$\frac{x - 2}{x + 1} - \frac{5}{1 - x} = \frac{x^2 + 27}{x^2 - 1}$
$\frac{x - 2}{x + 1} + \frac{5}{x - 1} - \frac{x^2 + 27}{(x - 1)(x + 1)} = 0$
$\frac{(x - 2)(x - 1) + 5(x + 1) - (x^2 + 27)}{(x - 1)(x + 1)} = 0$
$\frac{x^2 - 2x - x + 2 + 5x + 5 - x^2 - 27}{(x - 1)(x + 1)} = 0$
$\frac{2x - 20}{(x - 1)(x + 1)} = 0$
$\begin{equation*} \begin{cases} x - 1 ≠ 0 &\\ x + 1 ≠ 0 &\\ 2x - 20 = 0 & \end{cases} \end{equation*}$
$\begin{equation*} \begin{cases} x ≠ 1 &\\ x ≠ -1 &\\ 2x = 20 & \end{cases} \end{equation*}$
$\begin{equation*} \begin{cases} x ≠ 1 &\\ x ≠ -1 &\\ x = 10 & \end{cases} \end{equation*}$
Ответ: x = 10

$\frac{3x + 1}{3x - 1} - \frac{3x - 1}{3x + 1} = \frac{6}{1 - 9x^2}$
$\frac{3x + 1}{3x - 1} - \frac{3x - 1}{3x + 1} - \frac{6}{1 - 9x^2} = 0$
$\frac{3x + 1}{3x - 1} - \frac{3x - 1}{3x + 1} + \frac{6}{9x^2 - 1} = 0$
$\frac{3x + 1}{3x - 1} - \frac{3x - 1}{3x + 1} + \frac{6}{(3x - 1)(3x + 1)} = 0$
$\frac{(3x + 1)^2 - (3x - 1)^2 + 6}{(3x - 1)(3x + 1)} = 0$
$\frac{9x^2 + 6x + 1 - (9x^2 - 6x + 1) + 6}{(3x - 1)(3x + 1)} = 0$
$\frac{9x^2 + 6x + 1 - 9x^2 + 6x - 1 + 6}{(3x - 1)(3x + 1)} = 0$
$\frac{12x + 6}{(3x - 1)(3x + 1)} = 0$
$\begin{equation*} \begin{cases} 3x - 1 ≠ 0 &\\ 3x + 1 ≠ 0 &\\ 12x + 6 = 0 & \end{cases} \end{equation*}$
$\begin{equation*} \begin{cases} 3x ≠ 1 &\\ 3x ≠ -1 &\\ 12x = -6 & \end{cases} \end{equation*}$
$\begin{equation*} \begin{cases} x ≠ \frac{1}{3} &\\ x ≠ -\frac{1}{3} &\\ x = -\frac{1}{2} & \end{cases} \end{equation*}$
Ответ: $x = -\frac{1}{2}$

$\frac{4}{x - 3} + \frac{1}{x} = \frac{5}{x - 2}$
$\frac{4}{x - 3} + \frac{1}{x} - \frac{5}{x - 2} = 0$
$\frac{4x(x - 2) + (x - 3)(x - 2) - 5x(x - 3)}{x(x - 3)(x - 2)} = 0$
$\frac{4x^2 - 8x + x^2 - 3x - 2x + 6 - 5x^2 + 15x}{x(x - 3)(x - 2)} = 0$
$\frac{2x + 6}{x(x - 3)(x - 2)} = 0$
$\begin{equation*} \begin{cases} x ≠ 0 &\\ x - 3 ≠ 0 &\\ x - 2 ≠ 0 &\\ 2x + 6 = 0 & \end{cases} \end{equation*}$
$\begin{equation*} \begin{cases} x ≠ 0 &\\ x ≠ 3 &\\ x ≠ 2 &\\ 2x = -6 & \end{cases} \end{equation*}$
$\begin{equation*} \begin{cases} x ≠ 0 &\\ x ≠ 3 &\\ x ≠ 2 &\\ x = -3 & \end{cases} \end{equation*}$
Ответ: x = −3

$\frac{2x^2 - 2x}{x^2 - 4} + \frac{6}{x + 2} = \frac{x + 2}{x - 2}$
$\frac{2x^2 - 2x}{(x - 2)(x + 2)} + \frac{6}{x + 2} - \frac{x + 2}{x - 2} = 0$
$\frac{2x^2 - 2x + 6(x - 2) - (x + 2)^2}{(x - 2)(x + 2)} = 0$
$\frac{2x^2 - 2x + 6x - 12 - (x^2 + 4x + 4)}{(x - 2)(x + 2)} = 0$
$\frac{2x^2 + 4x - 12 - x^2 - 4x - 4}{(x - 2)(x + 2)} = 0$
$\frac{x^2 - 16}{(x - 2)(x + 2)} = 0$
$\begin{equation*} \begin{cases} x - 2 ≠ 0 &\\ x + 2 ≠ 0 &\\ x^2 - 16 = 0 & \end{cases} \end{equation*}$
$\begin{equation*} \begin{cases} x ≠ 2 &\\ x ≠ -2 &\\ x^2 = 16 & \end{cases} \end{equation*}$
$\begin{equation*} \begin{cases} x ≠ 2 &\\ x ≠ -2 &\\ x = ±4 & \end{cases} \end{equation*}$
Ответ: x = ±4

$\frac{7}{x^2 + 2x} + \frac{x + 1}{x^2 - 2x} = \frac{x + 4}{x^2 - 4}$
$\frac{7}{x^2 + 2x} + \frac{x + 1}{x^2 - 2x} - \frac{x + 4}{x^2 - 4} = 0$
$\frac{7}{x(x + 2)} + \frac{x + 1}{x(x - 2)} - \frac{x + 4}{(x - 2)(x + 2)} = 0$
$\frac{7(x - 2) + (x + 1)(x + 2) - x(x + 4)}{x(x - 2)(x + 2)} = 0$
$\frac{7x - 14 + x^2 + x + 2x + 2 - x^2 - 4x}{x(x - 2)(x + 2)} = 0$
$\frac{6x - 12}{x(x - 2)(x + 2)} = 0$
$\frac{6(x - 2)}{x(x - 2)(x + 2)} = 0$
$\frac{6}{x(x - 2)(x + 2)} = 0$
6 ≠ 0
Ответ: нет корней

$\frac{x^2 - 9x + 50}{x^2 - 5x} = \frac{x + 1}{x - 5} + \frac{x - 5}{x}$
$\frac{x^2 - 9x + 50}{x(x - 5)} - \frac{x + 1}{x - 5} - \frac{x - 5}{x} = 0$
$\frac{x^2 - 9x + 50 - x(x + 1) - (x - 5)^2}{x(x - 5)} = 0$
$\frac{x^2 - 9x + 50 - x^2 - x - (x^2 - 10x + 25)}{x(x - 5)} = 0$
$\frac{x^2 - 9x + 50 - x^2 - x - x^2 + 10x - 25}{x(x - 5)} = 0$
$\frac{-x^2 + 25}{x(x - 5)} = 0$
$\begin{equation*} \begin{cases} x ≠ 0 &\\ x - 5 ≠ 0 &\\ -x^2 + 25 = 0 & \end{cases} \end{equation*}$
$\begin{equation*} \begin{cases} x ≠ 0 &\\ x ≠ 5 &\\ x^2 = 25 & \end{cases} \end{equation*}$
$\begin{equation*} \begin{cases} x ≠ 0 &\\ x ≠ 5 &\\ x = ±5 & \end{cases} \end{equation*}$
Ответ: x = −5