Question

Buna, cum se rezolva acest tip de inecuatie cu modul:
|1- 2x|> 3- 2x

Answer 1

$|1- 2x| \ \textgreater \ 3- 2x \Rightarrow |1-2x| - 3 + 2x \ \textgreater \ 0 \\ \\ 1-2x = 0 \Rightarrow 2x = 1 \Rightarrow x = \dfrac{1}{2} \Rightarrow |1-2x| = \left\{ \begin{array}{c} 1-2x,\quad x\ \textless \ \dfrac{1}{2} \\ 0, \quad x = \dfrac{1}{2}\\-1+2x, \quad x\ \textgreater \ \dfrac{1}{2} \end{array} \right| \\ \\ \boxed{1}\quad x\ \textless \ \dfrac{1}{2} : \\ \\ 1-2x-3+2x\ \textgreater \ 0 \Rightarrow -2\ \textgreater \ 0 (F) \Rightarrow \boxed{x\in \emptyset}$

$\boxed{2}\quad x =\dfrac{1}{2}: \\ \\ 0-3+2x \ \textgreater \ 0 \Rightarrow 2x\ \textgreater \ 3 \Rightarrow x\ \textgreater \ \dfrac{3}{2} \Rightarrow Din \Big(\dfrac{3}{2}, \infty\Big)\cap \Big\{\dfrac{1}{2}\Big\} \Rightarrow \boxed{x\in \emptyset}\\ \\ \boxed{3}\quad x\ \textgreater \ \dfrac{1}{2}: \\ \\ -1+2x-3+2x\ \textgreater \ 0 \Rightarrow -4+4x\ \textgreater \ 0 \Rightarrow 4x\ \textgreater \ 4 \Rightarrow x\ \textgreater \ \dfrac{4}{4}\Rightarrow \\ \\ \Rightarrow x\ \textgreater \ 1 \Rightarrow Din (1,\infty) \cap \Big(\dfrac{1}{2},\infty\Big) \Rightarrow \boxed{x\in (1,\infty)}$

$\\ \\ \\ Din \boxed{1} \cup \boxed{2} \cup \boxed{3} \Rightarrow \boxed{S =(1,\infty)}$