Решите неравенство: |x-4|(x+2)>=4x.

Ответ:

\[|x - 4|(x + 2) \geq 4x\]

\[1)\ \left\{ \begin{matrix} x - 4 < 0\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ (4 - x)(x + 2) \geq 4x \\ \end{matrix}\text{\ \ \ \ \ \ \ } \right.\ \]

\[\left\{ \begin{matrix} x < 4\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ 4x + 8 - x^{2} - 2x \geq 4x\ \\ \end{matrix} \right.\ \text{\ \ }\]

\[\left\{ \begin{matrix} x < 4\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ - x^{2} - 2x + 8 \geq 0 \\ \end{matrix} \right.\ \]

\[- x^{2} - 2x + 8 = 0\]

\[x_{1} + x_{2} = - 2\ \ \ \ \ x_{1} = - 4\]

\[x_{1}x_{2} = - 8\ \ \ \ \ \ \ \ \ \ x_{2} = 2\]

\[\lbrack - 4;2\rbrack.\]

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

\[\left\{ \begin{matrix} x > 4\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ x^{2} - 2x - 8 \geq 4x \\ \end{matrix} \right.\ \text{\ \ \ }\]

\[\left\{ \begin{matrix} x > 4\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ x^{2} - 6x - 8 \geq 0 \\ \end{matrix} \right.\ \]

\[x^{2} - 6x - 8 = 0\]

\[D = 36 + 32 = 68\]

\[x = \frac{6 - 2\sqrt{17}}{2} = 3 - \sqrt{17}\]

\[x_{2} = 3 + \sqrt{17}\]