Решите уравнение: |x-2|+|x+3|=5.

Ответ:

\[|x - 2| + |x + 3| = 5\]

\[1)\ \left\{ \begin{matrix} x < - 3\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ 2 - x - x - 3 = 5 \\ \end{matrix} \right.\ \text{\ \ \ \ \ \ \ }\]

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

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

\[2)\ \left\{ \begin{matrix} - 3 \leq x \leq 2\ \ \ \ \ \ \ \ \ \ \ \ \ \\ 2 - x + x + 3 = 5 \\ \end{matrix} \right.\ \text{\ \ \ \ \ \ }\]

\[\left\{ \begin{matrix} - 3 \leq x \leq 2 \\ 0 \cdot x = 0\ \ \ \ \ \\ \end{matrix} \right.\ ,\ \ x \in \lbrack - 3;\ 2\rbrack\]

\[3)\ \left\{ \begin{matrix} x > 2\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ x - 2 + x + 3 = 5 \\ \end{matrix} \right.\ \text{\ \ \ \ \ \ }\]

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

\[\left\{ \begin{matrix} x > 2 \\ x = 2 \\ \end{matrix} \right.\ ,\ \ нет\ корней\]

\[Ответ:x \in \lbrack - 3;2\rbrack.\]