Решебник по алгебре 11 класс Никольский Параграф 11. Равносильность неравенств на множествах Задание 49

\[\boxed{\mathbf{49.}}\]

\[\textbf{а)}\ \left( x - \log_{3}75 \right)\left( x - \log_{2}22 \right) > 0\]

\[M = \lbrack 3;4\rbrack.\]

\[\log_{3}27 < \log_{3}75 < \log_{3}81\]

\[3 < \log_{3}75 < 4\]

\[\log_{3}75 \in (3;4).\]

\[\log_{2}16 < \log_{2}22 < \log_{2}32\]

\[4 < \log_{2}22 < 5\]

\[\log_{2}22 \in (4;5).\]

\[x > \log_{2}22.\]

\[1)\ \left\{ \begin{matrix} x - \log_{3}75 > 0 \\ x - \log_{2}22 > 0 \\ \end{matrix} \right.\ \]

\[\left\{ \begin{matrix} x > \log_{3}75 \\ x > \log_{2}22 \\ \end{matrix} \right.\ \]

\[2)\ \left\{ \begin{matrix} x - \log_{3}75 < 0 \\ x - \log_{2}22 < 0 \\ \end{matrix} \right.\ \]

\[\left\{ \begin{matrix} x < \log_{3}75 \\ x < \log_{2}22 \\ \end{matrix} \right.\ \]

\[x < \log_{3}75.\]

\[Решение\ неравенства:\]

\[x \in \left\lbrack 3;\log_{3}75 \right).\]

\[Ответ:x \in \left\lbrack 3;\log_{3}75 \right).\]

\[\textbf{б)}\ \left( x - \log_{2}17 \right)\left( x - \log_{2}71 \right) < 0\]

\[M = \lbrack 4\ ;5\rbrack.\]

\[\log_{2}16 < \log_{2}17 < \log_{2}32\]

\[4 < \log_{2}17 < 5;\]

\[\log_{2}17 \in (4;5).\]

\[\log_{2}64 < \log_{2}71 < \log_{2}128\]

\[6 < \log_{2}71 < 7\]

\[\log_{2}71 \in (6;7).\]

\[1)\ \left\{ \begin{matrix} x - \log_{2}17 > 0 \\ x - \log_{2}71 < 0 \\ \end{matrix} \right.\ \]

\[\left\{ \begin{matrix} x > \log_{2}17 \\ x < \log_{2}71 \\ \end{matrix} \right.\ \]

\[\log_{2}17 < x < \log_{2}71.\]

\[2)\ \left\{ \begin{matrix} x - \log_{2}17 > 0 \\ x - \log_{2}71 < 0 \\ \end{matrix} \right.\ \]

\[\left\{ \begin{matrix} x > \log_{2}17 \\ x < \log_{2}71 \\ \end{matrix} \right.\ \]

\[решений\ нет.\]

\[Решение\ неравенства:\]

\[x \in \left( \log_{2}17;5 \right\rbrack.\]

\[Ответ:x \in \left( \log_{2}17;5 \right\rbrack.\]