Question

1)Determinati cardinalul multimii A={x ∈ Z* | |x²-1|*(7-|2x-1|) > 0 }

2)Determinati valorile intrei ale nr ,,n" pentru care : ________
√n²-6n+34 ∈ N


URGEENT !DAU CORONITA + 12 PUNCTE

Answer 1

$1) |x^2-1|\cdot (7-|2x+1|)\ \textgreater \ 0 \\ \\ \ |x^2-1| \ \textgreater \ 0,\quad \forall x \in \mathbb_{Z}^* \\ deci, problema noastra ramane ca 7-|2x+1| \ \textgreater \ 0 \\ \\ \Rightarrow -|2x+1| \ \textgreater \ -7 \Rightarrow |2x+1| \ \textless \ 7 \Rightarrow -7 \ \textless \ 2x+1\ \textless \ 7\Big{|}-1 \Rightarrow \\ \Rightarrow -8\ \textless \ 2x\ \textless \ 6\Big{|}:2 \Rightarrow -4\ \textless \ x\ \textless \ 3 \Rightarrow x \in \{-3,-2,-1,1,2\} \\ \\ \Rightarrow card(A) = \{5\}$
$( fara zero, deoarece x \in \mathbb_{Z}^*, steluta inseamna ca ia toate valorile din \mathbb_{Z} \\ inafara de 0)$


$2) \sqrt{n^2-6n+34}\in \mathbb_{N} \Rightarrow \sqrt{n^2-6n+9+25}\in \mathbb_{N} \\ \\ \Rightarrow \sqrt{(n-3)^2+25}\in \mathbb_{N} \Rightarrow \ n=3$