Question

Fie expresia E(x)=(3xsupra x+1. -1):(1+ 3x patrat supra lui x patrat-1)
a)aratati ca E(x)=x-1 supra 1+2x
b)rezolvati ecuatia 2*E(x)-7=0.
AJUTATIMA VA ROG FRUMOS

Answer 1

a)

$\displaystyle C.E.\left \{ {{x+1\neq0} \atop {x^2-1\neq0}} \right. \Rightarrow x\in\mathbb{R}\backslash\{\pm1\} \\\displaystyle E(x)=\left(\frac{3x}{x+1}-1\right):\left(1+\frac{3x^2}{x^2-1}\right)=\frac{3x-(x+1)}{x+1}:\frac{x^2-1+3x^2}{x^2-1}=\frac{2x-1}{x+1}\cdot\frac{x^2-1}{4x^2-1}=\frac{2x-1}{x+1}\cdot\frac{(x+1)(x-1)}{(2x+1)(2x-1)}=\frac{x-1}{2x+1},\forall x\in\mathbb{R}\backslash\{\pm1\}$

b)

$2E(x)-7=0\overset{a)}{\Rightarrow}\displaystyle2\frac{x-1}{2x+1}=7\Rightarrow 2x-2=14x+7\Rightarrow x=-\frac34$