Решебник по алгебре и начала математического анализа 10 класс Алимов Задание 166

\[\boxed{\mathbf{166.}}\]

\[1)\ \sqrt{x} > 2\]

\[\left\{ \begin{matrix} x \geq 0\ \ \ \ \ \ \ \ \ \ \ \\ \left( \sqrt{x} \right)^{2} > 2^{2} \\ \end{matrix} \right.\ \text{\ \ \ \ \ \ \ \ \ }\left\{ \begin{matrix} x \geq 0 \\ x > 4 \\ \end{matrix} \right.\ \]

\[Ответ:\ \ x > 4.\]

\[2)\ \sqrt{x} < 3\]

\[\left\{ \begin{matrix} x \geq 0\ \ \ \ \ \ \ \ \ \ \ \\ \left( \sqrt{x} \right)^{2} < 3^{2} \\ \end{matrix} \right.\ \text{\ \ \ \ \ \ \ \ \ \ }\left\{ \begin{matrix} x \geq 0 \\ x < 9 \\ \end{matrix} \right.\ \]

\[Ответ:\ \ 0 \leq x < 9.\]

\[3)\ \sqrt[3]{x} \geq 1\]

\[\left( \sqrt[3]{x} \right)^{3} \geq 1^{3}\]

\[x \geq 1\]

\[Ответ:\ \ x \geq 1.\]

\[4)\ \sqrt[3]{2x} < 3\]

\[\left( \sqrt[3]{2x} \right)^{3} < 3^{3}\]

\[2x < 27\]

\[x < 13,5\]

\[Ответ:\ \ x < 13,5.\]

\[5)\ \sqrt{3x} > 1\]

\[\left\{ \begin{matrix} 3x \geq 0\ \ \ \ \ \ \ \ \ \ \ \\ \left( \sqrt{3x} \right)^{2} > 1^{2} \\ \end{matrix} \right.\ \text{\ \ \ \ \ \ }\left\{ \begin{matrix} x \geq 0\ \ \\ 3x > 1 \\ \end{matrix} \right.\ \text{\ \ \ \ \ \ }\]

\[\left\{ \begin{matrix} x \geq 0 \\ x > \frac{1}{3} \\ \end{matrix} \right.\ \]

\[Ответ:\ \ x > \frac{1}{3}.\]

\[6)\ \sqrt{2x} \leq 2\]

\[\left\{ \begin{matrix} 2x \geq 0\ \ \ \ \ \ \ \ \ \ \ \\ \left( \sqrt{2x} \right)^{2} \leq 2^{2} \\ \end{matrix} \right.\ \text{\ \ \ \ \ \ }\left\{ \begin{matrix} x \geq 0\ \ \\ 2x \leq 4 \\ \end{matrix} \right.\ \text{\ \ \ \ \ \ }\]

\[\left\{ \begin{matrix} x \geq 0 \\ x \leq 2 \\ \end{matrix} \right.\ \]

\[Ответ:\ \ 0 \leq x \leq 2.\]