Question

Cum se rezolvă ecuația de mai sus? Măcar o cheie. Mulțumesc anticipat

Answer 1

$[\frac{x+1}{3}]=\frac{x-1}{2} \\ \\ Notezi~cu~k=\frac{x-1}{2} \Rightarrow~x=2k+1.\\ \\ Inlocuiesti~in~partea~intreaga: \\ \\ Vei~avea:[\frac{2k+2}{3}]=k \\ \\ Din~proprietatea~partii~intregi~([x]\leq x<[x]+1),~ai: \\ \\ k\leq \frac{2k+2}{3}<k+1~\Leftrightarrow~3k\leq 2k+2<3k+3\\ \\ 1)~3k-2k\leq 2~\Leftrightarrow~k\leq 2 \Leftrightarrow~k\in~(-\infty,2].\\ \\ 2)~3k+3>2k+2~\Leftrightarrow~k>-1~\Leftrightarrow~k\in~(-1,+\infty).\\ \\ Din~1)~si~2)~ai~obtinut~ca,~k\in~(-\infty,2]~\cap~(-1,+\infty)=(-1,2] \\ \\ Deci,~k\in~(-1,2].\\ \\ Deoarece~k\in~Z~\Rightarrow~k=\{0,1,2\}.\\ \\ k=0~\Rightarrow~x=1.\\ \\ k=1~\Rightarrow~x=3.\\ \\ k=2~\Rightarrow~x=5.\\ \\ \boxed{Solutiile~sunt~x=\{1,3,5\}}$