Question

Rezolva ecuatia unde [a] reprezinta partea întreaga a numarului​

Answer 1

$\it \Big[\dfrac{x+1}{3}\Big]=\dfrac{x-1}{2}\ \ \ \ \ \ (1)\\ \\ \\ \Big[\dfrac{x+1}{3}\Big]\in\mathbb{Z}\ \stackrel{(1)}{\Longrightarrow}\ \dfrac{x-1}{2}\in\mathbb{Z}\\ \\ \\ Fie\ k\in\mathbb{Z}\ astfel\ \^{i}nc\hat at\ \dfrac{x-1}{2}=k \Rightarrow\ x=2k+1\ \ \ \ \ \ (2)$

$\it (1),\ (2) \Rightarrow \Big[\dfrac{2k+2}{3}\Big]=k \Rightarrow k\leq\dfrac{2k+2}{3}<k+1|_{\cdot3} \Rightarrow \\ \\ \\ \Rightarrow 3k\leq2k+2<3k+3|_{-3k} \Rightarrow 0\leq-k+2<3|_{-2} \Rightarrow -2\leq-k<1|_{\cdot(-1)} \Rightarrow \\ \\ \Rightarrow -1<k\leq2 \Rightarrow k\in\{0,\ 1,\ 2\}|_{\cdot2} \Rightarrow 2k\in\{0,\ 2,\ 4\}|_{+1} \Rightarrow \\ \\ \\ \Rightarrow 2k+1\in\{1,\ 3,\ 5\}\ \ \ \ \ (3)$

$\it (2),\ (3) \Rightarrow x\in\{1,\ \ 3,\ \ 5\}$