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

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Navigating the realm of inequalities can often feel like traversing a complex mathematical landscape. Among these, rational inequalities, which involve ratios of polynomials, present a unique set of challenges. In this comprehensive guide, we will embark on a journey to solve the rational inequality x2+9x2−36>0{ \frac{x^2+9}{x^2-36} > 0 } with clarity and precision. Our exploration will not only provide a solution to this specific problem but also equip you with the fundamental principles and techniques necessary to conquer a wide array of rational inequalities. Let's delve into the step-by-step process, unraveling the intricacies and empowering you with the tools to master these mathematical puzzles.

Understanding Rational Inequalities

Before we dive into the solution, let's first establish a firm understanding of rational inequalities. A rational inequality is an inequality that involves a rational expression, which is a fraction where the numerator and/or the denominator are polynomials. Solving these inequalities requires us to determine the values of the variable that make the expression either greater than, less than, greater than or equal to, or less than or equal to a certain value (often zero). The key to solving rational inequalities lies in identifying critical points, which are the values that make either the numerator or the denominator equal to zero. These points divide the number line into intervals, and we can then test values within each interval to determine where the inequality holds true.

Rational inequalities distinguish themselves from standard polynomial inequalities through the presence of a denominator that includes a variable. This crucial element introduces the possibility of undefined points, specifically when the denominator equals zero. These points, known as critical values, play a pivotal role in the solution process. They demarcate intervals on the number line, within which the inequality's solution may vary. Consequently, a thorough grasp of both numerator and denominator behavior is essential for accurately solving rational inequalities.

Key Concepts and Definitions

  • Rational Expression: A fraction where the numerator and denominator are polynomials.
  • Critical Points: Values that make the numerator or denominator of a rational expression equal to zero.
  • Interval Testing: A method of choosing test values within intervals defined by critical points to determine the sign of the rational expression.

Step-by-Step Solution for x2+9x2−36>0{ \frac{x^2+9}{x^2-36} > 0 }

Now, let's tackle the specific inequality at hand: x2+9x2−36>0{ \frac{x^2+9}{x^2-36} > 0 }. We'll break down the solution into manageable steps, ensuring a clear and methodical approach.

Step 1: Identify Critical Points

Critical points are the cornerstone of solving rational inequalities. They are the values of x that make either the numerator or the denominator of the rational expression equal to zero. These points act as dividers on the number line, separating intervals where the expression's sign remains constant. To identify them, we must set both the numerator and the denominator to zero and solve for x individually. This process allows us to pinpoint the exact locations where the expression might change its sign, which is crucial for determining the solution set of the inequality.

Numerator

We begin by setting the numerator, x2+9{ x^2 + 9 }, equal to zero and attempting to solve for x:

x2+9=0{ x^2 + 9 = 0 }

Subtracting 9 from both sides, we get:

x2=−9{ x^2 = -9 }

At this juncture, we encounter a significant observation: there exists no real number x that, when squared, yields a negative result. The square of any real number is invariably non-negative. Consequently, the equation x2=−9{ x^2 = -9 } has no real solutions. This implies that the numerator, x2+9{ x^2 + 9 }, never equals zero for any real value of x. Therefore, it does not contribute any critical points to our analysis.

Denominator

Next, we turn our attention to the denominator, x2−36{ x^2 - 36 }, setting it equal to zero to find its critical points:

x2−36=0{ x^2 - 36 = 0 }

This equation is readily recognizable as a difference of squares, a common algebraic form that can be factored to simplify the solving process. Factoring the left-hand side, we obtain:

(x−6)(x+6)=0{ (x - 6)(x + 6) = 0 }

Applying the zero-product property, which states that if the product of two factors is zero, then at least one of the factors must be zero, we set each factor equal to zero individually:

x−6=0orx+6=0{ x - 6 = 0 \quad \text{or} \quad x + 6 = 0 }

Solving these two linear equations yields:

x=6orx=−6{ x = 6 \quad \text{or} \quad x = -6 }

Thus, the denominator has two critical points: x = 6 and x = -6. These points are pivotal because they are the values at which the rational expression is undefined (division by zero is undefined in mathematics). They act as boundaries, dividing the number line into intervals where the expression's sign may vary.

In summary, the critical points for the given rational inequality are x = -6 and x = 6, stemming from the analysis of the denominator. The numerator, in this case, does not contribute any critical points as it never equals zero for real values of x. These critical points will be instrumental in our next step, where we divide the number line into intervals and test the sign of the rational expression within each.

Step 2: Divide the Number Line into Intervals

The critical points we identified in the previous step, x = -6 and x = 6, serve as dividers on the number line, partitioning it into distinct intervals. These intervals are crucial because the sign of the rational expression x2+9x2−36{ \frac{x^2+9}{x^2-36} } remains consistent within each interval. This consistency allows us to select a test value from each interval and evaluate the expression at that point to determine the sign for the entire interval. The intervals created by our critical points are:

  1. (−∞,−6){ (-\infty, -6) }: This interval encompasses all real numbers less than -6.
  2. (−6,6){ (-6, 6) }: This interval includes all real numbers between -6 and 6.
  3. (6,∞){ (6, \infty) }: This interval covers all real numbers greater than 6.

Each of these intervals represents a range of x-values that we will investigate to see how they affect the overall sign of the rational expression. The principle behind this approach is that the expression can only change signs at the critical points, where either the numerator or the denominator equals zero. Therefore, by testing a single point within each interval, we can confidently deduce the sign of the expression throughout that interval.

In the subsequent step, we will choose a test value from each interval and substitute it into the rational expression. This process will reveal whether the expression is positive or negative within that interval, providing us with the necessary information to determine the solution set of the inequality x2+9x2−36>0{ \frac{x^2+9}{x^2-36} > 0 }. The strategic division of the number line into these intervals is a cornerstone technique in solving rational inequalities, enabling a systematic and accurate approach to finding the solution.

Step 3: Test Values in Each Interval

With the number line divided into intervals by our critical points, the next crucial step is to determine the sign of the rational expression x2+9x2−36{ \frac{x^2+9}{x^2-36} } within each interval. To achieve this, we employ a technique known as interval testing. This method involves selecting a representative test value from each interval and substituting it into the expression. The resulting sign of the expression at the test value is indicative of the sign throughout the entire interval.

Let's proceed with testing each interval:

  1. Interval (−∞,−6){ (-\infty, -6) }: We choose a test value less than -6, such as x = -7. Substituting this value into the expression, we get:

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

    The result is a positive number. Therefore, the expression is positive in the interval (−∞,−6){ (-\infty, -6) }.

  2. Interval (−6,6){ (-6, 6) }: We select a test value between -6 and 6, such as x = 0. Substituting this value into the expression, we get:

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

    The result is a negative number. Therefore, the expression is negative in the interval (−6,6){ (-6, 6) }.

  3. Interval (6,∞){ (6, \infty) }: We choose a test value greater than 6, such as x = 7. Substituting this value into the expression, we get:

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

    The result is a positive number. Therefore, the expression is positive in the interval (6,∞){ (6, \infty) }.

By systematically testing a value within each interval, we have successfully determined the sign of the rational expression x2+9x2−36{ \frac{x^2+9}{x^2-36} } across the entire number line, excluding the critical points. This information is crucial for identifying the intervals that satisfy the original inequality, which requires the expression to be greater than zero.

Step 4: Determine the Solution Set

Having meticulously tested the intervals and determined the sign of the rational expression x2+9x2−36{ \frac{x^2+9}{x^2-36} } in each, we are now poised to identify the solution set for the inequality x2+9x2−36>0{ \frac{x^2+9}{x^2-36} > 0 }. Our goal is to find all values of x that make the expression strictly greater than zero, meaning we are looking for intervals where the expression is positive.

From our interval testing in the previous step, we found that the expression is positive in the following intervals:

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

These intervals represent the ranges of x-values for which the inequality holds true. However, it is crucial to consider the critical points themselves. Since the original inequality is strictly greater than zero (>0{ > 0 }), we must exclude any values of x that would make the expression equal to zero or undefined. The critical points x = -6 and x = 6 make the denominator zero, rendering the expression undefined. Therefore, these values must be excluded from the solution set.

Thus, the solution set for the inequality x2+9x2−36>0{ \frac{x^2+9}{x^2-36} > 0 } is the union of the two intervals where the expression is positive, excluding the critical points. We can express this solution set in interval notation as:

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

This notation signifies that the solution includes all real numbers less than -6 and all real numbers greater than 6. The union symbol (∪{ \cup }) indicates that we are combining these two intervals into a single solution set.

In conclusion, the solution set for the given rational inequality consists of all x-values that lie outside the closed interval [-6, 6]. This comprehensive solution is the culmination of our step-by-step analysis, which included identifying critical points, dividing the number line into intervals, testing values within each interval, and carefully considering the original inequality's requirements.

Conclusion

In this comprehensive guide, we've successfully navigated the process of solving the rational inequality x2+9x2−36>0{ \frac{x^2+9}{x^2-36} > 0 }. By systematically identifying critical points, dividing the number line into intervals, testing values within each interval, and carefully considering the inequality's conditions, we've arrived at the solution set: (−∞,−6)∪(6,∞){ (-\infty, -6) \cup (6, \infty) }. This solution encompasses all real numbers less than -6 and all real numbers greater than 6, effectively capturing the range of values for which the inequality holds true.

The techniques and principles explored in this guide are not limited to this specific example. They provide a robust framework for tackling a wide range of rational inequalities. The ability to solve these inequalities is a valuable skill in mathematics, with applications spanning various fields, including calculus, algebra, and mathematical analysis. Mastering these techniques empowers you to approach complex mathematical problems with confidence and precision.

As you continue your mathematical journey, remember the importance of a methodical approach. Break down complex problems into smaller, manageable steps, and apply the fundamental principles with care. With practice and perseverance, you'll develop the skills necessary to excel in mathematics and beyond. The world of rational inequalities, once a complex landscape, will become a familiar and navigable terrain.