Solve for x . If there is more than one solution, separate the \( \frac{8}{x-9}+\frac{x}{x+5}=\frac{-13 x+13}{x^{2}-4 x-45} \) \( x=\square \) Question Help: \( \square \) Video \( \square \) Message instructor

To solve the equation \( \frac{8}{x-9}+\frac{x}{x+5}=\frac{-13 x+13}{x^{2}-4 x-45} \), first, you can simplify the right side. Notice that \( x^2 - 4x - 45 \) factors to \( (x - 9)(x + 5) \). This gives us the equation \( \frac{8}{x-9} + \frac{x}{x+5} = \frac{-13x+13}{(x-9)(x+5)} \). Next, you can multiply through by \((x - 9)(x + 5)\) to eliminate the denominators. After distributing and collecting like terms, you’ll get a polynomial equation to solve for \( x \). Don’t forget to check for extraneous solutions! In terms of real-world applications, equations like this one can arise in fields such as physics when dealing with rates and motion, or in finance when calculating profit or loss based on varying conditions. Understanding how to manipulate and solve these equations can help in making informed decisions in real-life scenarios!