$\frac{x - 4}{x - 5} + \frac{x - 6}{x + 5} = 2$ |*(x − 5)(x + 5)
$\begin{equation*} \begin{cases} (x - 4)(x + 5) + (x - 6)(x - 5) = 2(x^2 - 25) &\\ x ≠ ±5 & \end{cases} \end{equation*}$
$x^2 + x - 20 + x^2 - 11x + 30 = 2x^2 - 50$
−10x = −60
x = 6
$\begin{equation*} \begin{cases} x = 6 &\\ x ≠ ±5 & \end{cases} \end{equation*}$
Ответ: x = 6

$\frac{1}{2 - x} - 1 = \frac{1}{x - 2} - \frac{6 - x}{3x^2 - 12}$
$-\frac{1}{2 - x} - 1 = \frac{1}{x - 2} - \frac{6 - x}{3x^2 - 12}$
$\frac{2}{x - 2} - \frac{6 - x}{3(x^2 - 4)} + 1 = 0$ |*3(x − 2)(x + 2)
$\begin{equation*} \begin{cases} 6(x + 2) - (6 - x) + 3(x^2 - 4) = 0 &\\ x ≠ ±2 & \end{cases} \end{equation*}$
$6x + 12 - 6 + x + 3x^2 - 12 = 0$
$3x^2 + 7x - 6 = 0$
(3x − 2)(x + 3) = 0
$x_1 = -3$
$x_2 = \frac{2}{3}$
$\begin{equation*} \begin{cases} x_1 = -3, x_2 = \frac{2}{3} &\\ x ≠ ±2 & \end{cases} \end{equation*}$
Ответ:
$x_1 = -3$
$x_2 = \frac{2}{3}$

$\frac{7y - 3}{y - y^2} = \frac{1}{y - 1} - \frac{5}{y(y - 1)}$
$-\frac{7y - 3}{y^2 - y} = \frac{1}{y - 1} - \frac{5}{y(y - 1)}$
$\frac{1}{y - 1} - \frac{5}{y(y - 1)} + \frac{7y - 3}{y(y - 1)} = 0$
$\frac{1}{y - 1} + \frac{7y - 8}{y(y - 1)} = 0$ |* y(y − 1)
$\begin{equation*} \begin{cases} y + 7y - 8 = 0 &\\ y ≠ {0;1} & \end{cases} \end{equation*}$
$\begin{equation*} \begin{cases} 8y = 8 &\\ y ≠ {0;1} & \end{cases} \end{equation*}$
$\begin{equation*} \begin{cases} y = 1 &\\ y ≠ {0;1} & \end{cases} \end{equation*}$
Ответ: y = ∅ − нет решений

$\frac{3}{y - 2} + \frac{7}{y + 2} = \frac{10}{y}$ |* y(y − 2)(y + 2)
$\begin{equation*} \begin{cases} 3y(y + 2) + 7y(y - 2) = 10(y^2 - 4) &\\ y ≠ {0;±2} & \end{cases} \end{equation*}$
$3y^2 + 6y + 7y^2 - 14y = 10y^2 - 40$
$3y^2 + 6y + 7y^2 - 14y - 10y^2 + 40 = 0$
−8y = −40
y = 5
$\begin{equation*} \begin{cases} y = 5 &\\ y ≠ {0;±2} & \end{cases} \end{equation*}$
Ответ: y = 5

$\frac{x + 3}{x - 3} + \frac{x - 3}{x + 3} = 3\frac{1}{3}$ |* 3(x − 3)(x + 3)
$\begin{equation*} \begin{cases} 3(x + 3)^2 + 3(x - 3)^2 = 10(x^2 - 9) &\\ x ≠ ±3 & \end{cases} \end{equation*}$
$3(x^2 + 6x + 9) + 3(x^2 - 6x + 9) = 10x^2 - 90$
$3x^2 + 18x + 27 + 3x^2 - 18x + 27 - 10x^2 + 90 = 0$
$-4x^2 = -144$
$x^2 = 36$
x = ±6
$\begin{equation*} \begin{cases} x = ±6 &\\ x ≠ ±3 & \end{cases} \end{equation*}$
Ответ:
$x_1 = -6$;
$x_2 = 6$.

$\frac{5x + 7}{x - 2} - \frac{2x + 21}{x + 2} = 8\frac{2}{3}$ |*3(x − 2)(x + 2)
$\begin{equation*} \begin{cases} 3(5x + 7)(x + 2) - 3(2x + 21)(x - 2) = 26(x^2 - 4) &\\ x ≠ ±2 & \end{cases} \end{equation*}$
$3(5x^2 + 17x + 14) - 3(2x^2 + 17x - 42) = 26x^2 - 104$
$15x^2 + 21x + 42 - 6x^2 - 51x + 126 = 26x^2 - 104$
$9x^2 + 168 = 26x^2 - 104$
$17x^2 = 272$
$x^2 = 16$
x = ±4
$\begin{equation*} \begin{cases} x = ±4 &\\ x ≠ ±2 & \end{cases} \end{equation*}$
Ответ:
$x_1 = -4$;
$x_2 = 4$.