$\frac{x - 6}{x - 4} = 0$
$\begin{equation*} \begin{cases} x - 6 = 0 &\\ x - 4 ≠ 0 & \end{cases} \end{equation*}$
$\begin{equation*} \begin{cases} x = 6 &\\ x ≠ 4 & \end{cases} \end{equation*}$
Ответ: x = 6

$\frac{x - 2}{x^2 - 4} = 0$
$\begin{equation*} \begin{cases} x - 2 = 0 &\\ x^2 - 4 ≠ 0 & \end{cases} \end{equation*}$
$\begin{equation*} \begin{cases} x = 2 &\\ x^2 ≠ 4 & \end{cases} \end{equation*}$
$\begin{equation*} \begin{cases} x = 2 &\\ x ≠ ±2 & \end{cases} \end{equation*}$
Ответ: нет корней

$\frac{x^2 - 4}{x - 2} = 0$
$\begin{equation*} \begin{cases} x^2 - 4 = 0 &\\ x - 2 ≠ 0 & \end{cases} \end{equation*}$
$\begin{equation*} \begin{cases} x^2 = 4 &\\ x ≠ 2 & \end{cases} \end{equation*}$
$\begin{equation*} \begin{cases} x = ±2 &\\ x ≠ 2 & \end{cases} \end{equation*}$
Ответ: x = −2

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

$\frac{2x^2 + 18}{x^2 + 9} = 2$
$\frac{2(x^2 + 9)}{x^2 + 9} = 2$
2 = 2
$\begin{equation*} \begin{cases} 2 = 2 &\\ x^2 + 9 ≠ 0 & \end{cases} \end{equation*}$
$\begin{equation*} \begin{cases} 2 = 2 &\\ x^2 ≠ -9 & \end{cases} \end{equation*}$
Ответ: x − любое число.

$\frac{x}{x - 5} + \frac{2x - 9}{x - 5} = 0$
$\frac{x + 2x - 9}{x - 5} = 0$
$\frac{3x - 9}{x - 5} = 0$
$\begin{equation*} \begin{cases} 3x - 9 = 0 &\\ x - 5 ≠ 0 & \end{cases} \end{equation*}$
$\begin{equation*} \begin{cases} 3x = 9 &\\ x ≠ 5 & \end{cases} \end{equation*}$
$\begin{equation*} \begin{cases} x = 3 &\\ x ≠ 5 & \end{cases} \end{equation*}$
Ответ: x = 3

$\frac{5x - 7}{x + 1} - \frac{x - 5}{x + 1} = 0$
$\frac{5x - 7 - (x - 5)}{x + 1} = 0$
$\frac{5x - 7 - x + 5}{x + 1} = 0$
$\frac{4x - 2}{x + 1} = 0$
$\begin{equation*} \begin{cases} 4x - 2 = 0 &\\ x + 1 ≠ 0 & \end{cases} \end{equation*}$
$\begin{equation*} \begin{cases} 4x = 2 &\\ x ≠ -1 & \end{cases} \end{equation*}$
$\begin{equation*} \begin{cases} x = 0,5 &\\ x ≠ -1 & \end{cases} \end{equation*}$
Ответ: x = 0,5

$\frac{2x + 16}{x + 3} - \frac{1 - 3x}{x + 3} = 0$
$\frac{2x + 16 - (1 - 3x)}{x + 3} = 0$
$\frac{2x + 16 - 1 + 3x}{x + 3} = 0$
$\frac{5x + 15}{x + 3} = 0$
$\frac{5(x + 3)}{x + 3} = 0$
5 ≠ 0
Ответ: нет корней

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

$\frac{3}{x - 2} = \frac{4}{x + 3}$
$\frac{3}{x - 2} - \frac{4}{x + 3} = 0$
$\frac{3(x + 3) - 4(x - 2)}{(x - 2)(x + 3)} = 0$
$\frac{3x + 9 - 4x + 8}{(x - 2)(x + 3)} = 0$
$\frac{17 - x}{(x - 2)(x + 3)} = 0$
$\begin{equation*} \begin{cases} x - 2 ≠ 0 &\\ x + 3 ≠ 0 &\\ 17 - x = 0 & \end{cases} \end{equation*}$
$\begin{equation*} \begin{cases} x ≠ 2 &\\ x ≠ -3 &\\ x = 17 & \end{cases} \end{equation*}$
Ответ: x = 17

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

$\frac{x - 4}{x - 3} = \frac{2x + 1}{2x - 1}$
$\frac{x - 4}{x - 3} - \frac{2x + 1}{2x - 1} = 0$
$\frac{(x - 4)(2x - 1) - (2x + 1)(x - 3)}{(x - 3)(2x - 1)} = 0$
$\frac{2x^2 - 8x - x + 4 - (2x^2 + x - 6x - 3)}{(x - 3)(2x - 1)} = 0$
$\frac{2x^2 - 9x + 4 - (2x^2 - 5x - 3)}{(x - 3)(2x - 1)} = 0$
$\frac{2x^2 - 9x + 4 - 2x^2 + 5x + 3}{(x - 3)(2x - 1)} = 0$
$\frac{7 - 4x}{(x - 3)(2x - 1)} = 0$
$\begin{equation*} \begin{cases} x - 3 ≠ 0 &\\ 2x - 1 ≠ 0 &\\ 7 - 4x = 0 & \end{cases} \end{equation*}$
$\begin{equation*} \begin{cases} x ≠ 3 &\\ 2x ≠ 1 &\\ 4x = 7 & \end{cases} \end{equation*}$
$\begin{equation*} \begin{cases} x ≠ 3 &\\ x ≠ \frac{1}{2} &\\ x = \frac{7}{4} = 1\frac{3}{4} & \end{cases} \end{equation*}$
Ответ: $x = 1\frac{3}{4}$

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

$\frac{2x}{x - 5} - \frac{x^2 + 15x}{x^2 - 25} = 0$
$\frac{2x}{x - 5} - \frac{x^2 + 15x}{(x - 5)(x + 5)} = 0$
$\frac{2x(x + 5) - (x^2 + 15x)}{(x - 5)(x + 5)} = 0$
$\frac{2x^2 + 10x - x^2 - 15x}{(x - 5)(x + 5)} = 0$
$\frac{x^2 - 5x}{(x - 5)(x + 5)} = 0$
$\frac{x(x - 5)}{(x - 5)(x + 5)} = 0$
$\frac{x}{x + 5} = 0$
$\begin{equation*} \begin{cases} x = 0 &\\ x + 5 ≠ 0 & \end{cases} \end{equation*}$
$\begin{equation*} \begin{cases} x = 0 &\\ x ≠ -5 & \end{cases} \end{equation*}$
Ответ: x = 0

$3 - \frac{2x^2 - 5x}{x^2 - 3x} = 0$
$\frac{3(x^2 - 3x) - (2x^2 - 5x)}{x^2 - 3x} = 0$
$\frac{3x^2 - 9x - 2x^2 + 5x}{x^2 - 3x} = 0$
$\frac{x^2 - 4x}{x^2 - 3x} = 0$
$\frac{x(x - 4)}{x(x - 3)} = 0$
$\frac{x - 4}{x - 3} = 0$
$\begin{equation*} \begin{cases} x - 4 = 0 &\\ x - 3 ≠ 0 & \end{cases} \end{equation*}$
$\begin{equation*} \begin{cases} x = 4 &\\ x ≠ 3 & \end{cases} \end{equation*}$
Ответ: x = 4