Solve the rational inequality.
$\frac{5}{x^2+9}>0$
The given inequality is \(\frac{5}{x^2+9} > 0\). To determine the set of all real numbers \(x\) that satisfy this condition, we examine the sign of the expression by considering its numerator and denominator separately.
The numerator of the expression is 5, which is a positive constant. For any real number \(x\), the value of the numerator is always 5, so it is strictly greater than 0. Now we look at the denominator, \(x^2+9\). For any real number \(x\), the square \(x^2\) is always greater than or equal to 0. Adding 9 to this non-negative value gives \(x^2 + 9 \geq 9\). This means that the denominator is always at least 9, and thus it is always positive and never zero for any real number \(x\).
Since the numerator is always positive and the denominator is always positive for every real number \(x\), the entire expression \(\frac{5}{x^2+9}\) is a quotient of two positive numbers. A quotient of two positive numbers is always positive. Therefore, the inequality \(\frac{5}{x^2+9} > 0\) holds true for every real number \(x\). The set of all solutions is all real numbers, denoted as \((-\infty, \infty)\).
In real-world applications, such as physics or engineering, we often encounter expressions where we need to determine the domain for which a physical property (like intensity, probability density, or power) is meaningful. For example, if the intensity of a signal is given by \(\frac{5}{x^2+9}\), this analysis tells us that the signal intensity is positive everywhere, which helps in understanding the coverage or range of the physical system. Understanding the behavior of rational functions and their domains is essential for modeling real-world phenomena accurately.