$\frac{x-6}{x-4}=0$
$\{\begin{array}{ll}x-6=0& \\ x-4\ne 0& \end{array}$
$\{\begin{array}{ll}x=6& \\ x\ne 4& \end{array}$
Ответ: x = 6

$\frac{x-2}{x^2-4}=0$
$\{\begin{array}{ll}x-2=0& \\ x^2-4\ne 0& \end{array}$
$\{\begin{array}{ll}x=2& \\ x^2\ne 4& \end{array}$
$\{\begin{array}{ll}x=2& \\ x\ne \pm 2& \end{array}$
Ответ: нет корней

$\frac{x^2-4}{x-2}=0$
$\{\begin{array}{ll}{x}^2-4=0& \\ x-2\ne 0& \end{array}$
$\{\begin{array}{ll}{x}^2=4& \\ x\ne 2& \end{array}$
$\{\begin{array}{ll}x=\pm 2& \\ x\ne 2& \end{array}$
Ответ: x = −2

$\frac{x-2}{x-2}=1$
1 = 1
$\{\begin{array}{ll}1=1& \\ x-2\ne 0& \end{array}$
$\{\begin{array}{ll}1=1& \\ x\ne 2& \end{array}$
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
$\{\begin{array}{ll}2=2& \\ x^2+9\ne 0& \end{array}$
$\{\begin{array}{ll}2=2& \\ x^2\ne -9& \end{array}$
Ответ: x − любое число.

$\frac{x}{x-5}+\frac{2x-9}{x-5}=0$
$\frac{x+2x-9}{x-5}=0$
$\frac{3x-9}{x-5}=0$
$\{\begin{array}{ll}3x-9=0& \\ x-5\ne 0& \end{array}$
$\{\begin{array}{ll}3x=9& \\ x\ne 5& \end{array}$
$\{\begin{array}{ll}x=3& \\ x\ne 5& \end{array}$
Ответ: x = 3

$\frac{5x-7}{x+1}-\frac{x-5}{x+1}=0$
$\frac{5x-7-(x-5)}{x+1}=0$
$\frac{5x-7-x+5}{x+1}=0$
$\frac{4x-2}{x+1}=0$
$\{\begin{array}{ll}4x-2=0& \\ x+1\ne 0& \end{array}$
$\{\begin{array}{ll}4x=2& \\ x\ne -1& \end{array}$
$\{\begin{array}{ll}x=0,5& \\ x\ne -1& \end{array}$
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

$\frac{2}{x-1}+\frac{1}{x+1}=0$
$\frac{2(x+1)+x-1}{(x-1)(x+1)}=0$
$\frac{2x+2+x-1}{x^2-1}=0$
$\frac{3x+1}{x^2-1}=0$
$\{\begin{array}{ll}3x+1=0& \\ x^2-1\ne 0& \end{array}$
$\{\begin{array}{ll}3x=-1& \\ x^2\ne 1& \end{array}$
$\{\begin{array}{ll}x=-\frac{1}{3}& \\ x\ne \pm 1& \end{array}$
Ответ: $x=-\frac{1}{3}$

$\frac{3}{x-2}=\frac{4}{x+3}$
$\frac{3}{x-2}-\frac{4}{x+3}=0$
$\frac{3(x+3)-4(x-2)}{(x-2)(x+3)}=0$
$\frac{3x+9-4x+8}{(x-2)(x+3)}=0$
$\frac{17-x}{(x-2)(x+3)}=0$
$\{\begin{array}{ll}x-2\ne 0& \\ x+3\ne 0& \\ 17-x=0& \end{array}$
$\{\begin{array}{ll}x\ne 2& \\ x\ne -3& \\ x=17& \end{array}$
Ответ: x = 17

$\frac{x}{x-6}=2$
$\frac{x}{x-6}-2=0$
$\frac{x-2(x-6)}{x-6}=0$
$\frac{x-2x+12}{x-6}=0$
$\frac{12-x}{x-6}=0$
$\{\begin{array}{ll}12-x=0& \\ x-6\ne 0& \end{array}$
$\{\begin{array}{ll}x=12& \\ x\ne 6& \end{array}$
Ответ: x = 12

$\frac{x-4}{x-3}=\frac{2x+1}{2x-1}$
$\frac{x-4}{x-3}-\frac{2x+1}{2x-1}=0$
$\frac{(x-4)(2x-1)-(2x+1)(x-3)}{(x-3)(2x-1)}=0$
$\frac{2x^2-8x-x+4-(2x^2+x-6x-3)}{(x-3)(2x-1)}=0$
$\frac{2x^2-9x+4-(2x^2-5x-3)}{(x-3)(2x-1)}=0$
$\frac{2x^2-9x+4-2x^2+5x+3}{(x-3)(2x-1)}=0$
$\frac{7-4x}{(x-3)(2x-1)}=0$
$\{\begin{array}{ll}x-3\ne 0& \\ 2x-1\ne 0& \\ 7-4x=0& \end{array}$
$\{\begin{array}{ll}x\ne 3& \\ 2x\ne 1& \\ 4x=7& \end{array}$
$\{\begin{array}{ll}x\ne 3& \\ x\ne \frac{1}{2}& \\ x=\frac{7}{4}=1\frac{3}{4}& \end{array}$
Ответ: $x=1\frac{3}{4}$

$\frac{x+8}{x}-\frac{6}{x-2}=0$
$\frac{(x+8)(x-2)-6x}{x(x-2)}=0$
$\frac{x^2+8x-2x-16-6x}{x(x-2)}=0$
$\frac{x^2-16}{x(x-2)}=0$
$\{\begin{array}{ll}x\ne 0& \\ x-2\ne 0& \\ x^2-16=0& \end{array}$
$\{\begin{array}{ll}x\ne 0& \\ x\ne 2& \\ x^2=16& \end{array}$
$\{\begin{array}{ll}x\ne 0& \\ x\ne 2& \\ x=\pm 4& \end{array}$
Ответ: x = ±4

$\frac{2x}{x-5}-\frac{x^2+15x}{x^2-25}=0$
$\frac{2x}{x-5}-\frac{x^2+15x}{(x-5)(x+5)}=0$
$\frac{2x(x+5)-(x^2+15x)}{(x-5)(x+5)}=0$
$\frac{2x^2+10x-x^2-15x}{(x-5)(x+5)}=0$
$\frac{x^2-5x}{(x-5)(x+5)}=0$
$\frac{x(x-5)}{(x-5)(x+5)}=0$
$\frac{x}{x+5}=0$
$\{\begin{array}{ll}x=0& \\ x+5\ne 0& \end{array}$
$\{\begin{array}{ll}x=0& \\ x\ne -5& \end{array}$
Ответ: x = 0

$3-\frac{2x^2-5x}{x^2-3x}=0$
$\frac{3(x^2-3x)-(2x^2-5x)}{x^2-3x}=0$
$\frac{3x^2-9x-2x^2+5x}{x^2-3x}=0$
$\frac{x^2-4x}{x^2-3x}=0$
$\frac{x(x-4)}{x(x-3)}=0$
$\frac{x-4}{x-3}=0$
$\{\begin{array}{ll}x-4=0& \\ x-3\ne 0& \end{array}$
$\{\begin{array}{ll}x=4& \\ x\ne 3& \end{array}$
Ответ: x = 4