Решите неравенство: |x+3|(x-6)>=4x.

Ответ:

\[|x + 3|(x - 6) \geq 4x\]

\[1.\ \ \left\{ \begin{matrix} (x + 3)(x - 6) \geq 4x \\ x + 3 \geq 0\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ \end{matrix}\text{\ \ \ \ \ } \right.\ \]

\[\left\{ \begin{matrix} x^{2} - 6x + 3x - 18 - 4x \geq 0 \\ x \geq - 3\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ \end{matrix}\text{\ \ \ \ \ \ } \right.\ \]

\[\left\{ \begin{matrix} x^{2} - 7x - 18 \geq 0 \\ x \geq - 3\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ \end{matrix} \right.\ \]

\[x^{2} - 7x - 18 \geq 0\]

\[x_{1} + x_{2} = 7,\ \ x_{1} \cdot x_{2} = - 18\]

\[x = 9,\ \ x = - 2\]

\[(x + 2)(x - 9) \geq 0\]

\[x \leq - 2;\ \ \ x \geq 9.\]

\[2.\ \ \left\{ \begin{matrix} ( - x - 3)(x - 6) \geq 4x \\ x + 3 < 0\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ \end{matrix}\text{\ \ \ \ \ \ \ } \right.\ \]

\[\left\{ \begin{matrix} - x^{2} + 6x - 3x + 18 - 4x \geq 0 \\ x < - 3\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ \end{matrix}\text{\ \ \ \ \ } \right.\ \]

\[\left\{ \begin{matrix} - x^{2} - x + 18 \geq 0 \\ x < - 3\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ \end{matrix} \right.\ \]

\[- x^{2} - x + 18 \geq 0\]

\[x^{2} + x - 18 \leq 0\]

\[D = 1 + 72 = 73\]

\[x = \frac{- 1 \pm \sqrt{73}}{- 2}\]

\[\frac{- 1 - \sqrt{73}}{2} \leq x \leq \frac{- 1 + \sqrt{73}}{2}.\]