Question

Considere a express˜ao $E(x) = \frac{3-2x}{|1-2x|-|x|}$
. Fa¸ca o que se pede:
a. Encontre os valores de x tais que E(x) = 0.
b. Encontre os valores de x tais que E(x) > 0.
c. Encontre os valores de x tais que E(x) < 0.

Answer 1

Dada a expressão

    $\mathsf{E(x)=\dfrac{3-2x}{|1-2x|-|x|}}$

O denominador deve ser diferente de zero:

    $\mathsf{|1-2x|-|x|\ne 0}\\\\ \mathsf{|1-2x|\ne |x|}\\\\ \begin{array}{rcl} \mathsf{1-2x\ne x}&\mathsf{~~~e~~}&\mathsf{1-2x\ne -x}\\\\ \mathsf{1\ne x+2x}&\mathsf{~~~e~~}&\mathsf{1\ne -x+2x}\\\\ \mathsf{3x\ne 1}&\mathsf{~~~e~~}&\mathsf{x\ne 1}\\\\ \mathsf{x\ne \dfrac{1}{3}}&\mathsf{~~~e~~}&\mathsf{x\ne 1} \end{array}$

Agora, encontrar os valores de x, tais que

a)  E(x) = 0:

    $\mathsf{\dfrac{3-2x}{|1-2x|-|x|}=0}$

Basta que o numerador se anule:

    $\mathsf{3-2x=0}\\\\ \mathsf{3=2x}\\\\ \mathsf{x=\dfrac{3}{2}}$

Para as letras b e c, vamos montar um quadro de sinais do numerador e do denominador.

    $\mathsf{E(x)=\dfrac{N(x)}{D(x)}}$

onde $\mathsf{N(x)=3-2x~~e~~D(x)=|1-2x|-|x|.}$

O denominador D(x) envolve módulos, e por isso D(x) pode ser descrito por mais de uma sentença. Analisando cada módulo envolvido,

    $\mathsf{|1-2x|}=\left\{ \begin{array}{rl} \mathsf{1-2x,}&\mathsf{se~~1-2x\ge 0}\\\\ \mathsf{-(1-2x),}&\mathsf{se~~1-2x<0} \end{array} \right.\\\\\\\\ \mathsf{|1-2x|}=\left\{ \begin{array}{rl} \mathsf{1-2x,}&\mathsf{se~~x\le \dfrac{1}{2}}\\\\ \mathsf{2x-1,}&\mathsf{se~~x>\dfrac{1}{2}} \end{array} \right.\qquad\mathsf{(i)}$

    $\mathsf{|x|}=\left\{ \begin{array}{rl} \mathsf{x,}&\mathsf{se~~x\ge 0}\\\\ \mathsf{-x,}&\mathsf{se~~x<0} \end{array} \right.\qquad\mathsf{(ii)}$

Os módulos mudam de sentença quando x = 1/2 e x = 0.

Subtraindo (i) e (ii), devemos ter,

    $\mathsf{|1-2x|-|x|}=\left\{ \begin{array}{rl} \mathsf{(1-2x)-(-x),}&\mathsf{se~~x<0\le \dfrac{1}{2}}\\\\ \mathsf{(1-2x)-(x),}&\mathsf{se~~0\le x \le \dfrac{1}{2}}\\\\ \mathsf{(2x-1)-(x),}&\mathsf{se~~0\le \dfrac{1}{2}<x} \end{array} \right.\\\\\\\\ \mathsf{|1-2x|-|x|}=\left\{ \begin{array}{rl} \mathsf{1-x,}&\mathsf{se~~x<0}\\\\ \mathsf{1-3x,}&\mathsf{se~~0\le x \le \dfrac{1}{2}}\\\\ \mathsf{x-1,}&\mathsf{se~~x>\dfrac{1}{2}} \end{array} \right.\qquad\mathsf{(iii)}$

    (3 sentenças)

Estudo do sinal de D(x):

Quando x < 0,

     $\mathsf{-x>0}\\\\ \mathsf{1-x>1>0}\\\\ \mathsf{D(x)>1>0}$

Quando 0 ≤ x < 1/3,

    $\mathsf{0\le 3x<1}\\\\ \mathsf{-1<-3x\le 0}\\\\ \mathsf{1-1<1-3x\le 1+0}\\\\ \mathsf{0<1-3x\le 1}\\\\ \mathsf{0<D(x)\le 1}$

Quando 1/3 < x ≤ 1/2,

    $\mathsf{1<3x\le \dfrac{3}{2}}\\\\\\ \mathsf{-\,\dfrac{3}{2}\le -3x<-1}\\\\\\ \mathsf{1-\dfrac{3}{2}\le 1-3x<1-1}\\\\\\ \mathsf{-\,\dfrac{1}{2}\le 1-3x<0}\\\\\\ \mathsf{-\,\dfrac{1}{2}\le D(x)<0}$

Quando 1/2 < x < 1,

    $\mathsf{\dfrac{1}{2}-1<x-1<1-1}\\\\\\ \mathsf{-\,\dfrac{1}{2}<x-1<0}\\\\\\ \mathsf{-\,\dfrac{1}{2}<D(x)<0}$

Quando x > 1,

    $\mathsf{x-1>1-1}\\\\ \mathsf{x-1>0}\\\\ \mathsf{D(x)>0}$

O numerador é mais simples de analisar, pois é uma função linear decrescente, cuja raiz é x = 3/2.

Montando o quadro de sinais:

    $\large\begin{array}{cl} \mathsf{N(x)=3-2x}&\mathsf{\quad \overset{+++++++++++++++++++++++++++++}{\textsf{------}\!\!\underset{0}{\bullet}\!\!\textsf{------}\!\!\underset{\frac{1}{3}}{\circ}\!\!\textsf{--------}\!\!\underset{\frac{1}{2}}{\bullet}\!\!\textsf{--------------}\!\!\underset{1}{\circ}\!\!\textsf{--------------}}\!\!\underset{\frac{3}{2}}{\overset{0}{\bullet}}\!\!\overset{---}{\textsf{--------}}}\!\!\!\footnotesize\begin{array}{l}\blacktriangleright\end{array}\\\\ \mathsf{D(x)=|1-2x|-|x|}&\mathsf{\quad \overset{+++++++}{\textsf{------}\!\!\underset{0}{\bullet}\!\!\textsf{------}}\!\!\underset{\frac{1}{3}}{\overset{0}{\circ}}\!\!\overset{------------}{\textsf{--------}\!\!\underset{\frac{1}{2}}{\bullet}\!\!\textsf{--------------}}\!\!\underset{1}{\overset{0}{\circ}}\!\!\overset{++++++++++++}{\textsf{--------------}\!\!\underset{\frac{3}{2}}{\bullet}\!\!\textsf{--------}}}\!\!\!\footnotesize\begin{array}{l}\blacktriangleright\end{array} \end{array}$

    $\large\begin{array}{ll} \mathsf{E(x)=\dfrac{3-2x}{|1-2x|-|x|}}&\mathsf{\quad \overset{+++++++}{\textsf{------}\!\!\underset{0}{\bullet}\!\!\textsf{------}}\!\!\underset{\frac{1}{3}}{\circ}\!\!\overset{------------}{\textsf{--------}\!\!\underset{\frac{1}{2}}{\bullet}\!\!\textsf{--------------}}\!\!\underset{1}{\circ}\!\!\overset{+++++++}{\textsf{--------------}}\!\!\underset{\frac{3}{2}}{\overset{0}{\bullet}}\!\!\overset{---}{\textsf{--------}}}\!\!\!\footnotesize\begin{array}{l}\blacktriangleright\end{array}\end{array}$

Pelo quadro acima,

b)  E(x) > 0   para  x < 1/3  ou  1 < x < 3/2

c)  E(x) < 0   para 1/3 < x < 1  ou  x > 3/2.

Dúvidas? Comente.

Bons estudos! :-)

Answer 2

Resposta:

Giuseppe Garibaldi foi um general, guerrilheiro, condotiero e patriota italiano. Foi alcunhado de "herói de dois mundos", devido à sua participação em conflitos na Europa e na América do Sul. Wikipédia

Nascimento: 4 de julho de 1807, Nice, França

Falecimento: 2 de junho de 1882, Ilha Caprera, Itália

Cônjuge: Francesca Armosino (de 1880 a 1882), Giuseppina Raimondi (de 1860 a 1860), Anita Garibaldi (de 1842 a 1849)

Filhos: Menotti Garibaldi, Teresa Garibaldi, Ricciotti Garibaldi, MAIS

Pais: Maria Rosa Nicoletta Raimondi, Giovanni Domenico Garibaldi