Question

achar todos os valores reais de x de modo que parte real do numero complexo z=x-i/x+i seja negativo? heeelllpppp :)

Answer 1

$\mathrm{Z=\dfrac{x-i}{x+i}\ \to\ Z=\dfrac{x-i}{x+i}.\dfrac{x-i}{x-i}\ \to\ Z=\dfrac{x^2-2xi+i^2}{x^2-i^2}}\\\\\\ \mathrm{Z=\dfrac{x^2-2xi-1}{x^2+1}\ \to\ \boxed{\mathrm{Z=\bigg(\dfrac{x^2-1}{x^2+1}\bigg)-\bigg(\dfrac{2x}{x^2+1}\bigg)i}}}\\\\\\ \mathrm{\Rightarrow \Re{(Z)}\ \textless \ 0\ \to\ \dfrac{x^2-1}{x^2+1}\ \textless \ 0\ \to\ (x+1)(x-1)\ \textless \ 0}\\\\ \mathrm{*\ x\ \textgreater \ menor\ raiz\ \to\ x\ \textgreater \ -1}\\ \mathrm{*\ x\ \textless \ maior\ raiz\ \to\ x\ \textless \ 1}\\\\ \boxed{\boxed{\mathbf{\mathbb{S}=\{x\in\mathbb{R}\ |\ -1\ \textless \ x\ \textless \ 1\}}}}$