Решите уравнение: |x|+|x-6|=8.

Ответ:

\[|x| + |x - 6| = 8\]

\[1)\ \left\{ \begin{matrix} x < 0\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ - x + 6 - x = 8 \\ \end{matrix} \right.\ \text{\ \ \ \ \ \ }\]

\[\left\{ \begin{matrix} x < 0\ \ \ \ \ \\ - 2x = 2 \\ \end{matrix} \right.\ \text{\ \ \ \ \ }\]

\[\left\{ \begin{matrix} x < 0\ \ \ \\ x = - 1 \\ \end{matrix} \right.\ \ \ \ x = - 1\]

\[2)\ \left\{ \begin{matrix} 0 \leq x \leq 6\ \ \ \ \ \ \\ x + 6 - x = 8 \\ \end{matrix} \right.\ \text{\ \ \ \ \ \ \ }\]

\[\left\{ \begin{matrix} 0 \leq x \leq 6 \\ 0 \cdot x = 2\ \ \\ \end{matrix} \right.\ \ \ \ \ нет\ корней\]

\[3)\ \left\{ \begin{matrix} x > 6\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ x + x - 6 = 8 \\ \end{matrix} \right.\ \text{\ \ \ \ \ \ \ }\]

\[\left\{ \begin{matrix} x > 6\ \ \ \ \\ 2x = 14 \\ \end{matrix} \right.\ \text{\ \ \ \ \ \ }\]

\[\left\{ \begin{matrix} x > 6 \\ x = 7 \\ \end{matrix} \right.\ \ \ \ \ x = 7\]

\[Ответ:\ - 1;7.\]