Question

Resolva a inequação [(-x²+x-20)³]/[x²(x-1)^5] < 0?

A) (1, ∞) B) (–∞, -1] C) (–∞, 1) D) [0, ∞) E) (–∞, 0)

no gabarito da letra A

Answer 1

$\displaystyle \frac{(-x^2+x-20)^3}{x^2(x-1)^5}\ \textless \ 0\iff \frac{-x^2+x-20}{x-1}\ \textless \ 0\wedge x\neq 0\\ \\ \frac{(-x^2+x-20)^3}{x^2(x-1)^5}\ \textless \ 0\iff \frac{x^2-x+20}{x-1}\ \textgreater \ 0\wedge x\neq 0\\ \\ \frac{(-x^2+x-20)^3}{x^2(x-1)^5}\ \textless \ 0\iff (x^2-x+20)(x-1)\ \textgreater \ 0\wedge x\neq 0\wedge x\neq 1\\ \\ \text{Calculemos el discriminante de }x^2-x+20:\\ \Delta=(-1)^2-4(20)=-79\ \textless \ 0\Longrightarrow x^2-x+20\ \textgreater \ 0 \text{ entonces:}$

$\displaystyle \frac{(-x^2+x-20)^3}{x^2(x-1)^5}\ \textless \ 0\iff x-1\ \textgreater \ 0\wedge x\neq 0\wedge x\neq 1\\ \\ \frac{(-x^2+x-20)^3}{x^2(x-1)^5}\ \textless \ 0\iff x\ \textgreater \ 1\\ \\ \\ \boxed{\frac{(-x^2+x-20)^3}{x^2(x-1)^5}\ \textless \ 0\iff x\in(1,+\infty)}$