Решите неравенство: |4x+3|-|x-2|>3.

Ответ:

\[|4x + 3| - |x - 2|\text{>}3\]

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

\[\left\{ \begin{matrix} x < - \frac{3}{4}\text{\ \ } \\ - 3x > 8 \\ \end{matrix}\ \right.\ \text{\ \ \ \ \ }\]

\[\left\{ \begin{matrix} x < - \frac{3}{4} \\ x < - \frac{8}{3} \\ \end{matrix} \right.\ \text{\ \ \ \ \ }\left( - \infty;\ - \frac{8}{3} \right)\]

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

\[\left\{ \begin{matrix} - \frac{3}{4} \leq x \leq 2 \\ 5x > 2\ \ \ \ \ \ \ \ \ \ \\ \end{matrix} \right.\ \text{\ \ \ \ }\]

\[\left\{ \begin{matrix} - \frac{3}{4} \leq x \leq 2 \\ x \geq 0,4\ \ \ \ \ \ \ \ \ \\ \end{matrix} \right.\ \ \ \ (0,4;2\rbrack\]

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

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

\[\left\{ \begin{matrix} x > 2\ \ \ \ \ \\ x > - \frac{2}{3} \\ \end{matrix} \right.\ \ \ \ (2; + \infty)\]

\[Ответ:\left( - \infty;\ - 2\frac{2}{3} \right) \cup (0,4; + \infty).\]