Question

Aflati x apartine z pentru care 2x+4/x+1 apartine Z

Answer 1

$\dfrac{2x+4}{x+1}\in \mathbb_{Z} \\ \\ \\ \dfrac{2x+4}{x+1} = \dfrac{2x+2+2}{x+1} = \dfrac{2(x+1)+2}{x+1} = \dfrac{2(x+1)}{x+1}+\dfrac{2}{x+1} = \\ \\ = \underset{\in \mathbb_{Z}}{2} +\dfrac{2}{x+1} \\ \\ \bullet \dfrac{2}{x+1}\in \mathbb_{Z} \Rightarrow (x+1) | 2 \Rightarrow (x+1)\in D_2 \Rightarrow \\ \Rightarrow (x+1)\in \Big\{-2,-1,1,2\Big\}\Big|-1 \Rightarrow \\ \Rightarrow x \in \Big\{-2-1,-1-1,1-1,2-1\Big\} \Rightarrow \boxed{x\in \Big\{-3,-2,0,1\Big\}}$

Answer 2

(2x+4)/(x+1)=(2x+2+2)/(x+1)=
2(x+1)/(x+1)+2/(x+1)=2+2/(x+1)
Pt ca   acest  numar   sa   apartina   lui   Z  trebuie ca   fractia   sa   fie    numar   intreg,Acest   lucru   se    intampla   daca   x+2 este   divizor  al   lui   2
x+1={-2,-1,1 2}
x+1=-2   x=-3
x+1=-1   x=-2
x+1=1   x=0
x+1=2  x=1
x={-3,-2,0,1}