Решите неравенство: |x+1|+|x-1|<=2.

Ответ:

\[|x + 1| + |x - 1| \leq 2\]

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

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

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

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

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

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

\[\left\{ \begin{matrix} x > 1\ \ \\ 2x \leq 2 \\ \end{matrix}\text{\ \ } \right.\ \text{\ \ \ \ \ \ \ \ }\]

\[\left\{ \begin{matrix} x > 1 \\ x \leq 1 \\ \end{matrix}\ \ \ нет\ решения. \right.\ \]

\[Ответ:\lbrack - 1;1\rbrack.\]