$\frac{5}{x^2-4}+\frac{2x}{x+2}=2$
$\frac{5}{(x-2)(x+2)}+\frac{2x}{x+2}-2=0$
$\frac{5+2x(x-2)-2(x^2-4)}{(x-2)(x+2)}=0$
$\frac{2x^2-4x+5-2x^2+8}{(x-2)(x+2)}=0$
$\frac{13-4x}{(x-2)(x+2)}=0$
$\{\begin{array}{ll}x-2\ne 0& \\ x+2\ne 0& \\ 13-4x=0& \end{array}$
$\{\begin{array}{ll}x\ne 2& \\ x\ne -2& \\ 4x=13& \end{array}$
$\{\begin{array}{ll}x\ne 2& \\ x\ne -2& \\ x=\frac{13}{4}=3\frac{1}{4}& \end{array}$
Ответ: $x=3\frac{1}{4}$

$\frac{2}{6x+1}+\frac{3}{6x-1}=\frac{30x+9}{36x^2-1}$
$\frac{2}{6x+1}+\frac{3}{6x-1}-\frac{30x+9}{(6x-1)(6x+1)}=0$
$\frac{2(6x-1)+3(6x+1)-(30x+9)}{(6x-1)(6x+1)}=0$
$\frac{12x-2+18x+3-30x-9}{(6x-1)(6x+1)}=0$
$\frac{-8}{(6x-1)(6x+1)}=0$
−8 ≠ 0
Ответ: нет корней

$\frac{6x+14}{x^2-9}+\frac{7}{x^2+3x}=\frac{6}{x-3}$
$\frac{6x+14}{(x-3)(x+3)}+\frac{7}{x(x+3)}-\frac{6}{x-3}=0$
$\frac{x(6x+14)+7(x-3)-6x(x+3)}{x(x-3)(x+3)}=0$
$\frac{6x^2+14x+7x-21-6x^2-18x}{x(x-3)(x+3)}=0$
$\frac{3x-21}{x(x-3)(x+3)}=0$
$\{\begin{array}{ll}x\ne 0& \\ x-3\ne 0& \\ x+3\ne 0& \\ 3x-21=0& \end{array}$
$\{\begin{array}{ll}x\ne 0& \\ x\ne 3& \\ x\ne -3& \\ 3x=21& \end{array}$
$\{\begin{array}{ll}x\ne 0& \\ x\ne 3& \\ x\ne -3& \\ x=7& \end{array}$
Ответ: x = 7

$\frac{2y^2+5}{1-y^2}+\frac{y+1}{y-1}=\frac{4}{y+1}$
$\frac{-(2y^2+5)}{y^2-1}+\frac{y+1}{y-1}-\frac{4}{y+1}=0$
$\frac{-2y^2-5}{(y-1)(y+1)}+\frac{y+1}{y-1}-\frac{4}{y+1}=0$
$\frac{-2y^2-5+(y+1{)}^2-4(y-1)}{(y-1)(y+1)}=0$
$\frac{-2y^2-5+y^2+2y+1-4y+4}{(y-1)(y+1)}=0$
$\frac{-y^2-2y}{(y-1)(y+1)}=0$
$\frac{-(y^2+2y)}{(y-1)(y+1)}=0$
$\frac{-y(y+2)}{(y-1)(y+1)}=0$
$\{\begin{array}{ll}y-1\ne 0& \\ y+1\ne 0& \\ -y=0& \\ y+2=0& \end{array}$
$\{\begin{array}{ll}y\ne 1& \\ y\ne -1& \\ y=0& \\ y=-2& \end{array}$
Ответ: y = −2, y = 0.

$\frac{2x-1}{2x+1}=\frac{2x+1}{2x-1}+\frac{4}{1-4x^2}$
$\frac{2x-1}{2x+1}-\frac{2x+1}{2x-1}-\frac{4}{1-4x^2}=0$
$\frac{2x-1}{2x+1}-\frac{2x+1}{2x-1}+\frac{4}{4x^2-1}=0$
$\frac{2x-1}{2x+1}-\frac{2x+1}{2x-1}+\frac{4}{(2x-1)(2x+1)}=0$
$\frac{(2x-1{)}^2-(2x+1{)}^2+4}{(2x-1)(2x+1)}=0$
$\frac{4x^2-4x+1-(4x^2+4x+1)+4}{(2x-1)(2x+1)}=0$
$\frac{4x^2-4x+1-4x^2-4x-1+4}{(2x-1)(2x+1)}=0$
$\frac{-8x+4}{(2x-1)(2x+1)}=0$
$\{\begin{array}{ll}2x-1\ne 0& \\ 2x+1\ne 0& \\ -8x+4=0& \end{array}$
$\{\begin{array}{ll}2x\ne 1& \\ 2x\ne -1& \\ -8x=-4& \end{array}$
$\{\begin{array}{ll}x\ne 0,5& \\ x\ne -0,5& \\ x=0,5& \end{array}$
Ответ: нет корней

$\frac{7}{(x+2)(x-3)}-\frac{4}{(x-3{)}^2}=\frac{3}{(x+2{)}^2}$
$\frac{7}{(x+2)(x-3)}-\frac{4}{(x-3{)}^2}-\frac{3}{(x+2{)}^2}=0$
$\frac{7(x+2)(x-3)-4(x+2{)}^2-3(x-3{)}^2}{(x+2{)}^2(x-3{)}^2}=0$
$\frac{7(x^2+2x-3x-6)-4(x^2+4x+4)-3(x^2-6x+9)}{(x+2{)}^2(x-3{)}^2}=0$
$\frac{7(x^2-x-6)-4(x^2+4x+4)-3(x^2-6x+9)}{(x+2{)}^2(x-3{)}^2}=0$
$\frac{7x^2-7x-42-4x^2-16x-16-3x^2+18x-27}{(x+2{)}^2(x-3{)}^2}=0$
$\frac{-5x-85}{(x+2{)}^2(x-3{)}^2}=0$
$\{\begin{array}{ll}x+2\ne 0& \\ x-3\ne 0& \\ -5x-85=0& \end{array}$
$\{\begin{array}{ll}x\ne -2& \\ x\ne 3& \\ -5x=85& \end{array}$
$\{\begin{array}{ll}x\ne -2& \\ x\ne 3& \\ x=-17& \end{array}$
Ответ: x = −17

$\frac{2x-1}{x+4}-\frac{3x-1}{4-x}=\frac{6x+64}{x^2-16}+4$
$\frac{2x-1}{x+4}+\frac{3x-1}{x-4}-\frac{6x+64}{(x-4)(x+4)}-4=0$
$\frac{(2x-1)(x-4)+(3x-1)(x+4)-(6x+64)-4(x^2-16)}{(x-4)(x+4)}=0$
$\frac{2x^2-x-8x+4+3x^2-x+12x-4-6x-64-4x^2+64}{(x-4)(x+4)}=0$
$\frac{x^2-4x}{(x-4)(x+4)}=0$
$\frac{x(x-4)}{(x-4)(x+4)}=0$
$\{\begin{array}{ll}x-4\ne 0& \\ x+4\ne 0& \\ x=0& \\ x-4=0& \end{array}$
$\{\begin{array}{ll}x\ne 4& \\ x\ne -4& \\ x=0& \\ x=4& \end{array}$
Ответ: x = 0

$\frac{2x-6}{x^2-36}-\frac{x-3}{x^2-6x}-\frac{x-1}{x^2+6x}=0$
$\frac{2x-6}{(x-6)(x+6)}-\frac{x-3}{x(x-6)}-\frac{x-1}{x(x+6)}=0$
$\frac{x(2x-6)-(x-3)(x+6)-(x-1)(x-6)}{x(x-6)(x+6)}=0$
$\frac{2x^2-6x-(x^2-3x+6x-18)-(x^2-x-6x+6)}{x(x-6)(x+6)}=0$
$\frac{2x^2-6x-x^2+3x-6x+18-x^2+x+6x-6}{x(x-6)(x+6)}=0$
$\frac{-2x+12}{x(x-6)(x+6)}=0$
$\frac{-2(x-6)}{x(x-6)(x+6)}=0$
$\frac{-2}{x(x+6)}=0$
−2 ≠ 0
Ответ: нет корней