Решите неравенство: |x^2-6x|>7.

Ответ:

\[\left| x^{2} - 6x \right| > 7\]

\[\left\{ \begin{matrix} x^{2} - 6x < - 7 \\ x^{2} - 6x > 7\ \ \ \\ \end{matrix}\text{\ \ \ \ \ \ \ } \right.\ \text{\ \ \ }\]

\[\left\{ \begin{matrix} x^{2} - 6x + 7 < 0 \\ x^{2} - 6x - 7 > 0 \\ \end{matrix} \right.\ \]

\[x^{2} - 6x + 7 < 0\]

\[D = 36 - 28 = 8\]

\[x = \frac{6 \pm 2\sqrt{2}}{2} = 3 \pm \sqrt{2}\]

\[3 - \sqrt{2} < x < 3 + \sqrt{2}.\]

\[x^{2} - 6x - 7 > 0\]

\[x_{1} + x_{2} = 6,\ \ \ x_{1} \cdot x_{2} = - 7\]

\[x = - 1,\ \ x = 7\]

\[(x + 1)(x - 7) > 0\]

\[x < - 1;\ \ x > 7.\]