Solving 6 / (x^2 + 9x + 8) < 0 A Step-by-Step Guide

In the realm of mathematics, solving inequalities involving rational expressions is a crucial skill. This article delves into the step-by-step process of solving the rational inequality 6 / (x^2 + 9x + 8) < 0, providing a comprehensive understanding of the underlying concepts and techniques. We will explore how to factor the quadratic expression, identify critical values, construct sign charts, and ultimately express the solution in interval notation. Understanding these methods is essential for anyone studying algebra, calculus, or related fields, as rational inequalities frequently appear in various mathematical problems and applications. This detailed guide aims to equip you with the knowledge and confidence to tackle similar problems effectively.

1. Factoring the Quadratic Expression

The first crucial step in solving the inequality 6 / (x^2 + 9x + 8) < 0 involves factoring the quadratic expression in the denominator, which is x^2 + 9x + 8. Factoring this quadratic will allow us to identify the critical values where the expression equals zero, which are essential for solving the inequality. To factor the quadratic expression x^2 + 9x + 8, we look for two numbers that multiply to 8 (the constant term) and add up to 9 (the coefficient of the x term). These numbers are 1 and 8, as 1 * 8 = 8 and 1 + 8 = 9. Therefore, we can rewrite the quadratic expression as follows:

x^2 + 9x + 8 = (x + 1)(x + 8)

This factorization is critical because it reveals the values of x that make the denominator equal to zero. The denominator cannot be zero because division by zero is undefined, which will be important in the next steps of solving the inequality. By understanding how to factor quadratic expressions, we lay the groundwork for identifying intervals where the original rational expression is positive, negative, or undefined. Factoring accurately is thus a vital skill for solving inequalities of this type.

2. Identifying Critical Values

Identifying the critical values is a pivotal step in solving rational inequalities. These critical values are the points at which the expression changes its sign, and they help define the intervals where the inequality is either true or false. In our inequality, 6 / (x^2 + 9x + 8) < 0, the critical values are derived from two sources: the zeros of the denominator and the zeros of the numerator. The numerator is 6, which is a constant and never equals zero, so it does not contribute to the critical values. However, the denominator, which we factored as (x + 1)(x + 8), provides critical values when it equals zero. Setting each factor to zero gives us:

x + 1 = 0 => x = -1 x + 8 = 0 => x = -8

Thus, the critical values are x = -1 and x = -8. These values are crucial because they divide the number line into intervals where the expression 6 / (x^2 + 9x + 8) maintains a consistent sign—either positive or negative. It is important to note that since these values make the denominator zero, they are not included in the solution set of the inequality, as the expression is undefined at these points. Recognizing and accurately identifying these critical values is a key step in the process of solving rational inequalities.

3. Constructing a Sign Chart

Constructing a sign chart is an essential method for solving rational inequalities. A sign chart helps to visually organize and determine the sign of the rational expression in different intervals defined by the critical values. For our inequality, 6 / (x^2 + 9x + 8) < 0, we've identified the critical values as x = -8 and x = -1. These values divide the number line into three intervals: (-∞, -8), (-8, -1), and (-1, ∞). To create the sign chart, we first list these intervals and then test a value within each interval to determine the sign of the expression 6 / (x^2 + 9x + 8) in that interval.

Let's pick test values:

• For the interval (-∞, -8), let's use x = -9. Substituting into the expression, we get 6 / ((-9)^2 + 9(-9) + 8) = 6 / (81 - 81 + 8) = 6 / 8, which is positive.
• For the interval (-8, -1), let's use x = -2. Substituting, we get 6 / ((-2)^2 + 9(-2) + 8) = 6 / (4 - 18 + 8) = 6 / (-6), which is negative.
• For the interval (-1, ∞), let's use x = 0. Substituting, we get 6 / ((0)^2 + 9(0) + 8) = 6 / 8, which is positive.

Now, we can create the sign chart:

This sign chart visually demonstrates the sign of the expression in each interval. Since we are solving for 6 / (x^2 + 9x + 8) < 0, we are interested in the intervals where the expression is negative. Constructing and interpreting the sign chart correctly is crucial for accurately identifying the solution set.

4. Expressing the Solution in Interval Notation

After constructing the sign chart, the final step in solving the rational inequality 6 / (x^2 + 9x + 8) < 0 is to express the solution in interval notation. The sign chart clearly indicates the intervals where the expression is negative, which is what we are looking for since the inequality is less than zero. From the sign chart, we observed that the expression 6 / (x^2 + 9x + 8) is negative in the interval (-8, -1). This means that for any x value within this interval, the inequality 6 / (x^2 + 9x + 8) < 0 holds true.

In interval notation, we represent this solution as (-8, -1). The parentheses indicate that the endpoints -8 and -1 are not included in the solution set. This is because these values make the denominator of the rational expression equal to zero, resulting in an undefined expression. Therefore, the solution includes all real numbers between -8 and -1, but not the numbers -8 and -1 themselves. Using interval notation to express the solution provides a concise and standardized way to represent the set of all x values that satisfy the original inequality. This notation is widely used in mathematics to clearly define solution sets, especially for inequalities and functions.

Conclusion

In conclusion, solving the rational inequality 6 / (x^2 + 9x + 8) < 0 involves several critical steps, each building upon the previous one. First, factoring the quadratic expression in the denominator allows us to identify the critical values. These critical values, in this case, -8 and -1, are crucial as they demarcate the intervals where the rational expression may change its sign. Next, constructing a sign chart helps visualize the sign of the expression in each interval, making it easier to determine where the inequality holds true. By testing a value within each interval, we can ascertain whether the expression is positive or negative.

Finally, expressing the solution in interval notation provides a clear and concise representation of the set of all x values that satisfy the original inequality. For our inequality, the solution is the interval (-8, -1), indicating all real numbers between -8 and -1, excluding the endpoints. Mastering these steps is essential for solving various rational inequalities and understanding their behavior. The techniques discussed in this article are applicable to a wide range of mathematical problems and are a fundamental part of algebraic problem-solving.