Question

rezolvati in R ecuatia
[x+1]=2x-1
unde [x] este partea intreaga al lui x

Answer 1

$[x+1] = 2x-1 \\ 2x-1 = t,\quad (t\in \mathbb{Z}) \Rightarrow 2x = t+1 \Rightarrow x = \dfrac{t+1}{2} \\ \\ \Big(~\text{Exemplu: }~~~~[x] = 3 \Rightarrow \boxed{3\leq x\ \textless \ 3+1}~\Big)\\ \\ \\ 2x-1\leq (x+1)\ \textless \ (2x-1)+1 \\ \\ t \leq \left(\dfrac{t+1}{2}+1\right)< t+1\Big|\cdot 2 \\ \\2t \leq t+1+2 < 2t+2 \\ 2t\leq t+3 < 2t+2 \Big|-2t \\ 0 \leq -t+3 < 2 \Big|-3$
$-3 \leq -t \ \textless \ -1 \Big|\cdot (-1) \\ 3\geq t \ \textgreater \ 1 \\ 1\ \textless \ t\leq 3 \Rightarrow t \in\{2,3\} \\ \\ \boxed{1} \quad t = 2 \Rightarrow x = \dfrac{2+1}{2} \Rightarrow x = \dfrac{3}{2} \\ \\ \boxed{2} \quad t = 3 \Rightarrow x = \dfrac{3+1}{2} \Rightarrow x = \dfrac{4}{2} \Rightarrow x = 2 \\ \\ \Rightarrow \boxed{S =\left\{\dfrac{3}{2},2\right\}}$