Solving (x+17)/(x-16) ≥ 1 Algebraically A Step-by-Step Guide
Solving inequalities involving rational expressions requires a slightly different approach than solving standard inequalities. The presence of variables in the denominator introduces the possibility of undefined points and sign changes, making it crucial to consider these factors. This comprehensive guide will walk you through the process of solving rational inequalities algebraically, using the example you provided: (x+17)/(x-16) ≥ 1. We'll break down each step, ensuring you understand the underlying principles and can apply them to various rational inequalities.
Understanding Rational Inequalities
Before diving into the solution, let's clarify what a rational inequality is. A rational inequality is an inequality that involves a rational expression, which is a fraction where the numerator and/or denominator are polynomials. These inequalities can take various forms, including greater than, less than, greater than or equal to, or less than or equal to. The key difference between solving rational inequalities and regular inequalities lies in the consideration of the denominator. Since division by zero is undefined, we need to identify any values of the variable that make the denominator zero and exclude them from our solution set. Furthermore, the sign of the rational expression can change at the points where the numerator or denominator equals zero, so these points must be considered carefully when determining the intervals that satisfy the inequality. In the given example, (x+17)/(x-16) ≥ 1, we have a rational expression on the left-hand side and a constant on the right-hand side. Our goal is to find all values of x that make the inequality true. This involves algebraic manipulation, critical value identification, interval testing, and expressing the solution in interval notation. By systematically working through these steps, we can confidently solve rational inequalities and gain a deeper understanding of their behavior.
Step 1: Rearrange the Inequality
The initial step in solving rational inequalities is to rearrange the inequality so that one side is zero. This makes it easier to analyze the sign of the expression. In our case, we have (x+17)/(x-16) ≥ 1. To get zero on one side, we subtract 1 from both sides:
(x+17)/(x-16) - 1 ≥ 0
Now, we need to combine the terms on the left-hand side into a single fraction. To do this, we find a common denominator, which in this case is (x-16). We rewrite 1 as (x-16)/(x-16) and subtract:
(x+17)/(x-16) - (x-16)/(x-16) ≥ 0
Combining the fractions, we get:
[(x+17) - (x-16)] / (x-16) ≥ 0
Simplifying the numerator, we have:
(x + 17 - x + 16) / (x-16) ≥ 0
This simplifies further to:
33 / (x-16) ≥ 0
This rearranged form is crucial because it allows us to identify the critical values and analyze the sign of the expression more effectively. By having zero on one side, we can focus on the sign of the rational expression on the other side. This step is a fundamental part of solving rational inequalities and sets the stage for the subsequent steps.
Step 2: Identify Critical Values
Critical values are the points where the expression on the left-hand side of the inequality can change its sign. These points occur where the numerator is equal to zero or where the denominator is equal to zero. In our simplified inequality, 33 / (x-16) ≥ 0, the numerator is 33, which is a constant and never equal to zero. Therefore, there are no critical values from the numerator in this specific case. However, the denominator, (x-16), can be equal to zero. Setting the denominator equal to zero, we have:
x - 16 = 0
Solving for x, we find:
x = 16
So, x = 16 is a critical value. This is a crucial point because it's the value of x that makes the denominator zero, and thus the rational expression undefined. This critical value divides the number line into two intervals, and the sign of the expression 33 / (x-16) may be different in each interval. Identifying critical values is a key step in solving rational inequalities because these values define the boundaries of the intervals that we need to test. Understanding where the expression can change sign is essential for determining the solution set of the inequality. In summary, the critical value in this case is x = 16, which comes from setting the denominator equal to zero.
Step 3: Create a Sign Chart
A sign chart is a visual tool that helps us determine the sign of the rational expression in the intervals created by the critical values. It's a systematic way to organize our analysis and ensure we consider all possible scenarios. We start by drawing a number line and marking the critical value(s) we found in the previous step. In our case, we have one critical value, x = 16, so we mark 16 on the number line. This critical value divides the number line into two intervals: (-∞, 16) and (16, ∞). Next, we choose a test value within each interval. A test value is any number within the interval that we can plug into the rational expression to determine its sign in that interval. For the interval (-∞, 16), we can choose a test value like x = 0. For the interval (16, ∞), we can choose a test value like x = 17. Now, we evaluate the expression 33 / (x-16) at each test value:
- For x = 0: 33 / (0-16) = 33 / (-16) = -33/16, which is negative.
- For x = 17: 33 / (17-16) = 33 / (1) = 33, which is positive.
We record these signs on the sign chart. We represent the interval (-∞, 16) with a negative sign (-) and the interval (16, ∞) with a positive sign (+). Additionally, we indicate on the sign chart that x = 16 is a critical value where the expression is undefined (often marked with a vertical line or an open circle). The sign chart provides a clear visual representation of the intervals where the expression is positive, negative, or undefined. This is a crucial step in solving rational inequalities, as it allows us to easily identify the intervals that satisfy the inequality condition.
Step 4: Determine the Solution Set
Now that we have our sign chart, we can determine the solution set for the inequality 33 / (x-16) ≥ 0. The inequality asks us to find the values of x for which the expression is greater than or equal to zero. Looking at our sign chart, we see that the expression is positive in the interval (16, ∞). This means that all values of x in this interval will satisfy the inequality. However, we also need to consider the "equal to" part of the inequality (≥). This means we should also include any values of x that make the expression equal to zero. In our case, the numerator is 33, which is never zero. The denominator, (x-16), is zero when x = 16, but since the expression is undefined at this point (division by zero), we cannot include x = 16 in our solution set. Therefore, the solution set consists only of the interval where the expression is positive. We write the solution in interval notation as (16, ∞). This notation indicates that the solution includes all real numbers greater than 16 but does not include 16 itself. In summary, by examining the sign chart and considering the original inequality, we have identified the interval (16, ∞) as the solution set. This final step is the culmination of our algebraic and analytical work, providing us with the answer to the problem.
Step 5: Express the Solution
The final step in solving rational inequalities is to express the solution clearly and concisely. We've already determined that the solution set for the inequality 33 / (x-16) ≥ 0 is the interval (16, ∞). This is the most common way to express the solution, using interval notation. In interval notation, parentheses indicate that the endpoint is not included in the solution, while brackets indicate that the endpoint is included. Since our solution includes all numbers greater than 16 but not 16 itself, we use a parenthesis on the left side of the interval. The infinity symbol (∞) always uses a parenthesis because infinity is not a specific number and cannot be included in the interval. Therefore, the solution (16, ∞) is a clear and standard way to represent the set of all real numbers greater than 16. We can also express the solution graphically by drawing a number line and shading the interval (16, ∞). We would use an open circle at 16 to indicate that it is not included in the solution. While interval notation is the most common way to express the solution, it's important to understand the other ways to represent it, as different contexts may call for different notations. However, for most algebraic solutions, interval notation is preferred for its clarity and conciseness. This final step ensures that our hard work in solving the inequality is communicated effectively and unambiguously.
Conclusion
Solving rational inequalities requires a systematic approach that takes into account the potential for undefined points and sign changes. By following the steps outlined in this guide – rearranging the inequality, identifying critical values, creating a sign chart, determining the solution set, and expressing the solution – you can confidently solve a wide range of rational inequalities. Remember to always pay close attention to the critical values, as these are the points where the expression can change sign. Using a sign chart is an invaluable tool for organizing your analysis and ensuring you consider all possible intervals. While this guide focused on a specific example, the principles and techniques discussed apply broadly to other rational inequalities. Practice is key to mastering these skills, so work through various examples to solidify your understanding. By carefully applying these steps, you'll be well-equipped to tackle any rational inequality that comes your way. Rational inequalities are a fundamental topic in algebra and precalculus, so mastering them will be a valuable asset in your mathematical journey.
