Question

ex 2,va rog.......................

Answer 1

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Answer 2

   
[tex]\displaystyle\\ \left[\frac{3x+2}{3}\right] =3\{x\}\\\\ \left[\frac{3x+2}{3}\right] \in Z\\\\ \Longrightarrow~~3\{x\} \in Z\\\\ \text{Avem 2 solutii:}\\ S_1:~~x = 0~\Longrightarrow~ 3\{x\}=3\cdot 0 = 0\in Z \\\\ \text{Verificam solutia in partea stanga a egalului.}\\\\ \left[\frac{3x+2}{3}\right]=\left[\frac{3\cdot 0+2}{3}\right]=\left[\frac{2}{3}\right]=0\\\\ \Longrightarrow~~\boxed{\bf x_1 = 0} [/tex]


[tex]\displaystyle\\ S_2: x= \frac{1}{3} ~\Longrightarrow~ 3\{x\}=3\cdot \frac{1}{3} = 1\in Z \\\\ \text{Verificam solutia in partea stanga a egalului.}\\\\ \left[\frac{3x+2}{3}\right]=\left[\frac{3\cdot \dfrac{1}{3}+2}{3}\right]=\left[\frac{1+2}{3}\right]=\left[\frac{3}{3}\right]=1\\\\ \Longrightarrow~~\boxed{x_2 = \frac{1}{3}}\\\\[/tex]