Question

Determinati x€Z stiind ca:
a)6 supra+1 €Z
b)5 supra 1-x
c)(x+5) (x-2)=18
d) -6 supra x-2€N
e)7x supra x+2€Z

Answer 1

rezolvarea in atașamente

Answer 2

$\it \dfrac{5}{1-x} \in \mathbb{Z} \Rightarrow 1-x|5 \Rightarrow 1-x \in D_5 \Rightarrow 1-x\in \{-5,\ -1,\ 1,\ 5\} |_{\cdot(-1)}\Rightarrow \\ \\ \\ \Rightarrow x-1\in \{-5,\ -1,\ 1,\ 5\} |_{+1} \Rightarrow x \in\{-4,\ 0,\ 2,\ 6\}$

c)

[tex]\it (x+5)(x-2) = 18 \Rightarrow x^2-2x+5x-10-18=0 \Rightarrow \\ \\ \Rightarrow x^2+3x-28=0 \Rightarrow x^2-4x+7x-28=0 \Rightarrow \\ \\ \Rightarrow x(x-4)+7(x-4)=0 \Rightarrow (x-4)(x+7) =0\Rightarrow \\ \\ \Rightarrow\begin{cases} \it x+7=0\Rightarrow x_1=-7 \\ \\ \it x-4=0\Rightarrow x_2=4\end{cases}[/tex]

d)

[tex]\it \dfrac{-6}{x-2} \in \mathbb{N} \Rightarrow x-2\ \textless \ 0\ \ \ \ (1) \\ \\ \\ x-2\in D_6 \ \ \ \ (2) \\ \\ \\ (1),\ (2) \Rightarrow x-2 \in\{-6,\ -3\ -2,\ -1 \}|_{+2} \Rightarrow x\in \{-4,\ -1,\ 0,\ 1\}[/tex]

e)

[tex]\it \dfrac{7x}{x+2} \in\mathbb{Z} \Rightarrow x+2|7x \ \ \ \ \ (1) \\ \\ \\ Dar,\ x+2|x+2 \Rightarrow x+2|(x+2)\cdot7 \Rightarrow x+2|7x+14 \ \ \ \ (2) \\ \\ \\ (1),\ (2) \Rightarrow x+2|7x+14-7x \Rightarrow x+2|14 \Rightarrow x+2 \in D_{14} \Rightarrow[/tex]

[tex]\it \Rightarrow x+2 \in \{\pm1,\ \ \pm2,\ \ \pm7,\ \ \pm14\} \Rightarrow \\ \\ \Rightarrow x+2\in \{-14, \ -7,\ -2,\ -1, \ 1, \ 2, \ 7, \ 14\}|_{-2} \Rightarrow \\ \\ \Rightarrow x \in \{-16,\ -9,\ -4,\ -3,\ -1,\ 0,\ 5,\ 12 \}[/tex]