Решите уравнение: |x|+|x-4|=5.

Ответ:

\[|x| + |x - 4| = 5\]

\[1)\ \left\{ \begin{matrix} x < 0\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ - x + 4 - x = 5 \\ \end{matrix}\text{\ \ \ \ \ } \right.\ \]

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

\[\left\{ \begin{matrix} x < 0\ \ \ \\ x = - \frac{1}{2} \\ \end{matrix} \right.\ \ \ \ \ \ x = - \frac{1}{2}\]

\[2)\ \left\{ \begin{matrix} 0 \leq x \leq 4\ \ \ \ \ \ \\ x + 4 - x = 5 \\ \end{matrix}\text{\ \ \ \ } \right.\ \text{\ \ \ \ }\]

\[\left\{ \begin{matrix} 0 \leq x \leq 4 \\ 0 \cdot x = 1\ \ \\ \end{matrix}\ \ \ \ \ \ \ \ \ \ нет\ корней. \right.\ \]

\[3)\ \left\{ \begin{matrix} x > 4\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ x + x - 4 = 5 \\ \end{matrix}\text{\ \ \ \ } \right.\ \text{\ \ \ \ \ }\]

\[\left\{ \begin{matrix} x > 4 \\ 2x = 9 \\ \end{matrix} \right.\ \text{\ \ \ \ }\]

\[\left\{ \begin{matrix} x > 4\ \ \ \\ x = 4,5 \\ \end{matrix}\ \ \ \ \ \ \ \ \ \ x = 4,5 \right.\ \]

\[Ответ:\ - \frac{1}{2};\ \ \ 4,5.\]