Question

Resolva as inequações:
$\frac{3x+2}{x} \ \textgreater \ 0\\\\\frac{x}{x+2} \ \textless \ 0$

Answer 1

a)

$\mathsf{\dfrac{3x+2}{x}\textless0}$

$\mathsf{f(x)=3x+2}\\\mathsf{f(x)=0\to~3x+2=0\to~x=-\dfrac{2}{3}}\\\mathsf{f(x)>0\to~x>-\dfrac{2}{3}}\\\mathsf{f(x)\textless0\to~x\textless-\dfrac{2}{3}}$

$\mathsf{g(x)=x}\\\mathsf{g(x)=0\to~x=0}\\\mathsf{g(x)>0\to~x>0}\\\mathsf{g(x)\textless0\to~x\textless0}$

$\mathsf{s=\{x\in\mathbb{R}|x\textless-\dfrac{2}{3}\}}$

$\mathsf{\dfrac{x}{x+2}\textless0}$

$\mathsf{f(x)=x}\\\mathsf{f(x)=0\to~x=0}\\\mathsf{f(x)\textless0\to~x\textless0}\\\mathsf{f(x)>0\to~x>0}$

$\mathsf{g(x)=x+2}\\\mathsf{g(x)=0\to~x+2=0\to~x=-2}\\\mathsf{g(x)>0\to~x>-2}\\\mathsf{g(x)\textless0\to~x\textless-2}$

$\boxed{\boxed{\boxed{\boxed{\mathsf{s=\{x\in\mathbb{R}|x\textless-2\}}}}}}$