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МатематикаРешебник по алгебре 11 класс Никольский Параграф 9. Равносильность уравнений и неравенств системам Задание 62
Задание 62
\[\boxed{\mathbf{62.}}\]
\[\textbf{а)}\log_{x}(9x + 1) < \log_{x}{|10x - 1|}\]
\[1)\ \left\{ \begin{matrix} 0 < 9x + 1 < |10x - 1| \\ x > 1\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ \end{matrix} \right.\ \]
\[\left\{ \begin{matrix} 10x - 1 > 9x + 1 \\ x > 1\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ \end{matrix} \right.\ \]
\[\left\{ \begin{matrix} x > 2 \\ x > 1 \\ \end{matrix} \right.\ \]
\[x > 2.\]
\[2)\ \left\{ \begin{matrix} 9x + 1 > |10x - 1| > 0 \\ 0 < x < 1\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ \end{matrix} \right.\ \]
\[10x - 1 \neq 0\]
\[x \neq 0,1.\]
\[1.\ \left\{ \begin{matrix} 9x + 1 > - 10x + 1 \\ 0 < x < 0,1\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ \end{matrix} \right.\ \text{\ \ }\]
\[\left\{ \begin{matrix} 19x > 0\ \ \ \ \ \ \ \\ 0 < x < 0,1 \\ \end{matrix} \right.\ \]
\[\left\{ \begin{matrix} x > 0\ \ \ \ \ \ \ \ \ \ \ \\ 0 < x < 0,1 \\ \end{matrix} \right.\ \]
\[0 < x < 0,1.\]
\[2.\ \left\{ \begin{matrix} - 9x - 1 < 10x - 1 \\ 0,1 < x < 1\ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ \end{matrix} \right.\ \]
\[\left\{ \begin{matrix} 19x > 0\ \ \ \ \ \ \ \\ 0,1 < x < 1 \\ \end{matrix} \right.\ \]
\[\left\{ \begin{matrix} x > 0\ \ \ \ \ \ \ \ \ \ \ \ \\ 0,1 < x < 1 \\ \end{matrix} \right.\ \]
\[0,1 < x < 1.\]
\[Объединим\ полученные\ \]
\[результаты:\]
\[x \in (0;0,1) \cup (0,1;1) \cup (2; + \infty).\]
\[Ответ:\ \]
\[x \in (0;0,1) \cup (0,1;1) \cup (2; + \infty)\text{.\ }\]
\[1)\ \left\{ \begin{matrix} 0 < 12x + 1 < |13x - 1| \\ x > 1\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ \end{matrix} \right.\ \]
\[\left\{ \begin{matrix} 13x - 1 > 12x + 1 \\ x > 1\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ \end{matrix} \right.\ \]
\[\left\{ \begin{matrix} x > 2 \\ x > 1 \\ \end{matrix} \right.\ \]
\[x > 2.\]
\[2)\ \left\{ \begin{matrix} 12x + 1 > |13x - 1| > 0 \\ 0 < x < 1\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ \end{matrix} \right.\ \]
\[1.\ \left\{ \begin{matrix} - 12x - 1 < 13x - 1 \\ 0 < x < \frac{1}{13}\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ } \\ \end{matrix} \right.\ \]
\[\left\{ \begin{matrix} 25x > 0\ \ \ \ \ \ \\ 0 < x < \frac{1}{13} \\ \end{matrix} \right.\ \]
\[\left\{ \begin{matrix} x > 0\ \ \ \ \ \ \ \ \ \\ 0 < x < \frac{1}{13} \\ \end{matrix} \right.\ \]
\[0 < x < \frac{1}{13}.\]
\[2.\ \left\{ \begin{matrix} 13x - 1 < 12x + 1 \\ \frac{1}{13} < x < 1\ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ \end{matrix} \right.\ \]
\[\left\{ \begin{matrix} x < 2\ \ \ \ \ \ \ \ \ \ \\ \frac{1}{13} < x < 1 \\ \end{matrix} \right.\ \]
\[\frac{1}{13} < x < 1.\]
\[Объединим\ полученные\ \]
\[результаты:\]
\[x \in \left( 0;\frac{1}{13} \right) \cup \left( \frac{1}{13};1 \right) \cup (2; + \infty).\]
\[Ответ:\ \]
\[x \in \left( 0;\frac{1}{13} \right) \cup \left( \frac{1}{13};1 \right) \cup (2; + \infty).\]
\[\textbf{в)}\log_{x}{(6x + 1)} < \log_{x}{|7x - 1|}\]
\[1)\ \left\{ \begin{matrix} 0 < 6x + 1 < |7x - 1| \\ x > 1\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ \end{matrix} \right.\ \]
\[\left\{ \begin{matrix} |7x - 1| > 6x + 1 \\ x > 1\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ \end{matrix} \right.\ \]
\[\left\{ \begin{matrix} 7x - 1 > 6x + 1 \\ x > 1\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ \end{matrix} \right.\ \]
\[\left\{ \begin{matrix} x > 2 \\ x > 1 \\ \end{matrix} \right.\ \]
\[x > 2.\]
\[2)\ \left\{ \begin{matrix} 6x + 1 > |7x - 1| > 0 \\ 0 < x < 1\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ \end{matrix} \right.\ \]
\[1.\ \left\{ \begin{matrix} - 6x - 1 < 7x - 1 \\ 0 < x < \frac{1}{7}\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ } \\ \end{matrix} \right.\ \]
\[\left\{ \begin{matrix} 13x > 0\ \ \ \ \\ 0 < x < \frac{1}{7} \\ \end{matrix} \right.\ \]
\[\left\{ \begin{matrix} x > 0\ \ \ \ \ \ \ \ \\ 0 < x < \frac{1}{7} \\ \end{matrix} \right.\ \]
\[0 < x < \frac{1}{7}.\]
\[2.\ \left\{ \begin{matrix} 7x - 1 < 6x + 1 \\ \frac{1}{7} < x < 1\ \ \ \ \ \ \ \ \ \ \ \\ \end{matrix} \right.\ \]
\[\left\{ \begin{matrix} x < 2\ \ \ \ \ \ \ \ \ \\ \frac{1}{7} < x < 1 \\ \end{matrix} \right.\ \]
\[\frac{1}{7} < x < 1.\]
\[Объединим\ полученные\ \]
\[результаты:\]
\[x \in \left( 0;\frac{1}{7} \right) \cup \left( \frac{1}{7};1 \right) \cup (2; + \infty).\]
\[Ответ:\ \]
\[x \in \left( 0;\frac{1}{7} \right) \cup \left( \frac{1}{7};1 \right) \cup (2; + \infty).\]
\[\textbf{г)}\log_{x}{(17x + 1)} < \log_{x}|18x - 1|\]
\[1)\ \left\{ \begin{matrix} 0 < 17x + 1 < |18x - 1| \\ x > 1\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ \end{matrix} \right.\ \]
\[\left\{ \begin{matrix} |18x - 1| > 17x + 1 \\ x > 1\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ \end{matrix} \right.\ \]
\[\left\{ \begin{matrix} 18x - 1 > 17x + 1 \\ x > 1\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ \end{matrix} \right.\ \]
\[\left\{ \begin{matrix} x > 2 \\ x > 1 \\ \end{matrix} \right.\ \]
\[x > 2.\]
\[2)\ \left\{ \begin{matrix} 17x + 1 > |18x - 1| > 0 \\ 0 < x < 1\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ \end{matrix} \right.\ \]
\[1.\ \left\{ \begin{matrix} - 17x - 1 < 18x - 1 \\ 0 < x < \frac{1}{18}\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ } \\ \end{matrix} \right.\ \]
\[\left\{ \begin{matrix} 35x > 0\ \ \ \ \ \ \\ 0 < x < \frac{1}{18} \\ \end{matrix} \right.\ \]
\[\left\{ \begin{matrix} x > 0\ \ \ \ \ \ \ \ \ \ \ \\ 0 < x < \frac{1}{18} \\ \end{matrix} \right.\ \]
\[0 < x < \frac{1}{18}.\]
\[2.\ \left\{ \begin{matrix} 18x - 1 < 17x + 1 \\ \frac{1}{18} < x < 1\ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ \end{matrix} \right.\ \]
\[\left\{ \begin{matrix} x < 2\ \ \ \ \ \ \ \ \ \ \\ \frac{1}{18} < x < 1 \\ \end{matrix} \right.\ \]
\[\frac{1}{18} < x < 1.\]
\[Объединим\ полученные\ \]
\[результаты:\]
\[x \in \left( 0;\frac{1}{18} \right) \cup \left( \frac{1}{18};1 \right) \cup (2; + \infty).\]
\[Ответ:\ \]
\[x \in \left( 0;\frac{1}{18} \right) \cup \left( \frac{1}{18};1 \right) \cup (2; + \infty).\]
