Solving Rational Inequalities A Step-by-Step Guide

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In the realm of mathematics, rational inequalities often present a unique challenge, requiring a blend of algebraic manipulation and careful consideration of domain restrictions. This article provides a comprehensive guide to solving the rational inequality (1/3) - (9/x^2) ≤ (-2/x), offering a step-by-step approach that not only arrives at the solution but also illuminates the underlying concepts and techniques. Whether you're a student grappling with homework or a math enthusiast seeking to deepen your understanding, this exploration will equip you with the tools to tackle similar problems with confidence. Understanding the intricacies of rational inequalities is crucial for various applications in calculus, engineering, and economics, making this a valuable skill for anyone pursuing quantitative fields. Let’s embark on this mathematical journey together and unravel the solution to this intriguing inequality.

1. Understanding Rational Inequalities

Before diving into the specifics of the inequality (1/3) - (9/x^2) ≤ (-2/x), let's first establish a solid understanding of what rational inequalities are and the general strategies for solving them. A rational inequality is an inequality that involves rational expressions, which are essentially fractions where the numerator and/or the denominator are polynomials. The key to solving these inequalities lies in identifying the critical values, which are the points where the expression equals zero or is undefined. These critical values divide the number line into intervals, and by testing a value within each interval, we can determine whether the inequality holds true or false. This method ensures we capture all possible solutions while accounting for any restrictions on the domain of the rational expressions involved. Mastering this technique is essential for handling more complex problems in advanced mathematics and real-world applications.

Key Concepts and Definitions

  • Rational Expression: A fraction where the numerator and denominator are polynomials.
  • Critical Values: Values of the variable that make the rational expression equal to zero or undefined.
  • Interval Testing: A method of determining the solution set of an inequality by testing values within intervals created by critical values.
  • Domain Restrictions: Values of the variable that make the denominator of a rational expression equal to zero, which must be excluded from the solution set.

2. Rewriting the Inequality

The first crucial step in solving the inequality (1/3) - (9/x^2) ≤ (-2/x) is to rewrite it in a form that is easier to analyze. This typically involves moving all terms to one side of the inequality, leaving zero on the other side. This rearrangement allows us to combine the rational expressions into a single fraction, which simplifies the process of identifying critical values. By manipulating the inequality in this way, we set the stage for a more straightforward application of interval testing. This initial transformation is not just a mere algebraic step; it's a strategic move that significantly streamlines the solution process, making the subsequent steps more manageable and intuitive. Let's see how this rearrangement unfolds for our specific inequality.

Step-by-Step Transformation

  1. Add (2/x) to both sides: This moves all terms to the left side of the inequality, leaving zero on the right side. (1/3) - (9/x^2) + (2/x) ≤ 0

  2. Find a common denominator: The common denominator for the fractions is 3x^2. We rewrite each term with this denominator. (x^2 / 3x^2) - (27 / 3x^2) + (6x / 3x^2) ≤ 0

  3. Combine the fractions: Now that all terms have the same denominator, we can combine them into a single fraction. (x^2 + 6x - 27) / (3x^2) ≤ 0

3. Identifying Critical Values

Once we have rewritten the inequality as (x^2 + 6x - 27) / (3x^2) ≤ 0, the next critical step is to identify the critical values. These values are the points where the expression either equals zero or is undefined. The points where the expression equals zero are the roots of the numerator, and the points where the expression is undefined are the roots of the denominator. Finding these critical values is paramount because they serve as the boundaries that divide the number line into intervals. Within each interval, the expression will maintain a consistent sign, either positive or negative, which is crucial for determining the solution set of the inequality. This methodical approach ensures that we account for all potential solutions and correctly interpret the inequality.

Finding the Critical Values

  1. Set the numerator equal to zero and solve: This will give us the values of x where the expression equals zero. x^2 + 6x - 27 = 0 (x + 9)(x - 3) = 0 x = -9, x = 3

  2. Set the denominator equal to zero and solve: This will give us the values of x where the expression is undefined. 3x^2 = 0 x = 0

Therefore, the critical values are x = -9, x = 0, and x = 3. These points will be the foundation for our next step: interval testing.

4. Interval Testing

With the critical values identified as x = -9, x = 0, and x = 3, we now proceed to the crucial step of interval testing. These critical values partition the number line into four distinct intervals: (-∞, -9), (-9, 0), (0, 3), and (3, ∞). Our goal is to determine the sign of the expression (x^2 + 6x - 27) / (3x^2) within each interval. To do this, we select a test value from each interval and substitute it into the expression. The sign of the result will indicate whether the expression is positive or negative within that entire interval. This methodical approach allows us to map out the behavior of the expression across the number line and pinpoint the intervals that satisfy our inequality. By carefully analyzing these intervals, we can construct the solution set for the rational inequality.

Performing the Interval Tests

  1. Interval (-∞, -9): Choose a test value, say x = -10. ((-10)^2 + 6(-10) - 27) / (3(-10)^2) = (100 - 60 - 27) / 300 = 13 / 300 > 0 The expression is positive in this interval.

  2. Interval (-9, 0): Choose a test value, say x = -1. ((-1)^2 + 6(-1) - 27) / (3(-1)^2) = (1 - 6 - 27) / 3 = -32 / 3 < 0 The expression is negative in this interval.

  3. Interval (0, 3): Choose a test value, say x = 1. ((1)^2 + 6(1) - 27) / (3(1)^2) = (1 + 6 - 27) / 3 = -20 / 3 < 0 The expression is negative in this interval.

  4. Interval (3, ∞): Choose a test value, say x = 4. ((4)^2 + 6(4) - 27) / (3(4)^2) = (16 + 24 - 27) / 48 = 13 / 48 > 0 The expression is positive in this interval.

5. Determining the Solution Set

Having completed the interval testing, we are now poised to determine the solution set for the inequality (x^2 + 6x - 27) / (3x^2) ≤ 0. We recall that we are looking for the intervals where the expression is either negative or equal to zero. From our interval testing, we found that the expression is negative in the intervals (-9, 0) and (0, 3). Additionally, the expression equals zero at x = -9 and x = 3, which were critical values derived from the numerator. However, we must also remember that x = 0 is a critical value where the expression is undefined, due to the denominator being zero. Therefore, we exclude x = 0 from our solution set. By carefully considering these factors, we can construct the final solution set in interval notation.

Constructing the Solution Set

  • The expression is negative in the intervals (-9, 0) and (0, 3).
  • The expression equals zero at x = -9 and x = 3.
  • The expression is undefined at x = 0.

Therefore, the solution set includes the intervals where the expression is negative, as well as the points where it equals zero, excluding the point where it is undefined. In interval notation, this is represented as:

[-9, 0) ∪ (0, 3]

This final interval notation encapsulates all the values of x that satisfy the original inequality, taking into account both the algebraic manipulations and the domain restrictions inherent in rational expressions.

Conclusion

In conclusion, we have successfully navigated the complexities of the rational inequality (1/3) - (9/x^2) ≤ (-2/x), arriving at the solution set [-9, 0) ∪ (0, 3]. This journey has underscored the importance of several key steps: rewriting the inequality, identifying critical values, performing interval testing, and carefully considering domain restrictions. By mastering these techniques, one can confidently tackle a wide array of rational inequalities. The systematic approach we've demonstrated not only provides a solution but also fosters a deeper understanding of the underlying mathematical principles. Rational inequalities, while seemingly intricate, are a fundamental concept in mathematics with applications in various fields, making the ability to solve them a valuable asset. We hope this comprehensive guide has illuminated the path to solving such inequalities and empowered you to approach similar problems with greater ease and confidence.