Решебник по алгебре 11 класс Никольский Параграф 10. Равносильность уравнений на множествах Задание 5

\[\boxed{\mathbf{5.}}\]

\[\textbf{а)}\ \sqrt[4]{x + 1} = \sqrt[4]{2x - 5}\]

\[1)\ x + 1 \geq 0\]

\[x \geq - 1.\]

\[2)\ 2x - 5 \geq 0\]

\[2x \geq 5\]

\[x \geq 2,5.\]

\[M = \lbrack 2,5; + \infty).\]

\[3)\ x + 1 = 2x - 5\]

\[x = 6.\]

\[Ответ:x = 6.\ \]

\[\textbf{б)}\ \sqrt[4]{x - 1} = \sqrt[4]{2x + 5}\]

\[1)\ x - 1 \geq 0\]

\[x \geq 1.\]

\[2)\ 2x + 5 \geq 0\]

\[x \geq - 2,5.\]

\[M = \lbrack 1; + \infty).\]

\[3)\ x - 1 = 2x + 5\]

\[x = - 6\]

\[нет\ решений.\]

\[Ответ:нет\ корней.\]

\[\textbf{в)}\ \sqrt{2x + 11} = \sqrt{4x + 1}\]

\[1)\ 2x + 11 \geq 0\]

\[x \geq - 5,5.\]

\[2)\ 4x + 1 \geq 0\]

\[x \geq - \frac{1}{4}\]

\[x \geq - 0,25.\]

\[M = \lbrack - 0,25; + \infty).\]

\[3)\ 2x + 11 = 4x + 1\]

\[2x = 10\]

\[x = 5.\]

\[Ответ:x = 5.\]

\[\textbf{г)}\ \sqrt{2x - 9} = \sqrt{4x + 3}\]

\[1)\ 2x - 9 \geq 0\]

\[x \geq 4,5.\]

\[2)\ 4x + 3 \geq 0\]

\[x \geq - \frac{3}{4}.\]

\[M = \lbrack 4,5; + \infty).\]

\[3)\ 2x - 9 = 4x + 3\]

\[2x = - 12\]

\[x = - 6.\]

\[нет\ корней.\]

\[Ответ:нет\ корней.\]