\(1)\ \frac{5}{x^2-4}+\frac{2x}{x+2}=2;\)
\(\phantom{1)\ }\frac{5}{ (x-2)(x+2)}+\frac{2x}{x+2}-2=0;\)
\(\phantom{1)\ }\frac{5+2x(x-2)-2(x-2)(x+2)}{ (x-2)(x+2)}=0;\)
\(\phantom{1)\ }\frac{5+2x^2-4x-2x^2+8}{ (x-2)(x+2)}=0;\)
\(\phantom{1)\ }\frac{\vphantom{^0}13-4x}{ (x-2)(x+2)}=0;\)
\(\phantom{1)\ }\begin{cases}13-4x=0,\\[-1ex]\\(x-2)(x+2)\ne0;\end{cases}\)
\(\phantom{1)\ }\begin{cases}x=\frac{13}{4}=3\frac{1}{4},\\[-1ex]\\x\ne2,\\[-1ex]\\x\ne{-}2.\end{cases}\)
Ответ: \(3\frac{1}{4}.\)
\(2)\ \frac{2}{6x+1}+\frac{3}{6x-1}=\frac{30x+9}{36x^2-1};\)
\(\phantom{2)\ }\frac{2}{6x+1}+\frac{3}{6x-1}-\frac{30x+9}{ (6x-1)(6x+1)}=0;\)
\(\phantom{2)\ }\frac{\vphantom{^0} 2(6x-1)+3(6x+1)-(30x+9)}{ (6x-1)(6x+1)}=0;\)
\(\phantom{2)\ }\frac{\vphantom{^0}12x-2+18x+3-30x-9}{ (6x-1)(6x+1)}=0;\)
\(\phantom{2)\ }\frac{\vphantom{^0}{-}8}{ (6x-1)(6x+1)}=0.\)
Ответ: корней нет.
\(3)\ \frac{6x+14}{x^2-9}+\frac{7}{x^2+3x}=\frac{6}{x-3};\)
\(\phantom{3)\ }\frac{6x+14}{ (x-3)(x+3)}+\frac{7}{ x(x+3)}-\frac{6}{x-3}=0;\)
\(\phantom{3)\ }\frac{ x(6x+14)+7(x-3)-6x(x+3)}{ x(x-3)(x+3)}=0;\)
\(\phantom{3)\ }\frac{6x^2+14x+7x-21-6x^2-18x}{ x(x-3)(x+3)}=0;\)
\(\phantom{3)\ }\frac{\vphantom{^0}3x-21}{ x(x-3)(x+3)}=0;\)
\(\phantom{3)\ }\begin{cases}3x-21=0,\\[-1ex]\\x(x-3)(x+3)\ne0;\end{cases}\)
\(\phantom{3)\ }\begin{cases}x=7,\\[-1ex]\\x\ne0,\\[-1ex]\\x\ne3,\\[-1ex]\\x\ne{-}3.\end{cases}\)
Ответ: \(7.\)
\(4)\ \frac{2y^2+5}{1-y^2}+\frac{y+1}{y-1}=\frac{4}{y+1};\)
\(\phantom{4)\ }\frac{2y^2+5}{ (1-y)(1+y)}-\frac{y+1}{1-y}-\frac{4}{1+y}=0;\)
\(\phantom{4)\ }\frac{2y^2+5-(1+y)^2-4(1-y)}{ (1-y)(1+y)}=0;\)
\(\phantom{4)\ }\frac{2y^2+5-1-2y-y^2-4+4y}{ (1-y)(1+y)}=0;\)
\(\phantom{4)\ }\frac{y^2+2y}{ (1-y)(1+y)}=0;\)
\(\phantom{4)\ }\begin{cases}y^2+2y=0,\\[-1ex]\\(1-y)(1+y)\ne0;\end{cases}\)
\(\phantom{4)\ }\begin{cases}y(y+2)=0,\\[-1ex]\\(1-y)(1+y)\ne0;\end{cases}\)
\(\phantom{4)\ }\begin{cases}y=0,\\[-1ex]\\y={-}2,\\[-1ex]\\y\ne1,\\[-1ex]\\y\ne{-}1.\end{cases}\)
Ответ: \({-}2,\ 0.\)
\(5)\ \frac{2x-1}{2x+1}=\frac{2x+1}{2x-1}+\frac{4}{1-4x^2};\)
\(\phantom{5)\ }\frac{2x-1}{2x+1}-\frac{2x+1}{2x-1}-\frac{4}{1-4x^2}=0;\)
\(\phantom{5)\ }\frac{\vphantom{ ()}2x-1}{2x+1}-\frac{2x+1}{2x-1}+\frac{4}{4x^2-1}=0;\)
\(\phantom{5)\ }\frac{\vphantom{^0}2x-1}{2x+1}-\frac{2x+1}{2x-1}+\frac{4}{ (2x-1)(2x+1)}=0;\)
\(\phantom{5)\ }\frac{ (2x-1)^2-(2x+1)^2+4}{ (2x-1)(2x+1)}=0;\)
\(\phantom{5)\ }\frac{4x^2-4x+1-(4x^2+4x+1)+4}{ (2x-1)(2x+1)}=0;\)
\(\phantom{5)\ }\frac{4x^2-4x+1-4x^2-4x-1+4}{ (2x-1)(2x+1)}=0;\)
\(\phantom{5)\ }\frac{{-}8x+4}{ (2x-1)(2x+1)}=0;\)
\(\phantom{5)\ }\frac{ {-}4(2x-1)}{ (2x-1)(2x+1)}=0;\)
\(\phantom{5)\ }\frac{{-}4}{2x+1}=0.\)
Ответ: корней нет.
\(6)\ \frac{7}{ (x+2)(x-3)}-\frac{4}{ (x-3)^2}=\frac{3}{ (x+2)^2};\)
\(\phantom{6)\ }\frac{7}{ (x+2)(x-3)}-\frac{4}{ (x-3)^2}-\frac{3}{ (x+2)^2}=0;\)
\(\phantom{6)\ }\frac{ 7(x+2)(x-3)-4(x+2)^2-3(x-3)^2}{ (x+2)^2(x-3)^2}=0;\)
\(\phantom{6)\ }\frac{ 7(x^2-3x+2x-6)-4(x^2+4x+4)-3(x^2-6x+9)}{ (x+2)^2(x-3)^2}=0;\)
\(\phantom{6)\ }\frac{7x^2-21x+14x-42-4x^2-16x-16-3x^2+18x-27}{ (x+2)^2(x-3)^2}=0;\)
\(\phantom{6)\ }\frac{{-}5x-85}{ (x+2)^2(x-3)^2}=0;\)
\(\phantom{6)\ }\begin{cases}{-}5x-85=0,\\[-1ex]\\(x+2)^2(x-3)^2\ne0;\end{cases}\)
\(\phantom{6)\ }\begin{cases}x={-}17,\\[-1ex]\\x\ne{-}2,\\[-1ex]\\x\ne3.\end{cases}\)
Ответ: \({-}17.\)
\(7)\ \frac{2x-1}{x+4}-\frac{3x-1}{4-x}=\frac{6x+64}{x^2-16}+4;\)
\(\phantom{7)\ }\frac{2x-1}{x+4}+\frac{3x-1}{x-4}-\frac{6x+64}{ (x-4)(x+4)}-4=0;\)
\(\phantom{7)\ }\frac{ (2x-1)(x-4)+(3x-1)(x+4)-(6x+64)-4(x-4)(x+4)}{ (x-4)(x+4)}=0;\)
\(\phantom{7)\ }\frac{2x^2-8x-x+4+3x^2+12x-x-4-6x-64-4x^2+64}{ (x-4)(x+4)}=0;\)
\(\phantom{7)\ }\frac{x^2-4x}{ (x-4)(x+4)}=0;\)
\(\phantom{7)\ }\frac{ x(x-4)}{ (x-4)(x+4)}=0;\)
\(\phantom{7)\ }\frac{x}{x+4}=0;\)
\(\phantom{7)\ }\begin{cases}x=0,\\[-1ex]\\x+4\ne0;\end{cases}\)
\(\phantom{7)\ }\begin{cases}x=0,\\[-1ex]\\x\ne{-}4.\end{cases}\)
Ответ: \(0.\)
\(8)\ \frac{\vphantom{^0}2x-6}{x^2-36}-\frac{x-3}{x^2-6x}-\frac{x-1}{x^2+6x}=0;\)
\(\phantom{8)\ }\frac{\vphantom{^0}2x-6}{ (x-6)(x+6)}-\frac{x-3}{ x(x-6)}-\frac{x-1}{ x(x+6)}=0;\)
\(\phantom{8)\ }\frac{\vphantom{^0} x(2x-6)-(x-3)(x+6)-(x-1)(x-6)}{ x(x-6)(x+6)}=0;\)
\(\phantom{8)\ }\frac{2x^2-6x-(x^2+6x-3x-18)-(x^2-6x-x+6)}{ x(x-6)(x+6)}=0;\)
\(\phantom{8)\ }\frac{2x^2-6x-x^2-6x+3x+18-x^2+6x+x-6}{ x(x-6)(x+6)}=0;\)
\(\phantom{8)\ }\frac{{-}2x+12}{ x(x-6)(x+6)}=0;\)
\(\phantom{8)\ }\frac{ {-}2(x-6)}{ x(x-6)(x+6)}=0;\)
\(\phantom{8)\ }\frac{{-}2}{ x(x+6)}=0.\)
Ответ: корней нет.
