Question

Determinati valorile intregi ale lui x pentru care 2x+9/×+1 €Z

Answer 1

$\displaystyle\dfrac{2x+9}{x+1}\in Z \\ \\ \dfrac{2(x+1)+7}{x+1}=2+\dfrac{7}{x+1}\implies x+1\in D_7 \\ \\ D_7=\{-7;-1;1;7\} \\ x+1\in\{-7;-1;1;7\} |-1 \\ x\in\{-8;-2;0;6\}$

Answer 2

$\dfrac{2x+9}{x+1} \in \mathbb_{Z} \\ \\ \dfrac{2x+9}{x+1} = \dfrac{2x+2+7}{x+1} = \dfrac{2(x+1)+7}{x+1} = \dfrac{2(x+1)}{x+1}+\dfrac{7}{x+1} =\\ \\ = \underset{\in\mathbb_{Z}}{2}+\dfrac{7}{x+1} \\ \\\\ \dfrac{7}{x+1}\in \mathbb _{Z}\Rightarrow (x+1) | 7 \Rightarrow (x+1)\in D_7 \Rightarrow (x+1)\in \Big\{\pm 1,\pm 7\Big\} \Rightarrow \\ \\ \Rightarrow (x+1)\in \Big\{-7,-1,1,7\Big\}\Big|-1 \Rightarrow \boxed{x\in \Big\{-8,-2,0,6\Big\}}$