A sign chart helps us determine the intervals where the inequality (-2x + 51)/(x-20) > 0 is satisfied. We place the critical values, 20 and 25.5, on the number line, which divides it into three intervals: (-∞, 20), (20, 25.5), and (25.5, ∞). We then pick a test value from each interval and plug it into the simplified inequality to determine the sign of the expression in that interval.
Interval (-∞, 20)
Let's choose a test value, say x = 0. Plugging this into the expression, we get:
(-2(0) + 51)/(0-20) = 51/-20 < 0
So, the expression is negative in this interval.
Interval (20, 25.5)
Let's choose a test value, say x = 23. Plugging this into the expression, we get:
(-2(23) + 51)/(23-20) = (-46 + 51)/3 = 5/3 > 0
So, the expression is positive in this interval.
Interval (25.5, ∞)
Let's choose a test value, say x = 26. Plugging this into the expression, we get:
(-2(26) + 51)/(26-20) = (-52 + 51)/6 = -1/6 < 0
So, the expression is negative in this interval.
The sign chart now shows that the expression is positive in the interval (20, 25.5). This interval is where the inequality (-2x + 51)/(x-20) > 0 holds true. The sign chart is an invaluable tool for visualizing the solution set of an inequality, as it clearly indicates the intervals where the expression satisfies the given condition.
Determining the Solution Set
From the sign chart, we see that the inequality (-2x + 51)/(x-20) > 0 is satisfied in the interval (20, 25.5). We need to exclude the critical value x = 20 because the denominator cannot be zero. The critical value x = 25.5 is also excluded because the inequality is strict (i.e., > and not ≥). Therefore, the solution set is the open interval (20, 25.5).
Expressing the Solution in Interval Notation
In interval notation, the solution set is written as (20, 25.5). This notation indicates all real numbers between 20 and 25.5, excluding the endpoints. The use of parentheses signifies that the endpoints are not included in the solution set, which is consistent with the strict inequality. Interval notation is a concise and standard way to represent solution sets, making it easy to communicate the range of values that satisfy the inequality.
Conclusion
The solution set for the inequality (x-9)/(x-20) > 3 is (20, 25.5). This was obtained by rewriting the inequality, combining terms, finding critical values, creating a sign chart, and determining the intervals where the inequality holds true. Each step is crucial in arriving at the correct solution, and understanding these steps is essential for solving more complex inequalities. Mastering these techniques not only enhances mathematical skills but also provides a solid foundation for tackling real-world problems that involve inequalities. Remember to always double-check your work and ensure that the solution set makes sense in the context of the original problem.
