Question

Aratati ca daca x,y sunt numere reale astfel incat |x|<1 si |y|<1 atunci |(x-y)/(1-x*y)| <1

Answer 1

$Pentru~a~demonstra~ca~ \big| \frac{x-y}{1-xy} \big |\ \textless \ 1,~trebuie~sa~aratam~ca \\ \\ \frac{x-y}{1-xy} \in (-1,1) . \\ \\ Observam~ca~x,y \in (-1,1)~de~unde~rezulta~ca~xy\ \textless \ 1~si~mai~apoi \\ \\ \boxed{1-xy\ \textgreater \ 0}~. \\ \\ 1.~Demonstram~ca~ \frac{x-y}{1-xy}\ \textgreater \ -1. \\ \\ \frac{x-y}{1-xy}\ \textgreater \ -1 \Leftrightarrow x-y\ \textgreater \ -1+xy \Leftrightarrow x-y+1-xy\ \textgreater \ 0 \Leftrightarrow \\ \\ \Leftrightarrow (x+1)(1-y)\ \textgreater \ 0,~evident!~(deoarece~x\ \textgreater \ -1,~si~deci~x+1\ \textgreater \ 0,~ \\ \\ iar~y\ \textless \ 1,~si~deci~1-y\ \textgreater \ 0).$

$2.~Demonstram~ca~ \frac{x-y}{1-xy}\ \textless \ 1. \\ \\ \frac{x-y}{1-xy}\ \textless \ 1 \Leftrightarrow x-y\ \textless \ 1-xy \Leftrightarrow 1-xy-x+y\ \textgreater \ 0 \Leftrightarrow \\ \\ \Leftrightarrow~(1-x)(1+y)\ \textgreater \ 0, ~evident! \\ \\ Din~1.~si~2.~rezulta~ca~ \big| \frac{x-y}{1-xy} \big | \ \textless \ 1.$