Иррациональные неравенства повышенной сложности. Задача 3

Задача 3. Решите неравенство \(\sqrt {11-5x} > x-1\)

Ответ

ОТВЕТ: \(\left( {-\infty ;\;2} \right).\)

Решение

\(\sqrt {11-5x} > x-1\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\left[ \begin{array}{l}\left\{ \begin{array}{l}11-5x \ge 0,\\x-1 < 0,\end{array} \right.\\\left\{ \begin{array}{l}x-1 \ge 0,\\11-5x > {x^2}-2x + 1\end{array} \right.\end{array} \right.\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\left[ \begin{array}{l}\left\{ \begin{array}{l}x \in \,\left( {-\infty ;2,2} \right],\\x \in \left( {-\infty ;1} \right),\end{array} \right.\\\left\{ \begin{array}{l}x \in \left[ {1;\infty } \right),\\x \in \left( {-5;2} \right)\end{array} \right.\end{array} \right.\,\,\,\,\,\, \Leftrightarrow \)

\( \Leftrightarrow \,\,\,\,\,\left[ \begin{array}{l}x \in \left( {-\infty ;1} \right),\\x \in \left[ {1;2} \right)\end{array} \right.\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,x \in \left( {-\infty ;2} \right).\)

Ответ: \(\left( {-\infty ;2} \right).\)