Question

Determinați:
B={ x € Z / 9x -5 supra 3x +2 € Z }

Answer 1

   
[tex]\displaystyle \\ B= \{x\in Z~\Big|~\frac{9x - 5}{3x+2} \in Z \} \\ \\ \\ \frac{9x - 5}{3x+2} = \frac{9x +6 - 6 - 5}{3x+2}=\frac{9x +6 - 11}{3x+2}= \\ \\ \\ =\frac{9x +6}{3x+2} + \frac{- 11}{3x+2}= \frac{3(3x +2)}{3x+2} - \frac{11}{3x+2}= 3- \frac{11}{3x+2} \\ \\ \\ \text{Ne ocupam de fractia: }~\frac{11}{3x+2} \\ \\ \\ D_{11} = \{-11;~-1;~1;~11 \} [/tex]


[tex]\text{Fractia este numar intreg daca (3x+2) este divizor al lui 11. } \\ \\ 3x+2 = -11 ~~\Longrightarrow~~ 3x = -13~~\Longrightarrow~~ x \notin Z \\ \\ 3x+2 = -1 ~~\Longrightarrow~~ 3x=-3 ~~\Longrightarrow~~ \boxed{x = -1} \\ \\ 3x+2 = 1 ~~\Longrightarrow~~ 3x=-1 ~~\Longrightarrow~~ x \notin Z \\ \\ 3x+2 = 11 ~~\Longrightarrow~~ 3x=9 ~~\Longrightarrow~~ \boxed{x = 3} \\ \\B= \{ -1;~ 3\}[/tex]}