Solving (x+11)(x-13)/(x+9) ≤ 0 A Step-by-Step Guide

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Introduction

In the realm of mathematics, inequalities form a fundamental concept, enabling us to compare the relative values of different expressions. Among the various types of inequalities, rational inequalities hold a unique position due to their involvement of rational functions – functions expressed as the ratio of two polynomials. This article delves into the intricacies of solving a specific rational inequality: (x+11)(x-13)/(x+9) ≤ 0. We will embark on a step-by-step journey, unraveling the techniques and strategies required to determine the solution set for this inequality. Understanding rational inequalities is crucial as they appear in various mathematical contexts, including calculus, pre-calculus, and real analysis, and have applications in real-world scenarios such as optimization problems and modeling physical systems.

Before diving into the specifics of our target inequality, it's important to grasp the underlying principles of rational inequalities. Rational inequalities are mathematical statements that compare two rational expressions using inequality symbols such as <, >, ≤, or ≥. The general form of a rational inequality is f(x)/g(x) < 0, f(x)/g(x) > 0, f(x)/g(x) ≤ 0, or f(x)/g(x) ≥ 0, where f(x) and g(x) are polynomial functions. The key to solving these inequalities lies in identifying the critical points, which are the values of x that make either the numerator or the denominator equal to zero. These critical points divide the number line into intervals, and the sign of the rational expression within each interval remains constant. By testing a value within each interval, we can determine whether the inequality holds true or false in that interval. This method, often called the interval testing method, allows us to systematically find the solution set of the inequality. Now, with a solid foundation in the concept of rational inequalities, we can proceed to tackle the specific inequality at hand.

Understanding the Inequality (x+11)(x-13)/(x+9) ≤ 0

The given inequality, (x+11)(x-13)/(x+9) ≤ 0, is a rational inequality where the rational expression is a product of two linear factors in the numerator divided by another linear factor in the denominator. To solve this inequality, we need to find all the values of x for which the expression is less than or equal to zero. This involves a systematic approach that combines algebraic manipulation and logical reasoning. The first step is to identify the critical points, which are the values of x that make either the numerator or the denominator equal to zero. The numerator, (x+11)(x-13), becomes zero when x = -11 or x = 13. These are the zeros of the numerator, and they represent the points where the expression can potentially change its sign. The denominator, (x+9), becomes zero when x = -9. This is a critical point because it makes the expression undefined, as division by zero is not allowed. These critical points are crucial as they divide the number line into intervals, and within each interval, the sign of the rational expression remains constant.

The next step in solving the inequality is to analyze the behavior of the rational expression around the critical points. The critical points, x = -11, x = -9, and x = 13, partition the number line into four intervals: (-∞, -11), (-11, -9), (-9, 13), and (13, ∞). Within each interval, the rational expression (x+11)(x-13)/(x+9) maintains a constant sign. To determine the sign in each interval, we can choose a test value within the interval and substitute it into the expression. For example, in the interval (-∞, -11), we can choose x = -12. Substituting this value into the expression, we get a positive numerator and a negative denominator, resulting in a negative overall value. In the interval (-11, -9), we can choose x = -10. This gives us a negative numerator and a negative denominator, resulting in a positive overall value. Similarly, we can test values in the remaining intervals to determine their signs. The critical point x = -9 is particularly important because it makes the denominator zero, and thus the expression is undefined at this point. This means that x = -9 cannot be included in the solution set, even if it satisfies the inequality. Now that we have identified the critical points and analyzed the sign of the expression in each interval, we are ready to assemble the solution set.

Step-by-Step Solution

1. Identify Critical Points

As discussed earlier, the critical points are the values of x that make the numerator or the denominator of the rational expression equal to zero. For the inequality (x+11)(x-13)/(x+9) ≤ 0, the critical points are found as follows:

  • Numerator: (x+11)(x-13) = 0 implies x = -11 or x = 13
  • Denominator: (x+9) = 0 implies x = -9

Therefore, the critical points are -11, -9, and 13. These points are the cornerstones of our solution, as they mark the boundaries where the expression's sign can potentially change. Recognizing these critical values is essential, as it forms the basis for the interval testing method, which we will use to determine the solution set of the inequality.

2. Create Intervals

The critical points divide the number line into intervals. In this case, the critical points -11, -9, and 13 divide the number line into four intervals:

  • (-∞, -11)
  • (-11, -9)
  • (-9, 13)
  • (13, ∞)

These intervals are the regions where the sign of the rational expression remains consistent. This is a crucial observation, as it allows us to test a single value within each interval to determine the sign of the entire interval. This simplifies the process of solving the inequality, as we don't need to evaluate the expression for every possible value of x. The intervals act as our guide, leading us through the number line to identify the regions that satisfy the inequality.

3. Test Values in Each Interval

To determine the sign of the rational expression (x+11)(x-13)/(x+9) in each interval, we choose a test value within each interval and substitute it into the expression. This process allows us to quickly assess whether the inequality holds true in that interval. Let's consider each interval:

  • Interval (-∞, -11): Choose x = -12. The expression becomes ((-12+11)(-12-13))/(-12+9) = ((-1)(-25))/(-3) = -25/3, which is negative. Therefore, the inequality holds true in this interval.
  • Interval (-11, -9): Choose x = -10. The expression becomes ((-10+11)(-10-13))/(-10+9) = ((1)(-23))/(-1) = 23, which is positive. Therefore, the inequality does not hold true in this interval.
  • Interval (-9, 13): Choose x = 0. The expression becomes ((0+11)(0-13))/(0+9) = ((11)(-13))/(9) = -143/9, which is negative. Therefore, the inequality holds true in this interval.
  • Interval (13, ∞): Choose x = 14. The expression becomes ((14+11)(14-13))/(14+9) = ((25)(1))/(23) = 25/23, which is positive. Therefore, the inequality does not hold true in this interval.

By testing these values, we have mapped out the sign of the expression across the entire number line, allowing us to identify the intervals that satisfy the inequality. This is a powerful technique that simplifies the process of solving rational inequalities.

4. Determine the Solution Set

Based on the test values, we can now determine the solution set for the inequality (x+11)(x-13)/(x+9) ≤ 0. The inequality holds true in the intervals (-∞, -11) and (-9, 13). However, we must also consider the critical points themselves. The inequality includes "≤", which means we need to include the values of x that make the expression equal to zero. The numerator is zero when x = -11 and x = 13, so these values are included in the solution set. However, the denominator is zero when x = -9, which makes the expression undefined. Therefore, x = -9 is not included in the solution set.

Combining these considerations, the solution set is the union of the interval (-∞, -11] and the interval (-9, 13]. The square bracket indicates that the endpoint is included in the solution set, while the parenthesis indicates that the endpoint is not included. This is a crucial detail in representing the solution set accurately. In interval notation, the solution set is written as (-∞, -11] ∪ (-9, 13]. This notation concisely captures all the values of x that satisfy the inequality, providing a complete and accurate solution.

Representing the Solution

The solution set can be represented in various ways, each offering a unique perspective on the set of values that satisfy the inequality. Understanding these representations is crucial for effectively communicating the solution and applying it in different contexts. Here are the common ways to represent the solution:

1. Interval Notation

As we have seen, interval notation is a concise and widely used method to represent sets of numbers. The solution set for (x+11)(x-13)/(x+9) ≤ 0 in interval notation is (-∞, -11] ∪ (-9, 13]. This notation clearly shows the intervals where the inequality holds true, as well as whether the endpoints are included or excluded. The use of parentheses and square brackets is essential in conveying this information accurately. Interval notation is particularly useful when dealing with continuous sets of numbers, as it provides a compact way to represent an infinite number of values.

2. Set-Builder Notation

Set-builder notation provides a more descriptive way to represent the solution set. In this notation, we define the set by specifying a condition that its elements must satisfy. For our inequality, the solution set in set-builder notation is {x | x ≤ -11 or -9 < x ≤ 13}. This notation explicitly states the conditions that x must meet to be included in the solution set. It is particularly useful when dealing with complex solution sets that may not be easily expressed in interval notation. Set-builder notation offers a precise and unambiguous way to define the solution set.

3. Graphical Representation

A graphical representation of the solution set can provide a visual understanding of the values that satisfy the inequality. We can represent the solution set on a number line by shading the intervals where the inequality holds true. For the inequality (x+11)(x-13)/(x+9) ≤ 0, we would shade the interval (-∞, -11] and the interval (-9, 13] on the number line. We would use a closed circle at x = -11 and x = 13 to indicate that these points are included in the solution set, and an open circle at x = -9 to indicate that this point is not included. The graphical representation provides an intuitive way to visualize the solution set, making it easier to grasp the range of values that satisfy the inequality. It is especially helpful for understanding the behavior of the rational expression and its relationship to the critical points.

Common Mistakes to Avoid

When solving rational inequalities, it's crucial to be aware of common mistakes that can lead to incorrect solutions. Avoiding these pitfalls will ensure accuracy and a deeper understanding of the concepts involved. Here are some common mistakes to watch out for:

1. Forgetting to Consider the Denominator

A frequent mistake is to only focus on the numerator and forget about the denominator. The denominator plays a crucial role in determining the critical points and the intervals where the expression is defined. Values of x that make the denominator zero must be excluded from the solution set, as they make the expression undefined. In our example, x = -9 is a critical point because it makes the denominator zero, and it cannot be included in the solution set. Ignoring the denominator can lead to including values that make the expression undefined, resulting in an incorrect solution.

2. Incorrectly Handling Critical Points

Critical points are the values of x that make either the numerator or the denominator zero. It's important to correctly identify these points and understand their significance. The zeros of the numerator are included in the solution set if the inequality includes "≤" or "≥", while the zeros of the denominator are always excluded. A common mistake is to include the zeros of the denominator in the solution set, which is incorrect. For example, in the inequality (x+11)(x-13)/(x+9) ≤ 0, x = -11 and x = 13 are included in the solution set because they make the numerator zero, but x = -9 is excluded because it makes the denominator zero. Misunderstanding how to handle critical points can lead to an inaccurate solution set.

3. Not Testing Intervals Correctly

The interval testing method is a key technique for solving rational inequalities. However, it's important to choose test values carefully and substitute them correctly into the expression. A common mistake is to choose test values that are critical points themselves, which will not provide information about the sign of the expression in the interval. It's also important to perform the substitution and evaluation accurately. A single arithmetic error can lead to an incorrect determination of the sign of the expression in the interval, resulting in an incorrect solution set. Double-checking the calculations and ensuring that the test values are within the correct intervals can help avoid these mistakes.

4. Multiplying Both Sides by an Expression Without Considering the Sign

When dealing with inequalities, it's generally not advisable to multiply both sides by an expression involving x without considering its sign. Multiplying by a negative expression reverses the inequality sign, and failing to account for this can lead to an incorrect solution. In the case of rational inequalities, multiplying by the denominator is a common temptation, but it's crucial to remember that the denominator can be positive or negative depending on the value of x. Instead of multiplying, it's better to rely on the interval testing method, which systematically analyzes the sign of the expression in different intervals. Avoiding this mistake will ensure that the solution set is accurate and complete.

Conclusion

Solving rational inequalities requires a systematic approach that combines algebraic skills with logical reasoning. In this article, we have explored the process of solving the inequality (x+11)(x-13)/(x+9) ≤ 0 step by step, from identifying the critical points to determining the solution set. We have also discussed common mistakes to avoid, ensuring a deeper understanding of the concepts involved. By mastering the techniques presented in this article, you will be well-equipped to tackle a wide range of rational inequalities and apply them in various mathematical contexts. Rational inequalities are not just abstract mathematical problems; they are powerful tools for modeling and solving real-world scenarios. From optimization problems to analyzing physical systems, rational inequalities provide a framework for understanding and predicting the behavior of complex phenomena. Therefore, a solid understanding of rational inequalities is essential for success in mathematics and related fields.

The solution to the inequality (x+11)(x-13)/(x+9) ≤ 0 is the set of all x values that satisfy the inequality. We found that the critical points are x = -11, x = -9, and x = 13. These points divide the number line into four intervals: (-∞, -11), (-11, -9), (-9, 13), and (13, ∞). By testing values in each interval, we determined that the inequality holds true in the intervals (-∞, -11] and (-9, 13]. The solution set is therefore the union of these intervals, which can be written in interval notation as (-∞, -11] ∪ (-9, 13]. This solution set represents all the values of x for which the rational expression is less than or equal to zero. By understanding the steps involved in solving rational inequalities and avoiding common mistakes, you can confidently tackle these types of problems and apply them in various mathematical and real-world applications. Remember, practice is key to mastering any mathematical concept, so continue to explore and solve different rational inequalities to solidify your understanding.