\(1)\ \frac{x-6}{x-4}=0;\)
\(\phantom{1)\ }\begin{cases}x-6=0,\\[-1ex]\\x-4\ne0;\end{cases}\)
\(\phantom{1)\ }\begin{cases}x=6,\\[-1ex]\\x\ne4.\end{cases}\)
Ответ: \(6.\)
\(2)\ \frac{x-2}{x^2-4}=0;\)
\(\phantom{2)\ }\begin{cases}x-2=0,\\[-1ex]\\x^2-4\ne0;\end{cases}\)
\(\phantom{2)\ }\begin{cases}x=2,\\[-1ex]\\x\ne2,\\[-1ex]\\x\ne{-}2.\end{cases}\)
Ответ: корней нет.
\(3)\ \frac{x^2-4}{x-2}=0;\)
\(\phantom{3)\ }\begin{cases}x^2-4=0,\\[-1ex]\\x-2\ne0;\end{cases}\)
\(\phantom{3)\ }\begin{cases}x=2,\\[-1ex]\\x={-}2,\\[-1ex]\\x\ne2.\end{cases}\)
Ответ: \({-}2.\)
\(4)\ \frac{x-2}{x-2}=1;\)
\(\phantom{4)\ }1=1;\)
\(\phantom{4)\ }\begin{cases}1=1,\\[-1ex]\\x-2\ne0;\end{cases}\)
\(\phantom{4)\ }\begin{cases}1=1,\\[-1ex]\\x\ne2.\end{cases}\)
Ответ: корнем может быть любое число, кроме \(2.\)
\(5)\ \frac{2x^2+18}{x^2+9}=2;\)
\(\phantom{5)\ }\frac{ 2(x^2+9)}{x^2+9}=2;\)
\(\phantom{5)\ }2=2;\)
\(\phantom{5)\ }\begin{cases}2=2,\\[-1ex]\\x^2+9\ne0;\end{cases}\)
\(\phantom{5)\ }\begin{cases}2=2,\\[-1ex]\\x^2\ne{-}9.\end{cases}\)
Ответ: корнем может быть любое число.
\(6)\ \frac{\vphantom{^0}x}{x-5}+\frac{2x-9}{x-5}=0;\)
\(\phantom{6)\ }\frac{\vphantom{^0}x+2x-9}{x-5}=0;\)
\(\phantom{6)\ }\frac{\vphantom{^0}3x-9}{x-5}=0;\)
\(\phantom{6)\ }\begin{cases}3x-9=0,\\[-1ex]\\x-5\ne0;\end{cases}\)
\(\phantom{6)\ }\begin{cases}x=3,\\[-1ex]\\x\ne5.\end{cases}\)
Ответ: \(3.\)
\(7)\ \frac{\vphantom{^0}5x-7}{x+1}-\frac{x-5}{x+1}=0;\)
\(\phantom{7)\ }\frac{\vphantom{^0}5x-7-(x-5)}{x+1}=0;\)
\(\phantom{7)\ }\frac{\vphantom{^0}5x-7-x+5}{x+1}=0;\)
\(\phantom{7)\ }\frac{\vphantom{^0}4x-2}{x+1}=0;\)
\(\phantom{7)\ }\begin{cases}4x-2=0,\\[-1ex]\\x+1\ne0;\end{cases}\)
\(\phantom{7)\ }\begin{cases}x=0{,}5,\\[-1ex]\\x\ne{-}1.\end{cases}\)
Ответ: \(0{,}5.\)
\(8)\ \frac{2x+16}{x+3}-\frac{1-3x}{x+3}=0;\)
\(\phantom{8)\ }\frac{2x+16-(1-3x)}{x+3}=0;\)
\(\phantom{8)\ }\frac{2x+16-1+3x}{x+3}=0;\)
\(\phantom{8)\ }\frac{5x+15}{x+3}=0;\)
\(\phantom{8)\ }\frac{ 5(x+3)}{x+3}=0;\)
\(\phantom{8)\ }5\ne0.\)
Ответ: корней нет.
\(9)\ \frac{2}{x-1}+\frac{1}{x+1}=0;\)
\(\phantom{9)\ }\frac{ 2(x+1)+x-1}{ (x-1)(x+1)}=0;\)
\(\phantom{9)\ }\frac{2x+2+x-1}{ (x-1)(x+1)}=0;\)
\(\phantom{9)\ }\frac{3x+1}{ (x-1)(x+1)}=0;\)
\(\phantom{9)\ }\begin{cases}3x+1=0,\\[-1ex]\\(x-1)(x+1)\ne0;\end{cases}\)
\(\phantom{9)\ }\begin{cases}x={-}\frac{1}{3},\\[-1ex]\\x\ne1,\\[-1ex]\\x\ne{-}1.\end{cases}\)
Ответ: \({-}\frac{1}{3}.\)
\(10)\ \frac{3}{x-2}=\frac{4}{x+3};\)
\(\phantom{10)\ }\frac{3}{x-2}-\frac{4}{x+3}=0;\)
\(\phantom{10)\ }\frac{ 3(x+3)-4(x-2)}{ (x-2)(x+3)}=0;\)
\(\phantom{10)\ }\frac{3x+9-4x+8}{ (x-2)(x+3)}=0;\)
\(\phantom{10)\ }\frac{17-x}{ (x-2)(x+3)}=0;\)
\(\phantom{10)\ }\begin{cases}17-x=0,\\[-1ex]\\(x-2)(x+3)\ne0;\end{cases}\)
\(\phantom{10)\ }\begin{cases}x=17,\\[-1ex]\\x\ne2,\\[-1ex]\\x\ne{-}3.\end{cases}\)
Ответ: \(17.\)
\(11)\ \frac{\vphantom{^0}x}{x-6}=2;\)
\(\phantom{11)\ }\frac{x}{x-6}-2=0;\)
\(\phantom{11)\ }\frac{x-2(x-6)}{x-6}=0;\)
\(\phantom{11)\ }\frac{x-2x+12}{x-6}=0;\)
\(\phantom{11)\ }\frac{12-x}{x-6}=0;\)
\(\phantom{11)\ }\begin{cases}12-x=0,\\[-1ex]\\x-6\ne0;\end{cases}\)
\(\phantom{11)\ }\begin{cases}x=12,\\[-1ex]\\x\ne6.\end{cases}\)
Ответ: \(12.\)
\(12)\ \frac{x-4}{x-3}=\frac{2x+1}{2x-1};\)
\(\phantom{12)\ }\frac{x-4}{x-3}-\frac{2x+1}{2x-1}=0;\)
\(\phantom{12)\ }\frac{ (x-4)(2x-1)-(2x+1)(x-3)}{ (x-3)(2x-1)}=0;\)
\(\phantom{12)\ }\frac{2x^2-x-8x+4-(2x^2-6x+x-3)}{ (x-3)(2x-1)}=0;\)
\(\phantom{12)\ }\frac{2x^2-x-8x+4-2x^2+6x-x+3}{ (x-3)(2x-1)}=0;\)
\(\phantom{12)\ }\frac{7-4x}{ (x-3)(2x-1)}=0;\)
\(\phantom{12)\ }\begin{cases}7-4x=0,\\[-1ex]\\(x-3)(2x-1)\ne0;\end{cases}\)
\(\phantom{12)\ }\begin{cases}x=\frac{7}{4}=1\frac{3}{4},\\[-1ex]\\x\ne3,\\[-1ex]\\x\ne\frac{1}{2}.\end{cases}\)
Ответ: \(1\frac{3}{4}.\)
\(13)\ \frac{\vphantom{^0}x+8}{x}-\frac{6}{x-2}=0;\)
\(\phantom{13)\ }\frac{\vphantom{^0}(x+8)(x-2)-6x}{ x(x-2)}=0;\)
\(\phantom{13)\ }\frac{x^2-2x+8x-16-6x}{ x(x-2)}=0;\)
\(\phantom{13)\ }\frac{x^2-16}{ x(x-2)}=0;\)
\(\phantom{13)\ }\begin{cases}x^2-16=0,\\[-1ex]\\x(x-2)\ne0;\end{cases}\)
\(\phantom{13)\ }\begin{cases}x=4,\\[-1ex]\\x={-}4,\\[-1ex]\\x\ne0,\\[-1ex]\\x\ne2.\end{cases}\)
Ответ: \(4,\ {-}4.\)
\(14)\ \frac{2x}{x-5}-\frac{x^2+15x}{x^2-25}=0;\)
\(\phantom{14)\ }\frac{2x}{x-5}-\frac{x^2+15x}{ (x-5)(x+5)}=0;\)
\(\phantom{14)\ }\frac{ 2x(x+5)-(x^2+15x)}{ (x-5)(x+5)}=0;\)
\(\phantom{14)\ }\frac{2x^2+10x-x^2-15x}{ (x-5)(x+5)}=0;\)
\(\phantom{14)\ }\frac{x^2-5x}{ (x-5)(x+5)}=0;\)
\(\phantom{14)\ }\frac{ x(x-5)}{ (x-5)(x+5)}=0;\)
\(\phantom{14)\ }\frac{x}{x+5}=0;\)
\(\phantom{14)\ }\begin{cases}x=0,\\[-1ex]\\x+5\ne0;\end{cases}\)
\(\phantom{14)\ }\begin{cases}x=0,\\[-1ex]\\x\ne{-}5.\end{cases}\)
Ответ: \(0.\)
\(15)\ 3-\frac{2x^2-5x}{x^2-3x}=0;\)
\(\phantom{15)\ }\frac{ 3(x^2-3x)-(2x^2-5x)}{x^2-3x}=0;\)
\(\phantom{15)\ }\frac{3x^2-9x-2x^2+5x}{x^2-3x}=0;\)
\(\phantom{15)\ }\frac{x^2-4x}{x^2-3x}=0;\)
\(\phantom{15)\ }\begin{cases}x^2-4x=0,\\[-1ex]\\x^2-3x\ne0;\end{cases}\)
\(\phantom{15)\ }\begin{cases}x(x-4)=0,\\[-1ex]\\x(x-3)\ne0;\end{cases}\)
\(\phantom{15)\ }\begin{cases}x=0,\\[-1ex]\\x=4,\\[-1ex]\\x\ne0,\\[-1ex]\\x\ne3.\end{cases}\)
Ответ: \(4.\)
