Решебник по алгебре 11 класс Никольский Параграф 10. Равносильность уравнений на множествах Задание 46

\[\boxed{\mathbf{46.}}\]

\[\textbf{а)}\log_{|x|}{(1 + x)} = \log_{|x|}\left( x^{2} - 5 \right)\]

\[1 + x > 0\]

\[x > - 1.\]

\[x^{2} - 5 > 0\]

\[\left( x + \sqrt{5} \right)\left( x - \sqrt{5} \right) > 0\]

\[x < - \sqrt{5};\ \ x > \sqrt{5}.\]

\[x \neq 0;\]

\[|x| \neq 1\]

\[x \neq \pm 1.\]

\[M = \left( \sqrt{5}; + \infty \right).\]

\[1 + x = x^{2} - 5\]

\[x^{2} - x - 6 = 0\]

\[x_{1} + x_{2} = 1;\ \ x_{1} \cdot x_{2} = - 6\]

\[x_{1} = 3;\ \ x_{2} = - 2 < \sqrt{5}.\]

\[Ответ:x = 3.\]

\[\textbf{б)}\log_{|x|}{(9 + x)} = \log_{|x|}\left( x^{2} + 7 \right)\]

\[9 + x > 0\]

\[x > - 9.\]

\[x^{2} + 7 > 0\]

\[\left( x + \sqrt{7} \right)\left( x - \sqrt{7} \right) > 0\]

\[x < - \sqrt{7};\ \ x > \sqrt{7}.\]

\[x \neq 0;\]

\[|x| \neq 1\]

\[x \neq \pm 1.\]

\[9 + x = x^{2} + 7\]

\[x^{2} - x - 2 = 0\]

\[x_{1} + x_{2} = 1;x_{1} \cdot x_{2} = - 2\]

\[x_{1} = 2 > 1;\]

\[x_{2} = - 1\ (не\ подходит).\]

\[Ответ:x = 2.\]