Problem 54 - 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. $$\frac{x^{4}+x^{2}+3}{x^{2}+2}<0$$

Step-by-Step Solution

Answer

The solution set is an empty set, \(\emptyset\), because the expression is always positive.

Step 1: Analyze the Numerator

Examine the numerator of the rational inequality, \(x^4 + x^2 + 3\). Since it consists of terms like \(x^4\) and \(x^2\), both of which are non-negative for all \(x\), and a constant positive term \(3\), the entire expression \(x^4 + x^2 + 3\) is always positive.

Step 2: Analyze the Denominator

Look at the denominator \(x^2 + 2\). It consists of \(x^2\), a non-negative term, plus \(2\), a positive constant. Thus, \(x^2 + 2\) is also always positive for all \(x\).

Step 3: Determine the Sign of the Rational Expression

Since both the numerator \(x^4 + x^2 + 3\) and the denominator \(x^2 + 2\) are positive for all real numbers \(x\), the quotient \(\frac{x^4 + x^2 + 3}{x^2 + 2}\) will always be positive.

Step 4: Solve the Inequality

The inequality is \(\frac{x^4 + x^2 + 3}{x^2 + 2} < 0\). Given that the rational expression is always positive as explained in Step 3, it can never satisfy the inequality, which requires it to be less than zero.

Step 5: Conclusion: Determine the Solution Set

Since the rational expression is always positive and can never be negative, the solution set for the inequality is an empty set, \(\emptyset\).

Key Concepts

Numerator and Denominator Analysis

When solving rational inequalities, it's crucial to first dissect both the numerator and the denominator. This analysis helps us understand their respective signs over a given domain of values. For example, in the expression \( \frac{x^4 + x^2 + 3}{x^2 + 2} \), let's break it down:

• The numerator \( x^4 + x^2 + 3 \) includes terms like \( x^4 \) and \( x^2 \), which are always non-negative. Adding \( 3 \), a positive constant, cements the entire numerator as a positive quantity for all real \( x \).
• The denominator \( x^2 + 2 \) consists of \( x^2 \), also non-negative, and \( 2 \), a positive number. Consequently, the denominator remains positive for every real number \( x \).

Understanding that both expressions don't change sign and remain positive simplifies solving the inequality.

Solution Set Determination

After analyzing the expressions involved, the next step is to determine the solution set of the inequality. With the rational inequality \( \frac{x^4 + x^2 + 3}{x^2 + 2} < 0 \), we leverage our analysis:

• Given that both the numerator and denominator are positive, the entire expression \( \frac{x^4 + x^2 + 3}{x^2 + 2} \) is positive for any real number \( x \).
• The inequality asks for places where this expression is negative—less than zero. However, since it's always positive, there's no possible 'x' that can satisfy the condition \( <0 \).
• Thus, we conclude the solution set is empty (\( \emptyset \)). There are no real numbers that satisfy the inequality.

Recognizing a no-solution scenario is just as vital as finding solutions, illustrating the need for deep analysis.

Positive and Negative Quotients

Understanding how quotients behave, especially in terms of positivity and negativity, is key in rational inequalities. Here's a quick breakdown:

• A quotient of two positive numbers remains positive. Therefore, if the numerator and the denominator of a fraction are both positive across a domain, their quotient is inevitably positive.
• Similarly, a quotient where one component is negative and the other is positive yields a negative quotient. If both are negative, the quotient turns positive again, following the double-negative rule.
• For \( \frac{x^4 + x^2 + 3}{x^2 + 2} \), since neither the numerator nor the denominator crosses into negative values, knowing this helps despite the inequality's request for negativity.

Applying these principles helps quickly judge the potential sign of a rational expression just by analyzing its parts.