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

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

$\frac{x+7}{x-7}-\frac{2x-3}{x-7}=0$
$\frac{x+7-(2x-3)}{x-7}=0$
$\frac{x+7-2x+3}{x-7}=0$
$\frac{10-x}{x-7}=0$
$\{\begin{array}{ll}10-x=0& \\ x-7\ne 0& \end{array}$
$\{\begin{array}{ll}x=10& \\ x\ne 7& \end{array}$
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

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

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

$\frac{x}{x+6}-\frac{36}{x^2+6x}=0$
$\frac{x}{x+6}-\frac{36}{x(x+6)}=0$
$\frac{x^2-36}{x(x+6)}=0$
$\{\begin{array}{ll}{x}^2-36=0& \\ x\ne 0& \\ x+6\ne 0& \end{array}$
$\{\begin{array}{ll}{x}^2=36& \\ x\ne 0& \\ x\ne -6& \end{array}$
$\{\begin{array}{ll}x=\pm 6& \\ x\ne 0& \\ x\ne -6& \end{array}$
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
$\{\begin{array}{ll}1=1& \\ 2x+1\ne 0& \end{array}$
$\{\begin{array}{ll}1=1& \\ 2x\ne -1& \end{array}$
$\{\begin{array}{ll}1=1& \\ x\ne -0,5& \end{array}$
Ответ: x − любое число, кроме −0,5.

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