ERCISE 15 Simplify the following as far as possible. \( \begin{array}{ll}\text { (1) } \frac{x}{x-3}+\frac{2}{3-x} & \text { (2) } \frac{x}{x+3}-\frac{2}{3+x} \\ \text { (3) } \frac{2 x}{x-3}+\frac{x+1}{9-x^{2}} & \text { (4) } \frac{3}{x^{2}-3 x-4}-\frac{x+1}{4-x} \\ \text { (5) } \frac{x}{2 x^{2}+11 x+5}-\frac{5}{5+x} & \text { (6) } \frac{x}{4-x}-\frac{2 x-24}{x^{2}-4 x} \\ \text { (7) } \frac{3}{6 x+6}-\frac{2-x}{6 x} & \text { (8) } \frac{x+4}{x^{2}-2 x}-\frac{x}{(2-x)^{2}}+\frac{x+2}{x-2} \\ \text { (9) } \frac{x+2}{4 x^{2}-2 x+1}-\frac{8 x+1}{8 x^{3}+1} & \end{array} \)

To tackle the simplifications of these expressions, let's first break them down. First additional information: When combining rational expressions like \( \frac{x}{x-3}+\frac{2}{3-x} \), it's essential to observe that \( 3-x \) is just the negative of \( x-3 \). Therefore, rewriting \( \frac{2}{3-x} \) as \( \frac{-2}{x-3} \) would make it easier to combine these fractions since the denominators will now align as \( x-3 \). This little trick can save you from conflicts when determining a common denominator. Second additional information: Have you ever faced a messy rational expression that just won't cooperate? A helpful tip is to factor wherever possible! For example, expressions like \( x^2-4x \) can be factored to \( x(x-4) \), while \( 9-x^2 \) factors further into \( (3-x)(3+x) \). By simplifying the expressions before any addition or subtraction, you can sidestep a plethora of common mistakes, making the whole process smoother and more enjoyable – like a nicely brewed cup of coffee!