Question

Vă rog să rezolvați acest exercițiu. Dau 60 de puncte.

Answer 1

[tex]|x-1|(|2x-1|-5)\ \textless \ 0\ \ \ \ (1) [/tex]

[tex]Stim\ ca\ |x-1| \geq0, \forall\ x\in \mathbb{R}. \\\;\\ \it Relatia (1) poate avea loc daca x-1 \ne 0 \Longrightarrow x \ne 1\ \ \ (2)[/tex]

[tex]\it In \ conditiile\ de\ mai\ sus, \ rezulta: [/tex]

[tex]|2x-1| - 5 \ \textless \ 0 \Leftrightarrow \ |2x-1| \ \textless \ 5 \ \Leftrightarrow \ -5\ \textless \ 2x-1\ \textless \ 5|_{+1} \\\;\\ \Leftrightarrow \ -4\ \textless \ 2x\ \textless \ 6|_{:2} \ \Leftrightarrow \ -2\ \textless \ x\ \textless \ 3 \ \Leftrightarrow \ x\in (-2,\ 3)\ \ \ (3)[/tex]

$(2), (3) \ \Longrightarrow \ x\in (-2,\ 1) \cup (1,\ 3)$