Question

Para quais valores de x vale $-2~\textless~x^{2}-3~\textless~\dfrac{1}{5}$ ?

Answer 1

$-2~\textless~x^{2}-3~\textless~\dfrac{1}{5}$

Somando 3 a todos os membros:

$-2+3~\textless~x^{2}-3+3~\textless~\dfrac{1}{5}+3\\\\\\1~\textless~x^{2}~\textless~\dfrac{1+15}{5}\\\\\\1~\textless~x^{2}~\textless~\dfrac{16}{5}$

Tirando a raiz quadrada de todos os membros:

$\sqrt{1}~\textless~\sqrt{x^{2}}~\textless~\sqrt{\dfrac{16}{5}}\\\\\\1~\textless~|x|~\textless~\dfrac{4}{\sqrt{5}}$

Se $x\ge0$, então $|x|=x$, e

$1~\textless~x~\textless~\dfrac{4}{\sqrt{5}}$

Se $x~\textless~0$, então $|x|=-x$, e

$1~\textless-x~\textless~\dfrac{4}{\sqrt{5}}$

Multiplicando todos os membros por -1 (e invertendo os sinais de desigualdade):

$-\dfrac{4}{\sqrt{5}}~\textless~x~\textless~-1$

Portanto, x deve ser tal que:

$\boxed{\boxed{x\in\bigg(-\dfrac{4}{\sqrt{5}},-1\bigg)\cup\bigg(1,\dfrac{4}{\sqrt{5}}\bigg)}}$

Answer 2

Ola Maria

-2 < x² - 3 < 1/5 

-2 + 3 < x² < 16/5 

1 < x² < 16/5 

1 < x < 4√5/5