Problem 50
Question
CONCEPT CHECK 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 nonnegative number, \(x^{2},\) and a positive number, 1. 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. $$\frac{5}{x^{2}+2}<0$$
Step-by-Step Solution
Verified Answer
The inequality \(\frac{5}{x^2 + 2} < 0\) has no solution because the fraction is always positive.
1Step 1: Analyze the Numerator
First, consider the numerator of the rational expression, which is 5. Since 5 is positive, this part of our expression will always contribute a positive value to the quotient. This simplifies our task, as we only need to evaluate under what conditions the entire fraction becomes negative.
2Step 2: Analyze the Denominator
Next, consider the denominator of the rational expression, which is \(x^2 + 2\). This expression is the sum of \(x^2\), a non-negative number for any real \(x\), and the positive number 2. Therefore, \(x^2 + 2\) must always be positive for all real numbers \(x\).
3Step 3: Determine if the Fraction Can Be Negative
Since the numerator, 5, is positive and the denominator, \(x^2 + 2\), is positive for all real \(x\), the quotient \(\frac{5}{x^2 + 2}\) is always positive. Since there are no conditions under which this fraction can be negative, it can never satisfy the inequality \(\frac{5}{x^2 + 2} < 0\).
4Step 4: Conclude the Solution Set
Because the fraction is always positive and can never be negative, the inequality \(\frac{5}{x^2 + 2} < 0\) has no solution. Thus, there are no values of \(x\) in the real number set that satisfy this inequality.
Key Concepts
Numerator AnalysisDenominator AnalysisSolution SetQuotients of Positive and Negative Numbers
Numerator Analysis
When solving rational inequalities, understanding the role of the numerator is crucial. In the rational expression \(\frac{5}{x^2+2}\), the numerator is 5. Since 5 is a positive number, it constantly contributes a positive value to the entire fraction. This is significant because in order for the entire fraction to possibly be negative, the numerator would need to at some point neutralize or be outweighed by a negative counterpart.
Yet, as 5 remains steadfastly positive, our focus shifts solely to the denominator to understand under which conditions, if any, the entire expression might become negative, if possible at all. Remember, positive numerators always initiate the fraction towards positivity unless offset by a negative denominator.
Yet, as 5 remains steadfastly positive, our focus shifts solely to the denominator to understand under which conditions, if any, the entire expression might become negative, if possible at all. Remember, positive numerators always initiate the fraction towards positivity unless offset by a negative denominator.
Denominator Analysis
The next step in solving rational inequalities is to analyze the denominator, \(x^2 + 2\). The denominator plays a pivotal role since it can determine the overall sign of the fraction.
In this particular inequality, the expression \(x^2 + 2\) is made up of \(x^2\), which is non-negative for any real number \(x\), and the constant 2, which is positive. Adding these together ensures that \(x^2 + 2\) will always be positive for all real values of \(x\).
Since the denominator is never negative, this confirms that the entire expression is deprived of any opportunity to transform into a negative value, ensuring its positivity.
In this particular inequality, the expression \(x^2 + 2\) is made up of \(x^2\), which is non-negative for any real number \(x\), and the constant 2, which is positive. Adding these together ensures that \(x^2 + 2\) will always be positive for all real values of \(x\).
Since the denominator is never negative, this confirms that the entire expression is deprived of any opportunity to transform into a negative value, ensuring its positivity.
Solution Set
The solution set of a rational inequality considers where the inequality's condition is met. In our case, we are exploring why \(\frac{5}{x^2 + 2} < 0\) cannot hold true.
The numerator is always positive, and as analyzed, the denominator is also unfailingly positive for real numbers \(x\). Therefore, the fraction \(\frac{5}{x^2 + 2}\) invariably yields a positive result.
Since \(\frac{5}{x^2 + 2}\) cannot be negative or zero under any real \(x\), the inequality \(\frac{5}{x^2 + 2} < 0\) has no solutions. This leads us to conclude that the solution set is empty. Simply put: there are no real numbers \(x\) that satisfy the inequality.
The numerator is always positive, and as analyzed, the denominator is also unfailingly positive for real numbers \(x\). Therefore, the fraction \(\frac{5}{x^2 + 2}\) invariably yields a positive result.
Since \(\frac{5}{x^2 + 2}\) cannot be negative or zero under any real \(x\), the inequality \(\frac{5}{x^2 + 2} < 0\) has no solutions. This leads us to conclude that the solution set is empty. Simply put: there are no real numbers \(x\) that satisfy the inequality.
Quotients of Positive and Negative Numbers
Understanding the results of dividing positive and negative numbers is essential in rational inequalities. Ratios involve two components: the numerator and the denominator, affecting the quotient's sign depending on their individual values.
A positive numerator combined with a positive denominator yields a positive quotient. Conversely, a positive numerator with a negative denominator would result in a negative quotient. In our inequality example, both the numerator and the denominator are positive. Hence, their quotient, \(\frac{5}{x^2 + 2}\), is always positive.
This established concept afirms the impossibility of a negative result in such an equation setup. Rational inequalities focusing on positivity or negativity depend fundamentally on these rules of quotient behavior.
A positive numerator combined with a positive denominator yields a positive quotient. Conversely, a positive numerator with a negative denominator would result in a negative quotient. In our inequality example, both the numerator and the denominator are positive. Hence, their quotient, \(\frac{5}{x^2 + 2}\), is always positive.
This established concept afirms the impossibility of a negative result in such an equation setup. Rational inequalities focusing on positivity or negativity depend fundamentally on these rules of quotient behavior.
Other exercises in this chapter
Problem 49
Sketch a graph of rational function. Your graph should include all asymptotes. Do not use a calculator. $$f(x)=\frac{x^{2}}{1-x^{2}}$$
View solution Problem 49
Evaluate \(f(x)\) at the given \(x\). Approximate each result to the nearest hundredth. $$f(x)=x^{3 / 2}-x^{1 / 2}, \quad x=50$$
View solution Problem 50
Sketch a graph of rational function. Your graph should include all asymptotes. Do not use a calculator. $$f(x)=\frac{1-x^{2}}{2 x^{2}}$$
View solution Problem 50
Use an analytic method to solve each equation in part (a). Support the solution with a graph. Then use the graph to solve the inequalities in parts (b) and (c).
View solution