Решите уравнение: |x+2|+|x-5|=7.

Ответ:

\[|x + 2| + |x - 5| = 7\]

\[1)\ \left\{ \begin{matrix} x < - 2\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ - x - 2 - x + 5 = 7 \\ \end{matrix} \right.\ \text{\ \ \ \ \ \ }\]

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

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

\[2)\ \left\{ \begin{matrix} - 2 \leq x \leq 5\ \ \ \ \ \ \ \ \ \ \ \\ x + 5 - x + 5 = 7 \\ \end{matrix} \right.\ \text{\ \ \ \ \ \ \ \ \ }\]

\[\left\{ \begin{matrix} - 2 \leq x \leq 5 \\ 0 \cdot x = 0\ \ \ \ \ \\ \end{matrix} \right.\ \ \ \ x \in \lbrack - 2;5\rbrack\]

\[3)\ \left\{ \begin{matrix} x > 5\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ x + 2 + x - 5 = 7 \\ \end{matrix} \right.\ \text{\ \ \ \ \ \ \ \ }\]

\[\left\{ \begin{matrix} x > 5\ \ \ \ \\ 2x = 10 \\ \end{matrix} \right.\ \text{\ \ \ \ \ }\]

\[\left\{ \begin{matrix} x > 5 \\ x = 5 \\ \end{matrix} \right.\ \ \ \ \ нет\ корней\]

\[Ответ:\ x \in \lbrack - 2;5\rbrack.\]