Solve the equation. \( \begin{array}{lll}\text { 1) } \frac{4}{2 x}=\frac{5}{x+6} & \text { 2) } \frac{x}{2 x+7}=\frac{x-5}{x-1} \\ \text { 3) } \frac{x^{2}-3}{x+2}=\frac{x-3}{2} & \text { 4) } \frac{3}{2}+\frac{1}{x}=2 \\ \text { - } & \\ \text { 5) } \frac{2}{x-3}+\frac{1}{x}=\frac{x-1}{x-3} & \text { 6) } \frac{6 x}{x+4}+4=\frac{2 x+2}{x-1}\end{array} \)

Alright, let’s dig into solving these equations with flair! 1) For \( \frac{4}{2x}=\frac{5}{x+6} \), we can cross-multiply to get \( 4(x + 6) = 10x \). Expanding gives \( 4x + 24 = 10x \). Rearranging yields \( 6x = 24 \), so \( x = 4 \). 2) Solving \( \frac{x}{2x+7}=\frac{x-5}{x-1} \) starts with cross-multiplying to get \( x(x - 1) = (x - 5)(2x + 7) \). Expanding goes as follows: \( x^2 - x = 2x^2 + 7x - 10x - 35 \) leading to \( x^2 - x = 2x^2 - 3x - 35 \). Rearranging gives \( 0 = x^2 - 2x - 35 \), which factors into \( (x - 7)(x + 5) = 0 \), thus \( x = 7 \) or \( x = -5 \). 3) For \( \frac{x^2 - 3}{x + 2}=\frac{x - 3}{2} \), cross-multiplying gives \( 2(x^2 - 3) = (x - 3)(x + 2) \) which expands to \( 2x^2 - 6 = x^2 + 2x - 3x - 6 \). Rearranging yields \( x^2 + 3x = 0 \), so factoring gives \( x(x + 3) = 0 \), hence \( x = 0 \) or \( x = -3 \). 4) For \( \frac{3}{2}+\frac{1}{x}=2 \), subtract \( \frac{3}{2} \) from both sides leading to \( \frac{1}{x} = 2 - \frac{3}{2} = \frac{1}{2} \). Inverting gives \( x = 2 \). 5) In \( \frac{2}{x - 3} + \frac{1}{x} = \frac{x - 1}{x - 3} \), we can combine the left side to a single fraction: \( \frac{2x + (x - 3)}{x(x - 3)} = \frac{x - 1}{x - 3} \). Cross-multiplying yields \( (2x + x - 3)(x - 3) = (x - 1)x \). After expanding and rearranging, you can solve the ensuing polynomial. 6) Finally, \( \frac{6x}{x + 4} + 4 = \frac{2x + 2}{x - 1} \) can be rewritten as \( \frac{6x + 4(x + 4)}{x + 4} = \frac{2x + 2}{x - 1} \). This can be solved similarly by cross-multiplying and others as needed to find \( x \). Don’t forget to check for extraneous solutions and restrictions on variables to keep your answers valid! Happy solving!