2 3 (2) x-1=x²-1 X x-4 (4)x²-9 x+3 2 1 1 = (6) x-3x29x+3 x²-13 1 1 (8) x2-5x+6x-2x-3

(2) \(\frac{2}{x - 1} = \frac{3}{x^2 - 1}\) \(\frac{2}{x - 1} = \frac{3}{(x - 1)(x + 1)}\) Умножаем обе части на (x - 1)(x + 1):\[2(x + 1) = 3\]\[2x + 2 = 3\]\[2x = 1\]\[x = \frac{1}{2}\]

Проверка:

\[\frac{2}{\frac{1}{2} - 1} = \frac{2}{-\frac{1}{2}} = -4\]\[\frac{3}{(\frac{1}{2})^2 - 1} = \frac{3}{\frac{1}{4} - 1} = \frac{3}{-\frac{3}{4}} = -4\] (4) \(\frac{x}{x^2 - 9} = \frac{x - 4}{x + 3}\) \(\frac{x}{(x - 3)(x + 3)} = \frac{x - 4}{x + 3}\) Умножаем обе части на (x - 3)(x + 3):\[x = (x - 4)(x - 3)\]\[x = x^2 - 7x + 12\]\[x^2 - 8x + 12 = 0\]\[(x - 6)(x - 2) = 0\]\[x = 6, 2\]

Проверка:

Для x = 6:\[\frac{6}{6^2 - 9} = \frac{6}{27} = \frac{2}{9}\]\[\frac{6 - 4}{6 + 3} = \frac{2}{9}\] Для x = 2:\[\frac{2}{2^2 - 9} = \frac{2}{-5} = -\frac{2}{5}\]\[\frac{2 - 4}{2 + 3} = \frac{-2}{5} = -\frac{2}{5}\] (6) \(\frac{2}{x - 3} - \frac{1}{x^2 - 9} = \frac{1}{x + 3}\) \(\frac{2}{x - 3} - \frac{1}{(x - 3)(x + 3)} = \frac{1}{x + 3}\) Умножаем обе части на (x - 3)(x + 3):\[2(x + 3) - 1 = (x - 3)\]\[2x + 6 - 1 = x - 3\]\[2x + 5 = x - 3\]\[x = -8\]

Проверка:

\[\frac{2}{-8 - 3} - \frac{1}{(-8)^2 - 9} = \frac{2}{-11} - \frac{1}{55} = \frac{-10 - 1}{55} = -\frac{11}{55} = -\frac{1}{5}\]\[\frac{1}{-8 + 3} = \frac{1}{-5} = -\frac{1}{5}\] (8) \(\frac{x^2 - 13}{x^2 - 5x + 6} = \frac{1}{x - 2} + \frac{1}{x - 3}\) \(\frac{x^2 - 13}{(x - 2)(x - 3)} = \frac{1}{x - 2} + \frac{1}{x - 3}\) Умножаем обе части на (x - 2)(x - 3):\[x^2 - 13 = (x - 3) + (x - 2)\]\[x^2 - 13 = 2x - 5\]\[x^2 - 2x - 8 = 0\]\[(x - 4)(x + 2) = 0\]\[x = 4, -2\]

Проверка:

Для x = 4:\[\frac{4^2 - 13}{4^2 - 5(4) + 6} = \frac{3}{2}\]\[\frac{1}{4 - 2} + \frac{1}{4 - 3} = \frac{1}{2} + 1 = \frac{3}{2}\] Для x = -2:\[\frac{(-2)^2 - 13}{(-2)^2 - 5(-2) + 6} = \frac{-9}{20}\]\[\frac{1}{-2 - 2} + \frac{1}{-2 - 3} = -\frac{1}{4} - \frac{1}{5} = \frac{-5 - 4}{20} = -\frac{9}{20}\]