Question

Resolva as seguintes inequações:

*(imagem)*

Answer 1

$\boxed{\begin{array}{l}\tt a)~\sf -x^2-x+12\leqslant0\\\sf fac_{\!\!,}a~f(x)=-x^2-x+12\\\sf devemos~ent\tilde ao~mostrar~qual~\acute e~o~intervalo\\\sf da~func_{\!\!,}\tilde ao~que~satisfaz~a~desigualdade~proposta.\end{array}}$

$\boxed{\begin{array}{l}\underline{\rm ra\acute izes\!:}\\\sf -x^2-x+12=0\\\sf\Delta=b^2-4ac\\\sf\Delta=(-1)^2-4\cdot(-1)\cdot12\\\sf\Delta=1+48\\\sf\Delta=49\\\sf x=\dfrac{-b\pm\sqrt{\Delta}}{2a}\\\\\sf x=\dfrac{-(-1)\pm\sqrt{49}}{2\cdot(-1)}\\\\\sf x=\dfrac{1\pm7}{-2}\begin{cases}\sf x_1=\dfrac{1+7}{-2}=\dfrac{8}{-2}=-4\\\\\sf x_2=\dfrac{1-7}{-2}=\dfrac{-6}{-2}=3\end{cases}\\\\\sf f(x)<0~para~x\leqslant-4~~ou~~x\geqslant3\\\sf portanto\\\sf S=\{x\in\mathbb{R}/x\leqslant-4~~ou~~x\geqslant3\} \end{array}}$

$\boxed{\begin{array}{l}\tt b)~\sf 2x^2-7x+3<0\\\sf fac_{\!\!,}a~g(x)=2x^2-7x+3\\\sf vamos~mostrar~qual~intervalo~da~func_{\!\!,}\tilde ao\\\sf corresponde~a~desigualdade.\end{array}}$

$\large\boxed{\begin{array}{l}\underline{\rm ra\acute izes\!:}\\\sf 2x^2-7x+3=0\\\sf\Delta=b^2-4ac\\\sf\Delta=(-7)^2-4\cdot2\cdot3\\\sf\Delta=49-24\\\sf\Delta=25\\\sf x=\dfrac{-b\pm\sqrt{\Delta}}{2a}\\\\\sf x=\dfrac{-(-7)\pm\sqrt{25}}{2\cdot2}\\\\\sf x=\dfrac{7\pm5}{4}\begin{cases}\sf x_1=\dfrac{7+5}{4}=\dfrac{12}{4}=3\\\\\sf x_2=\dfrac{7-5}{4}=\dfrac{2\div2}{4\div2}=\dfrac{1}{2}\end{cases}\\\\\sf g(x)<0~para~\dfrac{1}{2}<x<3\\\sf portanto\\\sf S=\bigg\{x\in\mathbb{R}/\dfrac{1}{2}<x<3\bigg\}\end{array}}$

$\boxed{\begin{array}{l}\tt c)~\sf x^2+4x+4>0\\\sf fac_{\!\!,}a~h(x)=x^2+4x+4\\\sf devemos~assinalar~o~intervalo~da~func_{\!\!,}\tilde ao\\\sf que~satisfaz~a~desigualdade.\end{array}}$

$\large\boxed{\begin{array}{l}\underline{\rm ra\acute izes\!:}\\\sf x^2+4x+4=0\\\sf (x+2)^2=0\\\sf x+2=0\\\sf x=-2\\\sf h(x)>0~para~todo~x\ne-2\\\sf portanto\\\sf S=\{x\in\mathbb{R}/x\ne-2\}\end{array}}$

$\boxed{\begin{array}{l}\tt d)~\sf 9x^2-12x+4\\\sf fac_{\!\!,}a~p(x)=9x^2-12x+4\\\sf devemos~assinalar~o~intervalo~da~func_{\!\!,}\tilde ao\\\sf que~corresponde~a~desigualdade~pedida.\end{array}}$

$\boxed{\begin{array}{l}\underline{\rm ra\acute izes\!:}\\\sf 9x^2-12x+4=0\\\sf(3x-2)^2=0\\\sf 3x-2=0\\\sf 3x=2\\\sf x=\dfrac{2}{3}\\\sf p(x)~jamais~ser\acute a~negativo\\\sf portanto\\\sf S=\bigg\{~~\bigg\}\end{array}}$

$\large\boxed{\begin{array}{l}\tt e)~\sf x^2-4x+4\geqslant0\\\sf fac_{\!\!,}a~m(x)=x^2-4x+4\\\sf devemos~assinalar~os~intervalos~da~func_{\!\!,}\tilde ao\\\sf que~satisfazem~a~desigualdade.\end{array}}$

$\large\boxed{\begin{array}{l}\underline{\rm ra\acute izes\!:}\\\sf x^2-4x+4=0\\\sf (x-2)^2=0\\\sf x-2=0\\\sf x=2\\\sf m(x)\geqslant0~\forall~x\in\mathbb{R}\\\sf portanto\\\sf S=\{x\in\mathbb{R}\}\end{array}}$