\(1)\ \frac{x^2-1}{x^2-2x+1}=0;\)
\(\phantom{1)\ }\frac{x^2-1}{ (x-1)^2}=0;\)
\(\phantom{1)\ }\begin{cases}x^2-1=0,\\[-1ex]\\(x-1)^2\ne0;\end{cases}\)
\(\phantom{1)\ }\begin{cases}x=1,\\[-1ex]\\x={-}1,\\[-1ex]\\x\ne1.\end{cases}\)
Ответ: \({-}1.\)
\(2)\ \frac{x^2-2x+1}{x^2-1}=0;\)
\(\phantom{2)\ }\frac{ (x-1)^2}{x^2-1}=0;\)
\(\phantom{2)\ }\begin{cases}(x-1)^2=0,\\[-1ex]\\x^2-1\ne0;\end{cases}\)
\(\phantom{2)\ }\begin{cases}x=1,\\[-1ex]\\x\ne1,\\[-1ex]\\x\ne{-}1.\end{cases}\)
Ответ: корней нет.
\(3)\ \frac{\vphantom{^0}x+7}{x-7}-\frac{2x-3}{x-7}=0;\)
\(\phantom{3)\ }\frac{\vphantom{^0}x+7-(2x-3)}{x-7}=0;\)
\(\phantom{3)\ }\frac{\vphantom{^0}x+7-2x+3}{x-7}=0;\)
\(\phantom{3)\ }\frac{\vphantom{^0}10-x}{x-7}=0;\)
\(\phantom{3)\ }\begin{cases}10-x=0,\\[-1ex]\\x-7\ne0;\end{cases}\)
\(\phantom{3)\ }\begin{cases}x=10,\\[-1ex]\\x\ne7.\end{cases}\)
Ответ: \(10.\)
\(4)\ \frac{10-3x}{x+8}+\frac{5x+6}{x+8}=0;\)
\(\phantom{4)\ }\frac{10-3x+5x+6}{x+8}=0;\)
\(\phantom{4)\ }\frac{2x+16}{x+8}=0;\)
\(\phantom{4)\ }\frac{ 2(x+8)}{x+8}=0;\)
\(\phantom{4)\ }2\ne0.\)
Ответ: корней нет.
\(5)\ \frac{x-6}{x-2}-\frac{x-8}{x}=0;\)
\(\phantom{5)\ }\frac{ x(x-6)-(x-8)(x-2)}{ x(x-2)}=0;\)
\(\phantom{5)\ }\frac{x^2-6x-(x^2-2x-8x+16)}{ x(x-2)}=0;\)
\(\phantom{5)\ }\frac{x^2-6x-x^2+2x+8x-16}{ x(x-2)}=0;\)
\(\phantom{5)\ }\frac{4x-16}{ x(x-2)}=0;\)
\(\phantom{5)\ }\begin{cases}4x-16=0,\\[-1ex]\\x(x-2)\ne0;\end{cases}\)
\(\phantom{5)\ }\begin{cases}x=4,\\[-1ex]\\x\ne0,\\[-1ex]\\x\ne2.\end{cases}\)
Ответ: \(4.\)
\(6)\ \frac{\vphantom{^0}2x-4}{x}-\frac{3x+1}{x}+\frac{x+5}{x}=0;\)
\(\phantom{6)\ }\frac{\vphantom{^0}2x-4-(3x+1)+x+5}{x}=0;\)
\(\phantom{6)\ }\frac{2x-4-3x-1+x+5}{x}=0;\)
\(\phantom{6)\ }\frac{\vphantom{^0}0}{x}=0;\)
\(\phantom{6)\ }\begin{cases}0=0,\\[-1ex]\\x\ne0.\end{cases}\)
Ответ: корнем может быть любое число, кроме \(0.\)
\(7)\ \frac{\vphantom{^0}x}{x+6}-\frac{36}{x^2+6x}=0;\)
\(\phantom{7)\ }\frac{\vphantom{^0}x}{x+6}-\frac{36}{ x(x+6)}=0;\)
\(\phantom{7)\ }\frac{x^2-36}{ x(x+6)}=0;\)
\(\phantom{7)\ }\frac{ (x-6)(x+6)}{ x(x+6)}=0;\)
\(\phantom{7)\ }\frac{x-6}{x}=0;\)
\(\phantom{7)\ }\begin{cases}x-6=0,\\[-1ex]\\x\ne0;\end{cases}\)
\(\phantom{7)\ }\begin{cases}x=6,\\[-1ex]\\x\ne0.\end{cases}\)
Ответ: \(6.\)
\(8)\ \frac{2x^2+3x+1}{2x+1}-x=1;\)
\(\phantom{8)\ }\frac{2x^2+3x+1}{2x+1}-x-1=0;\)
\(\phantom{8)\ }\frac{2x^2+3x+1-x(2x+1)-(2x+1)}{2x+1}=0;\)
\(\phantom{8)\ }\frac{2x^2+3x+1-2x^2-x-2x-1}{2x+1}=0;\)
\(\phantom{8)\ }\frac{0}{2x+1}=0;\)
\(\phantom{8)\ }\begin{cases}0=0,\\[-1ex]\\2x+1\ne0;\end{cases}\)
\(\phantom{8)\ }\begin{cases}0=0,\\[-1ex]\\x\ne{-}0{,}5.\end{cases}\)
Ответ: корнем может быть любое число, кроме \({-}0{,}5.\)
\(9)\ \frac{4}{x-1}-\frac{4}{x+1}=1;\)
\(\phantom{9)\ }\frac{4}{x-1}-\frac{4}{x+1}-1=0;\)
\(\phantom{9)\ }\frac{ 4(x+1)-4(x-1)-(x-1)(x+1)}{ (x-1)(x+1)}=0;\)
\(\phantom{9)\ }\frac{4x+4-4x+4-(x^2-1)}{ (x-1)(x+1)}=0;\)
\(\phantom{9)\ }\frac{4x+4-4x+4-x^2+1}{ (x-1)(x+1)}=0;\)
\(\phantom{9)\ }\frac{9-x^2}{ (x-1)(x+1)}=0;\)
\(\phantom{9)\ }\begin{cases}9-x^2=0,\\[-1ex]\\(x-1)(x+1)\ne0;\end{cases}\)
\(\phantom{9)\ }\begin{cases}x=3,\\[-1ex]\\x={-}3,\\[-1ex]\\x\ne1,\\[-1ex]\\x\ne{-}1.\end{cases}\)
Ответ: \(3,\ {-}3.\)
