For the function \( g(x)=\frac{x+1}{x+5} \), solve the following inequality. \( g(x)>0 \)

To solve the inequality \( g(x) > 0 \), we start by finding when the function \( g(x) = \frac{x + 1}{x + 5} \) is greater than zero. This fraction is positive when both the numerator and denominator are either both positive or both negative. 1. **Numerator**: \( x + 1 > 0 \) gives \( x > -1 \). 2. **Denominator**: \( x + 5 > 0 \) gives \( x > -5 \). Now, let's analyze the regions defined by these two critical points, \( -1 \) and \( -5 \). - Both numerator and denominator are positive for \( x > -1 \). - Both numerator and denominator are negative for \( x < -5 \). Thus, the solution to the inequality \( g(x) > 0 \) is \( x > -1 \) or \( x < -5 \). In interval notation, the solution is \( (-\infty, -5) \cup (-1, \infty) \). Remember to check for any values where the function is undefined; here \( g(x) \) is undefined at \( x = -5 \) but since we only care about when it's greater than zero, our intervals remain valid.