Question

Determine a soluçao da inequaçao simultania 0<-ײ-2×+3<3 no cobjunto dos numeros reais

Answer 1

$\large\boxed{\begin{array}{l}\sf0<-x^2-2x+3<3\implies\begin{cases}\sf -x^2-2x+3>0\\\sf -x^2-2x+3<3\end{cases}\end{array}}$

$\large\boxed{\begin{array}{l}\sf -x^2-2x+3>0\\\underline{\boldsymbol{fac_{\!\!,}a}}\\\sf f(x)=-x^2-2x+3\\\sf devemos~dizer~para~quais~valores~de~x\\\sf teremos~f(x)>0.\\\underline{\rm ra\acute izes~de~f(x):}\\\sf -x^2-2x+3=0\cdot(-1)\\\end{array}}$$\large\boxed{\begin{array}{l}\sf x^2+2x-3=0\\\sf\Delta=b^2-4ac\\\sf\Delta=2^2-4\cdot1\cdot(-3)\\\sf\Delta=4+12\\\sf\Delta=16\\\sf x=\dfrac{-b\pm\sqrt{\Delta}}{2a}\\\\\sf x=\dfrac{-2\pm\sqrt{16}}{2\cdot1}\\\\\sf x=\dfrac{-2\pm4}{2}\begin{cases}\sf x_1=\dfrac{-2+4}{2}=\dfrac{2}{2}=1\\\\\sf x_2=\dfrac{-2-4}{2}=-\dfrac{6}{2}=-3\end{cases}\\\sf f(x)>0\Longleftrightarrow -3<x<1\\\sf s_1=\{x\in\mathbb{R}/-3<x<1\}\end{array}}$

$\Large\boxed{\begin{array}{l}\sf-x^2-2x+3<3\\\sf -x^2-2x+\backslash\!\!\!3-\backslash\!\!\!3<0\\\sf -x^2-2x<0\\\underline{\boldsymbol{fac_{\!\!,}a}}\\\sf g(x)=-x^2-2x\\\sf devemos~dizer~para~quais~valores~de~x\\\sf teeremos~g(x)<0.\\\underline{\rm ra\acute izes~de~g(x):}\\\sf -x^2-2x=0\cdot(-1)\\\sf x^2+2x=0\\\sf x\cdot(x+2)=0\\\sf x=0\\\sf x+2=0\\\sf x=-2\\\sf g(x)<0\Longleftrightarrow x<0~ou~x>2\\\sf s_2=\{x\in\mathbb{R}/x<0~ou~x>2\}\end{array}}$ $\Large\boxed{\begin{array}{l}\underline{\rm Observe~a~figura~que~eu~anexei}\\\sf o~conjunto~soluc_{\!\!,}\tilde ao~da~inequac_{\!\!,}\tilde ao\\\sf \acute e~obtida~fazendo~a~intersecc_{\!\!,}\tilde ao\\\sf das~soluc_{\!\!,}\tilde oes~encontrada~anteriormente.\\\sf Desse~modo\\\sf S=s_1\cap s_2\\\sf S=\{x\in\mathbb{R}/-3<x<0\}\end{array}}$