Problem 53 - A Graphical Approach to College Algebra

Question

In some cases, it is possible to solve a rational inequality simply by deciding what sign the numerator and the denominator must have and then using the rules for quotients of positive and negative numbers to determine the solution set. For example, consider the rational inequality $$\frac{1}{x^{2}+1}>0$$ The numerator of the rational expression, \(1,\) is positive, and the denominator, \(x^{2}+1,\) must always be positive because it is the sum of a nomegative number, \(x^{2},\) and a positive number, i. Therefore, the rational expression is the quotient of two positive numbers, which is positive. Because the inequality requires that the rational expression be greater than 0 , and this will always be true, the solution set is ( \(-\infty, \infty\) ). Use similar reasoning to solve each inequality. $$\frac{x^{4}+2}{-6} \geq 0$$

Step-by-Step Solution

Answer

The solution set is \( \emptyset \).

Step 1: Identify Components of the Inequality

The inequality given is \( \frac{x^4 + 2}{-6} \geq 0 \). The numerator is \( x^4 + 2 \) and the denominator is \(-6\). This gives us insight into the sign of the rational function depending on \(x\).

Step 2: Analyze the Numerator

The numerator, \( x^4 + 2 \), is the sum of \( x^4 \) (which is always non-negative since any real number raised to the fourth power is non-negative) and 2 (which is positive). Thus, the entire numerator is always positive.

Step 3: Analyze the Denominator

The denominator is a constant \(-6\), which is always negative regardless of the value of \(x\).

Step 4: Determine the Sign of the Quotient

Since the numerator is positive and the denominator is negative, the quotient \( \frac{x^4 + 2}{-6} \) will always be negative.

Step 5: Solve the Inequality

We need the rational expression to be greater than or equal to 0. However, from the analysis, \( \frac{x^4 + 2}{-6} \) is always negative. There are no values of \( x \) which make the expression non-negative.

Step 6: Determine the Solution Set

Since no values of \( x \) satisfy the inequality, the solution set is empty. In mathematical terms, the solution set is \( \emptyset \).

Key Concepts

Numerator Analysis

When dealing with rational inequalities, a key step is understanding the behavior of the numerator. In the inequality \( \frac{x^4 + 2}{-6} \geq 0 \), the numerator is \( x^4 + 2 \). To analyze this, consider the nature of \( x^4 \). Any real number raised to the power of four, regardless of whether it is positive or negative, results in a non-negative value. This implies that \( x^4 \) itself is always zero or greater.
Adding 2 to \( x^4 \) makes the entire numerator \( x^4 + 2 \) strictly positive at all times. This is important because a positive numerator means the sign of the entire rational expression hinges on the sign of the denominator. Understanding the consistent positivity of the numerator helps pinpoint the next steps in determining the inequality's solution set.

Denominator Analysis

Let's turn our attention to the denominator in the rational inequality \( \frac{x^4 + 2}{-6} \geq 0 \). Here, the denominator is simply a constant value, \(-6\). Constants in inequalities are straightforward because they do not change with varying \( x \).
Given that \(-6\) is a negative number, the denominator remains negative for all real numbers \( x \). In the broader context of rational expressions, a negative denominator coupled with a positive numerator generally produces a negative quotient. Hence, performing a denominator analysis with constancy like \(-6\) helps infer the overall sign of the rational expression, leading toward understanding the solution set.

Solution Set

Once both the numerator and denominator have been analyzed, determining the solution set becomes clearer. We've identified that the numerator \( x^4 + 2 \) is always positive, and the denominator \(-6\) is always negative. Consequently, the rational expression \( \frac{x^4 + 2}{-6} \) is consistently negative because a positive number divided by a negative number results in a negative quotient.
Our original inequality asks for \( \frac{x^4 + 2}{-6} \geq 0 \), which means the expression needs to be zero or positive. However, since the quotient is never non-negative for any real \( x \), there are no \( x \)-values that satisfy this inequality. Therefore, the solution set is identified as empty – symbolically represented as \( \emptyset \).

Non-Negative Numbers

The concept of non-negative numbers plays an essential role in rational inequalities. Non-negative numbers include all positive numbers and zero. In this exercise, the term \( x^4 \) is always non-negative for all real \( x \) because raising a number to the fourth power results in either zero (when \( x = 0 \)) or a positive value.
This makes \( x^4 \) a reliable component, staying on the non-negative side in the equation. Adding a positive constant, such as 2, keeps the expression of the numerator, \( x^4 + 2 \), positive as a whole. Recognizing and leveraging the concept of non-negative numbers aids in thoroughly analyzing inequalities, leading you to safely deduce the nature and sign of mathematical expressions in similar contexts.