Question

Resolvam as seguintes inequações do 2º grau
em IR:
a) 3(x - 1) - 6x > 2 - 2x(x - 3)
b) 2(x - 1)2 < x
c) - 2x² - x + 1 < 0​

Answer 1

$\boxed{\begin{array}{l}\tt a)~\sf 3\cdot(x-1)>2-2x(x-3)\\\sf 3x-3>2-2x^2+6x\\\sf 2x^2-6x+3x-3-2>0\\\sf 2x^2-3x-5>0\\\sf f(x)=2x^2-3x-5\\\underline{\rm ra\acute izes\!:}\\\end{array}}$

$\large\boxed{\begin{array}{l}\sf 2x^2-3x-5=0\\\sf\Delta=b^2-4ac\\\sf\Delta=(-3)^2=-4\cdot2\cdot(-5)\\\sf\Delta=9+40\\\sf\Delta=49\\\sf x=\dfrac{-b\pm\sqrt{\Delta}}{2a}\\\\\sf x=\dfrac{-(-3)\pm\sqrt{49}}{2\cdot2}\\\\\sf x=\dfrac{3\pm7}{4}\begin{cases}\sf x_1=\dfrac{3+7}{4}=\dfrac{10\div2}{4\div2}=\dfrac{5}{2}\\\\\sf x_2=\dfrac{3-7}{4}=-\dfrac{4}{4}=-1\end{cases}\\\\\sf f(x)>0 ~se~x<-1~ou~x>\dfrac{5}{2}\\\sf S=\bigg\{x\in\mathbb{R}/x<-1~ou~x>\dfrac{5}{2}\bigg\}\end{array}}$

$\large\boxed{\begin{array}{l}\tt b)~\sf 2\cdot(x-1)^2<x\\\sf 2(x^2-2x+1)-x<0\\\sf 2x^2-4x+2-x<0\\\sf 2x^2-5x+2<0\\\sf g(x)=2x^2-5x+2\\\underline{\rm ra\acute izes\!:}\end{array}}$

$\large\boxed{\begin{array}{l}\sf 2x^2-5x+2=0\\\sf\Delta=b^2-4ac\\\sf\Delta=(-5)^2-4\cdot2\cdot2\\\sf\Delta=25-16\\\sf\Delta=9\\\sf x=\dfrac{-b\pm\sqrt{\Delta}}{2a}\\\\\sf x=\dfrac{-(-5)\pm\sqrt{9}}{2\cdot2}\\\\\sf x=\dfrac{5\pm3}{4}\begin{cases}\sf x_1=\dfrac{5+3}{4}=\dfrac{8}{4}=2\\\\\sf x_2=\dfrac{5-3}{2}=\dfrac{2\div2}{4\div2}=\dfrac{1}{2}\end{cases}\\\\\sf g(x)<0~se~\dfrac{1}{2}<x<2\\\\\sf S=\bigg\{x\in\mathbb{R}/\dfrac{1}{2}<x<2\bigg\}\end{array}}$

$\large\boxed{\begin{array}{l}\tt c)~\sf -2x^2-x+1<0\\\sf h(x)=-2x^2-x+1\\\underline{\rm ra\acute izes\!:}\end{array}}$

$\large\boxed{\begin{array}{l}\sf-2x^2-x+1=0\\\sf\Delta=b^2-4ac\\\sf\Delta=(-1)^2-4\cdot(-2)\cdot1\\\sf\Delta=1+8\\\sf\Delta=9\\\sf x=\dfrac{-b\pm\sqrt{\Delta}}{2a}\\\\\sf x=\dfrac{-(-1)\pm\sqrt{9}}{2\cdot(-2)}\\\\\sf x=\dfrac{1\pm3}{-4}\begin{cases}\sf x_1=\dfrac{1+3}{-4}=-\dfrac{4}{4}=-1\\\\\sf x_2=\dfrac{1-3}{-4}=\dfrac{-2\div2}{-4\div2}=\dfrac{1}{2}\end{cases}\\\\\sf h(x)<0~se~x<-1~ou~x>\dfrac{1}{2}\\\\\sf S=\bigg\{x\in\mathbb{R}/x<-1~ou~x>\dfrac{1}{2}\bigg\}\end{array}}$