Решите уравнение: |2x-6|-|x+4|=4x+10.

Ответ:

\[|2x - 6| - |x + 4| = 4x + 10\]

\[1)\ \left\{ \begin{matrix} x < - 4\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ 6 - 2x + x + 4 = 4x + 10 \\ \end{matrix} \right.\ \text{\ \ \ \ \ }\]

\[\left\{ \begin{matrix} x < - 4\ \ \\ - 5x = 0 \\ \end{matrix} \right.\ \text{\ \ \ \ \ }\]

\[\left\{ \begin{matrix} x < - 4 \\ x = 0\ \ \ \\ \end{matrix} \right.\ ,\ нет\ корней\]

\[2)\ \left\{ \begin{matrix} - 4 \leq x \leq 3\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ 6 - 2x - x - 4 = 4x + 10 \\ \end{matrix} \right.\ \text{\ \ \ \ \ }\]

\[\left\{ \begin{matrix} - 4 \leq x \leq 3 \\ - 7x = 8\ \ \ \ \ \ \\ \end{matrix} \right.\ ,\ \ x = - \frac{8}{7}\]

\[3)\ \left\{ \begin{matrix} x > 3\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ 2x - 6 - x - 4 = 4x + 10 \\ \end{matrix} \right.\ \text{\ \ \ \ \ }\]

\[\left\{ \begin{matrix} x > 3\ \ \ \ \ \ \ \ \\ - 3x = 20 \\ \end{matrix} \right.\ \text{\ \ \ \ \ \ }\]

\[\left\{ \begin{matrix} x > 3\ \ \ \ \ \ \ \\ x = - \frac{20}{3} \\ \end{matrix} \right.\ ,\ нет\ корней\]

\[Ответ:\ - \frac{8}{7}.\]