4. Solve the following equations: (a) \( x-3=\frac{18}{x} \) (c) \( \frac{x+6}{x^{2}-4}-\frac{2}{x-2}=\frac{-1}{x+2} \) (e) \( \frac{x}{x-2}=\frac{1}{x-3}-\frac{2}{2-x} \) (f) \( \frac{3 x+4}{x+6}=\frac{3 x-2}{x-3}-\frac{21 x}{x^{2}+3 x-18} \)

To solve these equations, it's often helpful to rearrange and simplify them step by step. For \( x - 3 = \frac{18}{x} \), start by multiplying both sides by \( x \) (where \( x \neq 0 \)) to eliminate the fraction: \( x^2 - 3x - 18 = 0 \). Factoring gives \( (x - 6)(x + 3) = 0 \), resulting in \( x = 6 \) or \( x = -3 \). For \( \frac{x+6}{x^{2}-4}-\frac{2}{x-2}=\frac{-1}{x+2} \), find a common denominator for all terms (factoring \( x^2 - 4 \) gives \( (x - 2)(x + 2) \)). After replacing the denominators and simplifying, you can solve for x. The common denominator is essential in combining fractions effectively and solving the resulting equation obtained. For \( \frac{x}{x-2}=\frac{1}{x-3}-\frac{2}{2-x} \), note that \( \frac{2}{2-x} = -\frac{2}{x-2} \). Thus, rearranging gives \( \frac{x}{x-2} = \frac{1}{x-3} + \frac{2}{x-2} \). Finding a common denominator and equating both sides leads towards potential solutions for x. Finally, for \( \frac{3x+4}{x+6}=\frac{3x-2}{x-3}-\frac{21x}{x^2+3x-18} \), it's crucial to factor \( x^2 + 3x - 18 = (x + 6)(x - 3) \). Combining the right side fractions and setting it equal to the left will allow for simplification to reach the final answer for x. Remember to check for extraneous solutions, especially when squaring or multiplying through by variables! Happy solving!