Решите уравнение: |3x+1|-|x-4|=2x-3.

\[|3x + 1| - |x - 4| = 2x - 3\]

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

\[\left\{ \begin{matrix} x < - \frac{1}{3}\ \\ - 4x = 2 \\ \end{matrix} \right.\ \text{\ \ \ \ \ \ }\]

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

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

\[\left\{ \begin{matrix} - \frac{1}{3} \leq x \leq 4 \\ 2x = 0\ \ \ \ \ \ \ \ \ \ \\ \end{matrix} \right.\ ,\ \ x = 0\]

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

\[\left\{ \begin{matrix} x > 4\ \ \ \ \ \ \ \ \ \\ 0 \cdot x = - 8 \\ \end{matrix} \right.\ \ \ \ \ нет\ корней\]

\[Ответ:\ - \frac{1}{2};0.\]