Problem 49 - 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. $$\frac{-1}{x^{2}+2}<0$$

Step-by-Step Solution

Answer

The solution set is \(( -\infty, \infty )\).

Step 1: Identify the Numerator and Denominator

The numerator of the rational expression is \(-1\), and the denominator is \(x^2 + 2\).

Step 2: Analyze the Numerator

The numerator \(-1\) is always negative since it is a constant negative number.

Step 3: Analyze the Denominator

The denominator \(x^2 + 2\) must always be positive. This is because \(x^2\) is always non-negative (zero or positive), and adding 2 makes it strictly positive for any real number \(x\).

Step 4: Determine the Quotient's Sign

The rational expression \(\frac{-1}{x^2 + 2}\) is a quotient of a negative number (numerator) and a positive number (denominator). According to the rules of division, this will always result in a negative number.

Step 5: Apply the Inequality Condition

The inequality \(\frac{-1}{x^2 + 2} < 0\) requires the expression to be negative. Since we determined that \(\frac{-1}{x^2 + 2}\) is always negative, the inequality is satisfied for all real numbers \(x\).

Step 6: Write the Solution Set

Since the inequality \(\frac{-1}{x^2 + 2} < 0\) holds true for all real \(x\), the solution set is \(( -\infty, \infty )\).

Key Concepts

Numerator and Denominator Analysis

When tackling rational inequalities, it's crucial to understand the role of both the numerator and the denominator in determining the overall sign of the expression. The numerator is the top part of the fraction, while the denominator is the bottom part. Together, they form a rational expression, which is essentially a fraction involving polynomials.

Let's consider the given inequality \[ \frac{-1}{x^2 + 2} < 0 \]. Here, the numerator is \(-1\), and the denominator is \(x^2 + 2\). The numerator is a constant negative value. This means it's always less than zero, regardless of \(x\). On the other hand, the denominator \(x^2 + 2\) is always positive for any real number \(x\).

• Analyze the sign of the numerator: negative if \(-1\), positive if a different positive constant.
• Analyze the sign of the denominator: always positive given \(x^2 + 2\) since \(x^2\) is non-negative.

Understanding these roles helps predict the positivity or negativity of the rational expression. This makes it much easier to apply the conditions laid out in an inequality.

Inequality Solution Sets

An inequality solution set consists of all values of \(x\) that make the inequality true. To solve rational inequalities like \[ \frac{-1}{x^2 + 2} < 0 \], analyzing the solution set involves determining where the entire expression (the rational function) satisfies the given inequality condition.

Once we have identified the sign of both the numerator and the denominator, we apply this understanding to the inequality. Because \( -1 \) is negative, and \( x^2 + 2 \) is positive, \(\frac{-1}{x^2 + 2}\) results in a negative value. Thus, the inequality condition, which requires the expression to be less than zero, is met.

• Check the sign of the entire rational expression based on our earlier analysis.
• If the inequality is true for all values within the domain, write the solution set as \((-\infty, \infty )\).

For the inequality \(\frac{-1}{x^2 + 2} < 0\), it's satisfied by any real number, but if the inequality was \(> 0\), there would be no solutions. Recognizing these patterns helps in quickly filling out the solution set.

Rational Expressions

Rational expressions are fractions where both the numerator and the denominator are polynomials. They behave like regular fractions, but they involve variables. Understanding rational expressions is crucial when solving inequalities, as you often need to think of them in terms of their algebraic behavior.

Take the expression \(\frac{-1}{x^2 + 2}\) as our example. The numerator is a constant polynomial (degree zero) while the denominator is a polynomial of degree two. When dealing with rational expressions, always consider the domains; here, it's all real numbers because there's no value of \(x\) that would make the denominator zero.

• Rational expressions can have restricted domains based on their denominators.
• The degree and structure of the polynomials in the numerator and denominator affect the complexity of the inequality.

An advantage of rational expressions is that much of their behavior, when it comes to inequalities, can often be determined just by analyzing the signs of their constitutive parts—as seen with numerator and denominator analysis.