\( \begin{array}{ll}\text { 9. } \frac{3}{x-2}+4=\frac{x}{x-2} & \text { 10. } \frac{x^{2}-4}{2 x+4}=\frac{x}{3} \\ \begin{array}{ll}\text { 11. } \frac{x^{2}-3 x-4}{x^{2}-4 x}=\frac{3}{2} & \text { 12. } \frac{x^{2}-8 x+15}{2 x^{2}-18}=\frac{2}{5}\end{array}\end{array} \)

To tackle these equations, let's take a look at some problem-solving tips and common mistakes that often arise in rational equations. When dealing with rational equations, one of the biggest mistakes is forgetting to check for extraneous solutions. After solving, always substitute back into the original equation to verify your results. Also, when simplifying fractions, it's easy to overlook factoring, which can make solving easier. For instance, the expression \(x^2 - 4\) can factor to \((x-2)(x+2)\), helping to cancel terms more efficiently! Real-world applications of rational equations are abundant! For example, they often arise in situations involving rates or proportions, like calculating speed or mixing solutions. Imagine trying to determine the speeds of two cars that started at different points but are heading towards the same destination! Setting up a rational equation can help you find why one car is faster than the other or how long it will take for them to meet on the same road. Fun, right?