Question

1.Aflati n∈N, daca urmatoarele fractii sunt subunitare
$\frac{2n+1}{n+5}$
$\frac{20}{3n+1}$
2.Aflati n∈N, daca urmatoarele fractii sunt supraunitare
a) $\frac{4n+3}{n+9}$
b) $\frac{7}{5n-13}$
c) $\frac{15}{n+3}$

Answer 1

$1)\quad \dfrac{2n+1}{n+5} \rightarrow fractie subunitara \\ \\ \Rightarrow n+5 \ \textgreater \ 2n+1 \Rightarrow 0 \ \textgreater \ 2n-n+1-5 \Rightarrow 0 \ \textgreater \ n-4 \Rightarrow n-4\ \textless \ 0 \Rightarrow \\ \Rightarrow n \ \textless \ 4 \Rightarrow \boxed{n\in\Big\{0,1,2,3\Big\}}$

$\dfrac{20}{3n+1} \rightarrow fractie subunitara \\ \\ \Rightarrow 3n+1 \ \textgreater \ 20 \Rightarrow 3n \ \textgreater \ 20-1 \Rightarrow 3n \ \textgreater \ 19 \Rightarrow n \ \textgreater \ \dfrac{19}{3} \Rightarrow n\ \textgreater \ 6,(3) \Rightarrow \\ \Rightarrow \boxed{n\in\Big\{7,8,9,10,11,...,\infty\big\} }$



$2) \\ a)\quad \dfrac{4n+3}{n+9} \rightarrow fractie supraunitara \\ \\ \Rightarrow 4n+3 \ \textgreater \ n+9 \Rightarrow 4n-n \ \textgreater \ 9-3 \Rightarrow 3n\ \textgreater \ 6 \Rightarrow n \ \textgreater \ \dfrac{6}{3} \Rightarrow n \ \textgreater \ 2 \Rightarrow \\ \Rightarrow \boxed{n\in\Big\{3,4,5,6,7,...,\infty\Big\}}$

$b) \quad \dfrac{7}{5n-13} \rightarrow fractie supraunitara \\ \\ \Rightarrow 7 \ \textgreater \ 5n-13 \Rightarrow 7 + 13 \ \textgreater \ 5n \Rightarrow 20 \ \textgreater \ 5n \Rightarrow 5n \ \textless \ 20 \Rightarrow n \ \textless \ \dfrac{20}{5} \Rightarrow \\ \Rightarrow n \ \textless \ 4 \Rightarrow \boxed{n\in\Big\{0,1,2,3\Big\}}$

$c) \quad \dfrac{15}{n+3} \rightarrow fractie supraunitara \\ \\ \Rightarrow 15 \ \textgreater \ n+3 \Rightarrow 15-3 \ \textgreater \ n \Rightarrow 12 \ \textgreater \ n \Rightarrow n \ \textless \ 12 \Rightarrow \\ \\ \Rightarrow \boxed{n\in\Big\{0,1,2,3,4,5,6,7,8,9,10,11,12\Big\}}$