Solving Rational Inequalities A Step-by-Step Guide To (x^2+9)/(x^2-36) > 0

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#Solving Rational Inequalities is a crucial skill in algebra, and this article provides a step-by-step guide to solving the inequality x2+9x2−36>0{\frac{x^2+9}{x^2-36} > 0}. This comprehensive guide will walk you through the process, ensuring you understand each step thoroughly. Rational inequalities, which involve comparing a rational function to zero, are essential in various fields, including calculus and mathematical analysis. Understanding how to solve them is fundamental for advanced mathematical studies and practical applications. This article not only provides the solution but also aims to enhance your problem-solving skills and deepen your understanding of algebraic concepts. Let's dive into the intricacies of solving this inequality and uncover the underlying principles that make it work. By the end of this guide, you will be well-equipped to tackle similar problems with confidence and precision.

Understanding Rational Inequalities

A rational inequality is an inequality that contains a rational expression, which is a fraction where the numerator and/or the denominator are polynomials. The key to solving these inequalities lies in identifying the critical values and testing intervals. Critical values are the points where the expression equals zero or is undefined. These points divide the number line into intervals, and we test each interval to determine where the inequality holds true. For the given inequality, x2+9x2−36>0{\frac{x^2+9}{x^2-36} > 0}, we need to find the values of x that make the expression greater than zero. This involves analyzing both the numerator and the denominator. The numerator, x2+9{x^2 + 9}, is always positive because the square of any real number is non-negative, and adding 9 makes it strictly positive. The denominator, x2−36{x^2 - 36}, can be positive, negative, or zero, depending on the value of x. Understanding the behavior of each part of the rational expression is crucial for solving the inequality correctly. This section sets the stage for a detailed, step-by-step solution that follows, ensuring clarity and a solid grasp of the underlying concepts.

Step-by-Step Solution

1. Identify Critical Values

The first step in solving the rational inequality x2+9x2−36>0{\frac{x^2+9}{x^2-36} > 0} is to identify the critical values. These are the values of x that make either the numerator or the denominator equal to zero. The numerator is x2+9{x^2 + 9}, and setting it to zero gives us x2+9=0{x^2 + 9 = 0}. This equation has no real solutions because x2{x^2} is always non-negative, and adding 9 makes the expression always positive. Therefore, the numerator does not contribute any critical values. The denominator is x2−36{x^2 - 36}. Setting this to zero gives us x2−36=0{x^2 - 36 = 0}, which can be factored as (x−6)(x+6)=0{(x - 6)(x + 6) = 0}. This yields two critical values: x = 6 and x = -6. These values are crucial because they are the points where the expression can change its sign. The critical values divide the number line into intervals that we will test in the next steps. Identifying critical values is a foundational step in solving any rational inequality, as it helps to define the regions where the inequality may hold true or false. In this case, the critical values of 6 and -6 will be pivotal in determining the solution set.

2. Determine Intervals

The critical values x = -6 and x = 6 divide the number line into three intervals:

  1. (−∞,−6){(-\infty, -6)}
  2. (−6,6){(-6, 6)}
  3. (6,∞){(6, \infty)}

These intervals represent the regions on the number line where the expression x2+9x2−36{\frac{x^2+9}{x^2-36}} will have a consistent sign. By testing a value within each interval, we can determine whether the inequality x2+9x2−36>0{\frac{x^2+9}{x^2-36} > 0} holds true in that interval. The critical values themselves are not included in the solution because the inequality is strict (greater than, not greater than or equal to). If the inequality were non-strict (i.e., ≥0{\geq 0}), we would need to consider whether the values that make the numerator zero should be included in the solution. However, since the numerator in this case is always positive, we only need to focus on the denominator. This step of determining intervals is crucial because it sets the stage for testing values and finding the solution set. Understanding how critical values partition the number line is essential for solving rational inequalities effectively.

3. Test Intervals

Now, we test a value from each interval in the inequality x2+9x2−36>0{\frac{x^2+9}{x^2-36} > 0} to determine where the inequality holds true.

  1. Interval (−∞,−6){(-\infty, -6)}: Choose a test value, such as x = -7. Plug this value into the expression:

    (−7)2+9(−7)2−36=49+949−36=5813>0{ \frac{(-7)^2 + 9}{(-7)^2 - 36} = \frac{49 + 9}{49 - 36} = \frac{58}{13} > 0 }

    Since the result is positive, the inequality holds true in this interval.

  2. Interval (−6,6){(-6, 6)}: Choose a test value, such as x = 0. Plug this value into the expression:

    (0)2+9(0)2−36=9−36=−14<0{ \frac{(0)^2 + 9}{(0)^2 - 36} = \frac{9}{-36} = -\frac{1}{4} < 0 }

    Since the result is negative, the inequality does not hold true in this interval.

  3. Interval (6,∞){(6, \infty)}: Choose a test value, such as x = 7. Plug this value into the expression:

    (7)2+9(7)2−36=49+949−36=5813>0{ \frac{(7)^2 + 9}{(7)^2 - 36} = \frac{49 + 9}{49 - 36} = \frac{58}{13} > 0 }

    Since the result is positive, the inequality holds true in this interval.

Testing intervals is a critical step because it confirms the regions where the inequality is satisfied. By plugging in test values, we can determine the sign of the expression in each interval and thus identify the solution set. The positive results in the intervals (−∞,−6){(-\infty, -6)} and (6,∞){(6, \infty)} indicate that these are the regions where the inequality x2+9x2−36>0{\frac{x^2+9}{x^2-36} > 0} is true. This thorough testing process ensures an accurate determination of the solution.

4. Write the Solution

Based on the test results, the inequality x2+9x2−36>0{\frac{x^2+9}{x^2-36} > 0} holds true for the intervals (−∞,−6){(-\infty, -6)} and (6,∞){(6, \infty)}. Therefore, the solution to the inequality is:

x<−6 or x>6{ x < -6 \text{ or } x > 6 }

This can also be written in interval notation as:

(−∞,−6)∪(6,∞){ (-\infty, -6) \cup (6, \infty) }

This solution indicates that any value of x less than -6 or greater than 6 will satisfy the given inequality. The values x = -6 and x = 6 are excluded because they make the denominator zero, which makes the expression undefined. Writing the solution clearly and accurately is the final step in solving the rational inequality. The solution set provides a precise answer to the problem and demonstrates a complete understanding of the inequality.

Final Answer

The solution to the rational inequality x2+9x2−36>0{\frac{x^2+9}{x^2-36} > 0} is:

D. x<−6{x < -6} or x>6{x > 6}

This final answer is derived from the step-by-step solution provided above, which includes identifying critical values, determining intervals, testing those intervals, and writing the solution. The correct answer, x < -6 or x > 6, aligns with the intervals where the inequality holds true. The other options are incorrect because they include values that either make the denominator zero or do not satisfy the inequality. Option A, x ≤ -6 or x ≥ 6, includes the critical values, which are not part of the solution due to the strict inequality. Options B and C suggest that the solution lies between -6 and 6, which contradicts our test results. Thus, option D is the only correct solution. This detailed explanation reinforces the importance of each step in the solving process and ensures a clear understanding of how the final answer is obtained.

Additional Practice Problems

To master the skill of solving rational inequalities, it’s essential to practice with a variety of problems. Here are a few additional practice problems that you can try:

  1. Solve x−2x+3>0{\frac{x-2}{x+3} > 0}
  2. Solve x2−4x−1<0{\frac{x^2-4}{x-1} < 0}
  3. Solve 2x+1x−4≥0{\frac{2x+1}{x-4} \geq 0}
  4. Solve x2−9x2−16≤0{\frac{x^2-9}{x^2-16} \leq 0}

Working through these problems will help you solidify your understanding of the steps involved in solving rational inequalities. Each problem presents a slightly different scenario, requiring you to apply the same principles in a variety of contexts. As you solve these problems, pay close attention to identifying critical values, determining intervals, and testing values within those intervals. Check your solutions by graphing the rational function and observing where it is above or below the x-axis. Consistent practice is key to building confidence and proficiency in solving rational inequalities. These additional problems provide an excellent opportunity to hone your skills and deepen your understanding of this important algebraic concept.

Conclusion

In conclusion, solving the rational inequality x2+9x2−36>0{\frac{x^2+9}{x^2-36} > 0} involves a systematic approach that includes identifying critical values, determining intervals, testing those intervals, and writing the solution. The solution to this particular inequality is x<−6{x < -6} or x>6{x > 6}, which corresponds to option D. Understanding the underlying concepts and following a step-by-step method is crucial for solving any rational inequality. Rational inequalities are a fundamental topic in algebra and calculus, and mastering them is essential for further studies in mathematics. By practicing more problems and applying the techniques discussed in this guide, you can enhance your problem-solving skills and achieve a deeper understanding of rational inequalities. Remember, the key to success lies in consistent practice and a thorough understanding of the principles involved. With the knowledge and practice, you will be well-prepared to tackle a wide range of rational inequalities and excel in your mathematical endeavors.