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

$\frac{x^2 - 2x + 1}{x^2 - 1} = 0$
$\frac{(x - 1)^2}{(x - 1)(x + 1)} = 0$
$\frac{x - 1}{x + 1} = 0$
$\begin{equation*} \begin{cases} x - 1 = 0 &\\ x^2 - 1 ≠ 0 & \end{cases} \end{equation*}$
$\begin{equation*} \begin{cases} x = 1 &\\ x^2 ≠ 1 & \end{cases} \end{equation*}$
$\begin{equation*} \begin{cases} x = 1 &\\ x ≠ ±1 & \end{cases} \end{equation*}$
Ответ: нет корней

$\frac{x + 7}{x - 7} - \frac{2x - 3}{x - 7} = 0$
$\frac{x + 7 - (2x - 3)}{x - 7} = 0$
$\frac{x + 7 - 2x + 3}{x - 7} = 0$
$\frac{10 - x}{x - 7} = 0$
$\begin{equation*} \begin{cases} 10 - x = 0 &\\ x - 7 ≠ 0 & \end{cases} \end{equation*}$
$\begin{equation*} \begin{cases} x = 10 &\\ x ≠ 7 & \end{cases} \end{equation*}$
Ответ: x = 10

$\frac{10 - 3x}{x + 8} + \frac{5x + 6}{x + 8} = 0$
$\frac{10 - 3x + 5x + 6}{x + 8} = 0$
$\frac{2x + 16}{x + 8} = 0$
$\frac{2(x + 8)}{x + 8} = 0$
2 ≠ 0
Ответ: нет корней

$\frac{x - 6}{x - 2} - \frac{x - 8}{x} = 0$
$\frac{x(x - 6) - (x - 8)(x - 2)}{x(x - 2)} = 0$
$\frac{x^2 - 6x - (x^2 - 8x - 2x + 16)}{x(x - 2)} = 0$
$\frac{x^2 - 6x - x^2 + 8x + 2x - 16}{x(x - 2)} = 0$
$\frac{4x - 16}{x(x - 2)} = 0$
$\begin{equation*} \begin{cases} 4x - 16 = 0 &\\ x - 2 ≠ 0 &\\ x ≠ 0 & \end{cases} \end{equation*}$
$\begin{equation*} \begin{cases} 4x = 16 &\\ x ≠ 2 &\\ x ≠ 0 & \end{cases} \end{equation*}$
$\begin{equation*} \begin{cases} x = 4 &\\ x ≠ 2 &\\ x ≠ 0 & \end{cases} \end{equation*}$
Ответ: x = 4

$\frac{2x - 4}{x} - \frac{3x + 1}{x} + \frac{x + 5}{x} = 0$
$\frac{2x - 4 - (3x + 1) + x + 5}{x} = 0$
$\frac{2x - 4 - 3x - 1 + x + 5}{x} = 0$
$\frac{0}{x} = 0$
$\begin{equation*} \begin{cases} 0 = 0 &\\ x ≠ 0 & \end{cases} \end{equation*}$
Ответ: x − любое число, кроме 0.

$\frac{x}{x + 6} - \frac{36}{x^2 + 6x} = 0$
$\frac{x}{x + 6} - \frac{36}{x(x + 6)} = 0$
$\frac{x^2 - 36}{x(x + 6)} = 0$
$\begin{equation*} \begin{cases} x^2 - 36 = 0 &\\ x ≠ 0 &\\ x + 6 ≠ 0 & \end{cases} \end{equation*}$
$\begin{equation*} \begin{cases} x^2 = 36 &\\ x ≠ 0 &\\ x ≠ -6 & \end{cases} \end{equation*}$
$\begin{equation*} \begin{cases} x = ±6 &\\ x ≠ 0 &\\ x ≠ -6 & \end{cases} \end{equation*}$
Ответ: x = 6

$\frac{2x^2 + 3x + 1}{2x + 1} - x = 1$
$\frac{2x^2 + 3x + 1 - x(2x + 1)}{2x + 1} = 1$
$\frac{2x^2 + 3x + 1 - 2x^2 - x}{2x + 1} = 1$
$\frac{2x + 1}{2x + 1} = 1$
1 = 1
$\begin{equation*} \begin{cases} 1 = 1 &\\ 2x + 1 ≠ 0 & \end{cases} \end{equation*}$
$\begin{equation*} \begin{cases} 1 = 1 &\\ 2x ≠ -1 & \end{cases} \end{equation*}$
$\begin{equation*} \begin{cases} 1 = 1 &\\ x ≠ -0,5 & \end{cases} \end{equation*}$
Ответ: x − любое число, кроме −0,5.

$\frac{4}{x - 1} - \frac{4}{x + 1} = 1$
$\frac{4}{x - 1} - \frac{4}{x + 1} - 1 = 0$
$\frac{4(x + 1) - 4(x - 1) - (x - 1)(x + 1)}{(x - 1)(x + 1)} = 0$
$\frac{4x + 4 - 4x + 4 - (x^2 - 1)}{(x - 1)(x + 1)} = 0$
$\frac{8 - x^2 +1}{(x - 1)(x + 1)} = 0$
$\frac{9 - x^2}{x^2 - 1} = 0$
$\begin{equation*} \begin{cases} 9 - x^2 = 0 &\\ x^2 - 1 ≠ 0 & \end{cases} \end{equation*}$
$\begin{equation*} \begin{cases} x^2 = 9 &\\ x^2 ≠ 1 & \end{cases} \end{equation*}$
$\begin{equation*} \begin{cases} x = ±3 &\\ x ≠ ±1 & \end{cases} \end{equation*}$
Ответ: x = ±3