For the function \( g(x)=\frac{x+9}{x-5} \), solve the following inequality. \( g(x)>0 \)

To solve the inequality \( g(x) > 0 \) for the function \( g(x) = \frac{x + 9}{x - 5} \), we need to determine when the fraction is positive. This occurs when both the numerator and the denominator have the same sign. 1. **Numerator**: \( x + 9 > 0 \) leads to \( x > -9 \). 2. **Denominator**: \( x - 5 > 0 \) leads to \( x > 5 \). Now, we incorporate both inequalities: - The numerator is positive when \( x > -9 \). - The denominator is positive when \( x > 5 \). Thus, for \( g(x) \) to be positive overall, we need \( x > 5 \). However, we also have to consider the scenario when both the numerator and denominator are negative. 1. **Numerator**: \( x + 9 < 0 \) leads to \( x < -9 \). 2. **Denominator**: \( x - 5 < 0 \) leads to \( x < 5 \). Here, the numerator is negative for \( x < -9\) and the denominator is negative for \( x < 5\). Both conditions are satisfied when \( x < -9 \). Finally, the solution to \( g(x) > 0 \) can be summarized as: \[ x < -9 \quad \text{or} \quad x > 5. \] Therefore, the solution to the inequality \( g(x) > 0 \) is: \[ (-\infty, -9) \cup (5, \infty). \]