Решебник по алгебре 11 класс Никольский Параграф 9. Равносильность уравнений и неравенств системам Задание 44

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\[\textbf{а)}\ \sqrt{2x - 1} < x - 2\]

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

\[\left\{ \begin{matrix} 2x - 1 < x^{2} - 4x + 4 \\ x \geq 0,5\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ x > 2\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ \end{matrix} \right.\ \]

\[x^{2} - 6x + 5 > 0\]

\[D_{1} = 9 - 5 = 4\]

\[x_{1} = 3 + 2 = 5;\]

\[x_{2} = 3 - 2 = 1;\]

\[(x - 1)(x - 5) < 0\]

\[x < 1;\ \ \ x > 5.\]

\[\left\{ \begin{matrix} x < 1\ \ \ \\ x > 5\ \ \ \\ x \geq 0,5 \\ x > 2\ \ \ \\ \end{matrix} \right.\ \]

\[x > 5.\]

\[Ответ:x > 5.\]

\[\textbf{б)}\ \sqrt{2x + 1} < x - 1\]

\[\left\{ \begin{matrix} 2x + 1 < (x - 1)^{2} \\ 2x + 1 \geq 0\ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ x - 1 > 0\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ \end{matrix} \right.\ \]

\[\left\{ \begin{matrix} 2x + 1 < x^{2} - 2x + 1 \\ x \geq - 0,5\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ x > 1\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ \end{matrix} \right.\ \]

\[x^{2} - 4x > 0\]

\[x(x - 4) > 0\]

\[x < 0;\ \ \ x > 4.\]

\[\left\{ \begin{matrix} x < 0\ \ \ \ \ \ \ \\ x > 4\ \ \ \ \ \ \ \\ x \geq - 0,5 \\ x > 1\ \ \ \ \ \ \ \\ \end{matrix} \right.\ \]

\[x > 4.\]

\[Ответ:x > 4.\]