Question

Rezolvati ecuatia : $[ \frac{2x-1}{5} ]=x+3$ , stiind ca am notat cu [a] este partea intreaga .

Answer 1

$\Big[ \frac{2x-1}{5} \Big]=x+3=k \in Z \Rightarrow x=k-3 \in Z. \\ \\ Ecuatia~devine:~ \Big[ \frac{2k-7}{5} \Big]=k \Rightarrow k+1 \ \textgreater \ \frac{2k-7}{5} \geq k . \\ \\ Din~k+1\ \textgreater \ \frac{2k-7}{5} \Rightarrow k\ \textgreater \ -4. \\ \\ Din~ \frac{2k-7}{5} \geq k \Rightarrow -\frac{7}{3} \geq k. \\ \\ Deci~ k \in \Big (-4; -\frac{7}{3} \Big] ,~dar~ k \in Z \Rightarrow k =-3 \Rightarrow x=-6. \\ \\ Intr-adevar:~\Big[ \frac{2 \cdot (-6)-1}{5} \Big] = -6+3 \Rightarrow Solutie:~x=-6.$