(MACKENZIE-1974) Se o número $\,{\large x}\,$ é solução da equação $\;\sqrt[{\large 3}]{x + 9}\, -\, \sqrt[{\large 3}]{x\, -\, 9}\, =\, 3\;$, então $\;x^{\large 2}\;$ está entre:
✓ mostrar resposta ... FGV-1974) Resolver a desigualdade
$\phantom{XX}1\,-\,3x\, >\, \sqrt{2\,+\,x^2\,-\,3x}\;$:
a)
$x < \dfrac{3-\sqrt{41}}{16}$
c)
$x < 1\;\;$ ou $\;\;x > 2$
d)
$\dfrac{1}{3}\leqslant x \leqslant \dfrac{3 + \sqrt{41}}{16}$
e)
$x < \dfrac{3 - \sqrt{41}}{16}\;\;$ ou $\;\; x > \dfrac{3 + \sqrt{41}}{16}$
✓ mostrar resposta ... (ITA - 1973) A respeito da equação
$\phantom{XX}{\large 3x^2\,-\,4x\,+\,\sqrt{3x^2\,-\,4x\,-6}\,=\,18}\;$
podemos dizer:
a)
${\large \frac{2\pm\sqrt{70}}{3}}\;$ são raízes
c)
A única raiz é $x=2+\sqrt{10}$
d)
tem 2 raízes reais e 2 imaginárias
✓ mostrar resposta ... (MACKENZIE - 1976) Todas as raízes da equação $\;{\large 2\sqrt{x}\,+\,2x^{(-\frac{1}{2})}\,=\,5}\;$ estão no intervalo:
c)
$\,[\dfrac{1}{5},\dfrac{9}{2}]$
✓ mostrar resposta ... (FEI-1968) Seja V o conjunto dos números reais que são solução da equação irracional $\; \sqrt{2x} - \sqrt{7 + x} = 1\;$
a) $V = \{2,18\}$
b) $V=\{2\}$
c) $V=\{18\}$
d) $V=\varnothing$
e) nenhuma das anteriores
✓ mostrar resposta ... (FUVEST - 1977) Resolva (em $\,\mathbb{R}\,$) a inequação
$\phantom{XXX}\dfrac{\large x^2\,-\,x\,-\,1}{\large \sqrt{x^2\,-\,3x}}\,\geqslant\,0\phantom{X}$
✓ mostrar resposta ... resposta:
$\phantom{X}\dfrac{\large x^2\,-\,x\,-\,1}{\large \sqrt{x^2\,-\,3x}}\,>\,0\phantom{X}\Longleftrightarrow\;\left\{ \begin{array}{rcr} x^{\large 2}\,-\,3x\,>\,0\phantom{XX}&(I) \\ x^{\large 2}\,-\,x\,-\,1\,\geqslant\,0\,&(II) \\ \end{array}\right.$
Solução de (I) $\,x^2\,-\,3x\,>\,0\;\Rightarrow\;x(x\,-\,3)\,>\,0\;\Longleftrightarrow\;x\,<\,0\,$ ou $\,x\,>\,3\,$
O gráfico de $\;f(x)\,=\,x^2\,-\,3x\;$ é uma parábola como na figura:
Temos então do gráfico que a solução de
(I) é $\;S_1\,=\,\left\{\,x\,\in\,\mathbb{R}\;|\;x\,<\,0\;\mbox{ou}\;x\,>\,3\,\right\}\,$
Solução de (II) Como o gráfico $\,f(x)\,=\,x^2\,-\,x\,-\,1\,$ é uma parábola do tipo:
então: $\,x^2\,-\,x\,-1\,\geqslant\,0\;\Longleftrightarrow\;x\,\leqslant\,\dfrac{1\,-\,\sqrt{5}}{2}\;\mbox{ou}\;x\,\geqslant\,\dfrac{1\,+\,\sqrt{5}}{2}\;\,$ e temos o conjunto temporário da situação
(II) $\;S_2\,=\,\left\{\,x\,\in\,\mathbb{R}\;|\;x\,\leqslant\,\dfrac{1\,-\,\sqrt{5}}{2}\;\mbox{ou}\;x\,\geqslant\,\dfrac{1\,+\,\sqrt{5}}{2}\,\right\}\;$
Solução da questão (Conjunto Verdade) A solução é o conjunto Verdade, a intersecção dos dois conjuntos $\,S_1\,$ e $\,S_2\,$
$\;V\,=\,S_1\,\cap\,S_2\,=\,\left\{\,x\,\in\,\mathbb{R}\;|\;x\,\leqslant\,\dfrac{1\,-\,\sqrt{5}}{2}\;\mbox{ou}\;x\,>\,3\,\right\}\;$ conforme o diagrama abaixo:
RESPOSTA: $\,V\,=\,\lbrace\,x\,\in\,\mathbb{R}\,|\,x\,\leqslant\,\dfrac{1\,-\,\sqrt{5}}{2}\;\mbox{ou}\;x\,>\,3 \rbrace\,$
× Resolver as inequações:
a)
$\,4\,\lt\,x^2\,-\,12\,\leqslant\,4x\,$
b)
$\,x^2\,+\,1\,\lt\,2x^2\,-\,3\,\leqslant\,-5x\,$
c)
$\,0\,\leqslant\,x^2\,-\,3x\,+\,2\,\leqslant\,6\,$
d)
$\,7x\,+\,1\,\lt\,x^2\,+\,3x\,-\,4\,\leqslant\,2x\,+\,2\,$
e)
$\,0\,\lt\,x^2\,+\,x\,+\,1\,\lt\,1\,$
e)
$\,4x^2\,-\,5x\,+\,4\,\lt\,3x^2\,-\,6x\,+\,6\,\lt\,x^2\,+\,3x\,-\,4\,$
✓ mostrar resposta ... resposta: a)$\,\mathbb{S}\,=\,\lbrace x\,\in\,{\rm I\!R}\;|\;4\,\lt\,x\,\leqslant\,6\rbrace$
b)$\,\mathbb{S}\,=\,\lbrace x\,\in\,{\rm I\!R}\;|\;-3\,\leqslant\,x\,\lt\,-2\rbrace$
c)$\,\mathbb{S}\,=\,\lbrace x\,\in\,{\rm I\!R}\;|\;-1\,\leqslant\,x\,\leqslant\,1\phantom{X}{\text ou}\phantom{X}2\,\leqslant\,x\,\leqslant\,4\rbrace$
d)$\,\mathbb{S}\,=\,\lbrace x\,\in\,{\rm I\!R}\;|\;-3\,\leqslant\,x\,\lt\,-1\rbrace$
e)$\,\mathbb{S}\,=\,\lbrace x\,\in\,{\rm I\!R}\;|\;-1\,\lt\,x\,\lt\,0\rbrace$
f)$\,\mathbb{S}\,=\,\varnothing\;$
× Resolver a inequação $\phantom{X}x^2\,-\,2x\,+\,2\;\gt\;0\phantom{X}$.
✓ mostrar resposta ... resposta: $\,\mathbb{S}\,=\,{\rm I\!R}\,$
× Resolver a inequação $\phantom{X}x^2\,-\,2x\,+\,1\;\leqslant\;0\phantom{X}$.
✓ mostrar resposta ... resposta: $\,\mathbb{S}\,=\,\lbrace1\rbrace\,$
× Resolver a inequação $\phantom{X}-2x^2\,+\,3x\,+\,2\;\geqslant\;0\phantom{X}$.
✓ mostrar resposta ... resposta: $\,\mathbb{S}\,=\,\lbrace x\,\in\,{\rm I\!R}\;|\;-\frac{1}{2}\,\leqslant\,x\,\leqslant\,2\rbrace\;$
× Resolver as inequações do segundo grau a seguir em $\,{\rm I\!R}\,$:
a)
$\,x^2\,-\,3x\,+\,2\;\gt\;0\,$
b)
$\,-x^2\,+\,x\,+\,6\;\gt\;0\,$
c)
$\,-3x^2\,-\,8x\,+\,3\;\leqslant\;0\,$
d)
$\,-x^2\,+\,\dfrac{\,3\,}{\,2\,}x\,+\,10\;\geqslant\;0\,$
e)
$\,8x^2\,-\,14x\,+\,3\;\leqslant\;0\,$
f)
$\,4x^2\,-\,4x\,+\,1\;\gt\;0\,$
g)
$\,x^2\,-\,6x\,+\,9\;\geqslant\;0\,$
h)
$\,-4x^2\,+\,12x\,-\,9\;\geqslant\;0\,$
i)
$\,x^2\,+\,3x\,+\,7\;\gt\;0\,$
j)
$\,-3x^2\,+\,3x\,-\,3\;\lt\;0\,$
k)
$\,2x^2\,-\,4x\,+\,5\;\lt\;0\,$
l)
$\,-\dfrac{\,1\,}{\,3\,}x^2\,+\,\dfrac{\,1\,}{\,2\,}x\,-\,\dfrac{\,1\,}{\,4\,}\;\gt\;0\,$
✓ mostrar resposta ... resposta:
a)
$\,\mathbb{S}\,=\,\lbrace x\,\in\,{\rm I\!R}\;|\;x\,\lt\,1\phantom{X}{\text ou}\phantom{X}x\,\gt\,2\rbrace\;$
b)
$\,\mathbb{S}\,=\,\lbrace x\,\in\,{\rm I\!R}\;|-2\,\lt\,x\,\lt\,3\rbrace\;$
c)
$\,\mathbb{S}\,=\,\lbrace x\,\in\,{\rm I\!R}\;|\;x\,\leqslant\,-3\phantom{X}{\text ou}\phantom{X}x\,\geqslant\,\frac{\,1\,}{\,3\,}\rbrace\;$
d)
$\,\mathbb{S}\,=\,\lbrace x\,\in\,{\rm I\!R}\;|\;-\frac{\,5\,}{\,2\,}\,\leqslant\,x\,\leqslant\,4\rbrace\;$
e)
$\,\mathbb{S}\,=\,\lbrace x\,\in\,{\rm I\!R}\;|\;\frac{\,1\,}{\,4\,}\,\leqslant\,x\,\leqslant\,\frac{\,3\,}{\,2\,}\rbrace\;$
f)
$\,\mathbb{S}\,=\,{\rm I\!R}\;-\;\lbrace \frac{\,1\,}{\,2\,}\rbrace\,$
g)
$\,\mathbb{S}\,=\,{\rm I\!R}\,$
h)
$\,\mathbb{S}\,=\,\lbrace \frac{\,3\,}{\,2\,}\rbrace\;$
i)
$\,\mathbb{S}\,=\,{\rm I\!R}\,$
j)
$\,\mathbb{S}\,=\,{\rm I\!R}\,$
k)
$\,\mathbb{S}\,=\,\varnothing\,$
l)
$\,\mathbb{S}\,=\,\varnothing\,$
× Resolver a inequação $\phantom{X}(x^2\,-\,x\,-\,2)(-x^2\,+\,4x\,-\,3)\;\lt\;0\phantom{X}$ em $\,{\rm I\!R}\,$.
✓ mostrar resposta ... resposta: $\,\mathbb{S}\,=\,\lbrace x\,\in\,{\rm I\!R}\;|\;-1\,\lt\,x\,\lt\,1\phantom{X}{\text ou}\phantom{X}2\,\lt\,x\,\lt\,3\rbrace\;$
× Resolver em $\,{\rm I\!R}\,$ as inequações:
a)
$\,(1\,-\,4x^2)\centerdot(2x^2\,+\,3)\,\gt\,0\,$
b)
$\,(2x^2\,-\,7x\,+\,6)\centerdot(2x^2\,-\,7x\,+\,5)\,\leqslant\,0\,$
c)
$\,(x^2\,-\,x\,-\,6)\centerdot(-x^2\,+\,2x\,-1)\,\gt\,0\,$
d)
$\,(x^2\,+\,x\,-\,6)\centerdot(-x^2\,-2x\,+\,3)\,\geqslant\,0\,$
e)
$\,x^3\,-\,2x^2\,-\,x\,+\,2\,\gt\,0\,$
f)
$\,2x^3\,-\,6x^2\,+\,x\,-\,3\,\leqslant\,0\,$
✓ mostrar resposta ... resposta:
a)
$\,\mathbb{S}\,=\,\lbrace x\,\in\,{\rm I\!R}\;|\;-\frac{3}{2}\,\lt\,x\,\lt\,-\frac{1}{2}\phantom{X}{\text ou}\phantom{X}0\,\lt\,x\,\lt\,\frac{1}{2}\rbrace\;$
b)
$\,\mathbb{S}\,=\,\lbrace x\,\in\,{\rm I\!R}\;|\;1\,\leqslant\,x\,\leqslant\,\frac{3}{2}\phantom{X}{\text ou}\phantom{X}2\,\leqslant\,x\,\leqslant\,\frac{5}{2}\rbrace\;$
c)
$\,\mathbb{S}\,=\,\lbrace x\,\in\,{\rm I\!R}\;|\;-2\,\lt\,x\,\lt\,3\phantom{X}{\text e}\phantom{X}x\,\ne\,1\rbrace\;$
d)
$\,\mathbb{S}\,=\,\lbrace x\,\in\,{\rm I\!R}\;|\;x\,=\,-3\phantom{X}{\text ou}\phantom{X}1\,\leqslant\,x\,\leqslant\,2\rbrace\;$
e)
$\,\mathbb{S}\,=\,\lbrace x\,\in\,{\rm I\!R}\;|\;-1\,\lt\,x\,\lt\,1\phantom{X}{\text ou}\phantom{X}x\,\gt\,2\rbrace\;$
e)
$\,\mathbb{S}\,=\,\lbrace x\,\in\,{\rm I\!R}\;|\;x\,\leqslant\,3\rbrace\;$
× (MAPOFEI - 1971) É dada a função $\phantom{X}y\,=\,(2x^2\,-\,9x\,-\,5)(x^2\,-\,2x\,+2)\phantom{X}$
Determinar:
a)
os pontos de intersecção do gráfico da função com o eixo das abscissas.
b)
o conjunto dos valores de x para os quais $\,y\,\leqslant\,0\,$.
✓ mostrar resposta ... resposta:
a) P1 = (5, 0) e P2 = (-1/2, 0)
b) $\,\mathbb{S}\,=\,\lbrace x\,\in\,{\rm I\!R}\;|\;-\frac{1}{2}\,\leqslant\,x\,\leqslant\,5\rbrace\;$
× Resolver a inequação $\phantom{X}\dfrac{\;2x^2\,+\,x\,-\,1\;}{2x\,-\,x^2}\;\geqslant\;0\phantom{X}$
✓ mostrar resposta ... resposta: $\,\mathbb{S}\,=\,\lbrace x\,\in\,{\rm I\!R}\;|\;x\leqslant\,-1\phantom{X}{\text ou}\phantom{X}0\,\lt\,x\,\leqslant\,\frac{1}{2}\phantom{X}{\text ou}\phantom{X}x\,\gt\,2\rbrace\;$
× Resolver as inequações em $\,{\rm I\!R}\,$:
a)
$\,\dfrac{\;4x^2\,+\,x\,-\,5\;}{2x^2\,-3x\,-\,2}\;\gt\;0\,$
b)
$\,\dfrac{\;-9x^2\,+\,9x\,-\,2\;}{\;3x^2\,+\,7x\,+\,2\;}\;\geqslant\;0\,$
c)
$\,\dfrac{\;x^2\,+\,2x\;}{\;x^2\,+\,5x\,+\,6\;}\;\geqslant\;0\,$
d)
$\,\dfrac{\;2\,-\,3x\;}{\;2x^2\,+\,3x\,-\,2\;}\;\lt\;0\,$
e)
$\,\dfrac{\;x^2\,+\,3x\,-\,16\;}{\;-x^2\,+\,7x\,-\,10\;}\;\geqslant\;1\,$
f)
$\,\dfrac{\;2x^2\,+\,4x\,+\,5\;}{\;3x^2\,+\,7x\,+\,2\;}\;\lt\;-2\,$
g)
$\,\dfrac{\;6x^2\,+\,12x\,+\,17\;}{\;-2x^2\,+\,7x\,-\,5\;}\;\geqslant\;-1\,$
h)
$\,\dfrac{\;(x\,+\,1)^3\,-\,1\;}{\;(x\,-\,1)^3\,+\,1\;}\;\gt\;1\,$
✓ mostrar resposta ... resposta: a)$\,\mathbb{S}\,=\,\lbrace x\,\in\,{\rm I\!R}\;|\;x\,\lt\,-\frac{5}{4}\phantom{X}{\text ou}\phantom{X}-\frac{1}{2}\,\lt\,x\,\lt\,1\phantom{X}{\text ou}\phantom{X}x\,\gt\,2\rbrace$
b)$\,\mathbb{S}\,=\,\lbrace x\,\in\,{\rm I\!R}\;|\;x\,\lt\,-2\phantom{X}{\text ou}\phantom{X}-\frac{1}{3}\,\lt\,x\,\leqslant\,\frac{1}{3}\phantom{X}{\text ou}\phantom{X}x\,\geqslant\,\frac{2}{3}\rbrace$
c)$\,\mathbb{S}\,=\,\lbrace x\,\in\,{\rm I\!R}\;|\;x\,\lt\,-3\phantom{X}{\text ou}\phantom{X}x\,\geqslant\,0\rbrace$
d)$\,\mathbb{S}\,=\,\lbrace x\,\in\,{\rm I\!R}\;|\;-2\,\lt\,x\,\lt\,\frac{1}{2}\phantom{X}{\text ou}\phantom{X}x\,\gt\,\frac{2}{3}\rbrace$
e)$\,\mathbb{S}\,=\,\lbrace x\,\in\,{\rm I\!R}\;|\;-1\,\leqslant\,x\,\lt\,2\phantom{X}{\text ou}\phantom{X}3\,\leqslant\,x\,\lt\,5\rbrace$
f)$\,\mathbb{S}\,=\,\lbrace x\,\in\,{\rm I\!R}\;|\;-2\,\lt\,x\,\lt\,-\frac{3}{2}\phantom{X}{\text ou}\phantom{X}-\frac{3}{4}\,\lt\,x\,\lt\,-\frac{1}{3}\rbrace$
g)$\,\mathbb{S}\,=\,\lbrace x\,\in\,{\rm I\!R}\;|\;-4\,\leqslant\,x\,\leqslant\,-\frac{3}{4}\phantom{X}{\text ou}\phantom{X}1\,\lt\,x\,\lt\,\frac{5}{2}\rbrace$
f)$\,\mathbb{S}\,=\,\lbrace x\,\in\,{\rm I\!R}\;|\;x\,\gt\,0\rbrace$
× Resolver os sistemas de inequações:
a)
$\,\left\{\begin{array}{rcr} x^2\,+\,x\,-\,2\;\gt\;0 & \\ 3x\,-\,x^2\,\lt\,0\phantom{XX}& \\ \end{array} \right.\,$
b)
$\,\left\{\begin{array}{rcr} x^2\,+\,x\,-\,20\;\leqslant\;0\;\;& \\ x^2\,-\,4x\,-\,21\,\gt\,0\;& \\ \end{array} \right.\,$
c)
$\,\left\{\begin{array}{rcr} 1\,+\,2x\;\geqslant\;0\phantom{XXXX}& \\ -4x^2\,+\,8x\,-\,3\,\lt\,0\;& \\ \end{array} \right.\,$
d)
$\,\left\{\begin{array}{rcr} -2x^2\,-\,x\,+\,1\,\geqslant\,0\phantom{X}& \\ 4x^2\,-\,8x\,+\,3\;\leqslant\,0\phantom{X}& \\ \end{array} \right.\,$
✓ mostrar resposta ... resposta:
a)$\,\mathbb{S}\,=\,\lbrace x\,\in\,{\rm I\!R}\;|\;x\,\lt\,-2\,\phantom{X}{\text ou}\phantom{X}x\,\gt\,3\rbrace$ × Resolver a inequação $\phantom{X}x^4\,-\,5x^2\,+\,4\,\geqslant\,0\phantom{X}$ em $\,{\rm I\!R}\,$.
✓ mostrar resposta ... resposta: $\,\mathbb{S}\,=\,\lbrace x\,\in\,{\rm I\!R}\;|\;x\,\leqslant\,-2\,\phantom{X}{\text ou}\phantom{X}-1\,\leqslant\,x\,\leqslant\,1\phantom{X}{\text ou}\phantom{X}x\,\geqslant\,2\rbrace$
× Resolver em $\,{\rm I\!R}\,$ as inequações a seguir:
a)
$\,x^4\,-\,10x^2\,+\,9\,\leqslant\,0\,$
b)
$\,x^4\,-\,3x^2\,-\,9\,\gt\,0\,$
c)
$\,x^4\,+\,8x^2\,-\,9\,\lt\,0\phantom{X}$
d)
$\,2x^4\,-\,3x^2\,+\,4\,\lt\,0\,$
e)
$\,x^6\,-\,7x^3\,-\,8\,\geqslant\,0\phantom{X}$
f)
$\,3x^4\,-\,5x^2\,+\,4\,\gt\,0\,$
✓ mostrar resposta ... resposta: a) $\,\mathbb{S}\,=\,\lbrace x\,\in\,{\rm I\!R}\;|\;-3\,\leqslant\,x\,\leqslant\,-1\,\phantom{X}{\text ou}\phantom{X}1\,\leqslant\,x\,\leqslant\,3\rbrace$
b) $\,\mathbb{S}\,=\,\lbrace x\,\in\,{\rm I\!R}\;|\;x\,\lt\,-2\,\phantom{X}{\text ou}\phantom{X}x\,\gt\,2\rbrace$
c) $\,\mathbb{S}\,=\,\lbrace x\,\in\,{\rm I\!R}\;|\;-1\,\lt\,x\,\lt\,1\rbrace$
d) $\,\mathbb{S}\,=\,\varnothing\,$
e) $\,\mathbb{S}\,=\,\lbrace x\,\in\,{\rm I\!R}\;|\;x\,\leqslant\,-1\,\phantom{X}{\text ou}\phantom{X}x\,\geqslant\,2\rbrace$
a) $\,\mathbb{S}\,=\,{\rm I\!R}\,$
× 
