Question

[x+1]=2x-1, aflați x, parantezele reprezintă partea intreaga.

Answer 1

$\it Fie\ k\in\mathbb{Z},\ pentru\ care\ 2x-1=k \Rightarrow x=\dfrac{k+1}{2} \ \ \ \ \ (*)\\ \\ \\ Ecua\c{\it t}ia\ se\ scrie:\\ \\ \Big[\dfrac{k+1}{2}+1\Big]=k \Rightarrow \Big[\dfrac{k+3}{2}\Big]=k \Rightarrow k\leq\dfrac{k+3}{2}<k+1|_{\cdot2} \Rightarrow 2k\leq k+3<2k+2$

$\it I)\ 2k\leq k+3\Rightarrow k\leq3\ \ \ \ (1)\\ \\ II)\ k+3<2k+2 \Rightarrow k>1\ \ \ \ \ (2)\\ \\ (1),\ (2) \Rightarrow k\in\{2,\ 3\} \stackrel{(*)}{\Longrightarrow} \begin{cases} \it x=\dfrac{2+1}{2}=\dfrac{3}{2}\\ \\ \\ \it x=\dfrac{3+1}{2}=2 \end{cases}$