Решите уравнение: |x+1|+|x-3|=4.

Ответ:

\[|x + 1| + |x - 3| = 4\]

\[1)\ \left\{ \begin{matrix} x < - 1\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ - x - 1 + 3 - x = 4 \\ \end{matrix}\text{\ \ \ \ } \right.\ \ \]

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

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

\[2)\ \left\{ \begin{matrix} - 1 \leq x \leq 3\ \ \ \ \ \ \ \ \ \ \ \ \\ x + 1 + 3 - x = 4 \\ \end{matrix}\text{\ \ \ \ } \right.\ \text{\ \ \ \ \ }\]

\[\left\{ \begin{matrix} - 1 \leq x \leq 3 \\ 0 \cdot x = 0\ \ \ \ \ \\ \end{matrix}\ \ \ \ \ \ x \in \lbrack - 1;3\rbrack \right.\ \]

\[3)\ \left\{ \begin{matrix} x > 3\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ x + 1 + x - 3 = 4 \\ \end{matrix}\text{\ \ } \right.\ \text{\ \ \ \ \ }\]

\[\left\{ \begin{matrix} x > 3\ \ \\ 2x = 6 \\ \end{matrix}\text{\ \ \ } \right.\ \text{\ \ \ \ }\]

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

\[Ответ:\ x\ \in \lbrack - 1;3\rbrack.\]