Question

qual resultado dessas 2 inequaçao de 1 grau?

m-2+5m<2(m-1)

$\frac{m}{2} - \frac{4}{3} \ \textgreater \ \frac{2m}{3}$

Answer 1

$\texttt{Primeira inequa\c{c}\~ao:}$

$m-2+5m~\ \textless \ ~2(m-1)~~\texttt{(distributiva)}\\\\6m-2\ \textless \ 2m-2~~\texttt{(divida ambos os lados por 2)}\\\\3m-1\ \textless \ m-1~~\texttt{(diminua m em ambos os lados)}\\\\2m-1\ \textless \ -1~~\texttt{(some +1 em ambos os lados)}\\\\2m\ \textless \ 0~~\texttt{(divida ambos os lados por 2)}\\\\\ \fbox{\fbox{ m\ \textless \ 0 }}\\ \fbox{\fbox{ ]-\infty,0[~=m\in\mathbb{R}~|~m\ \textless \ 0 }}$

$\texttt{Segunda inequa\c{c}\~ao:}$

$\frac{m}{2}-\frac{4}{3}\ \textgreater \ \frac{2m}{3}~~\texttt{(reduza~o~lado~esquerdo~ao~mesmo~denominador)}\\\\ \frac{3m}{6}-\frac{8}{6}\ \textgreater~\frac{2m}{3}~~\texttt{(aqui~o~lado~esquerdo~j\'a~est\'a~reduzido)}\\\\ \frac{3m-8}{6} \ \textgreater \ \frac{2m}{3}~~\texttt{(multiplique~ambos~os~lados~por~6)} \\\\ 6\hspace{-6}\diagup\cdot \frac{3m-8}{6\hspace{-6}\diagup}\ \textgreater \ \frac{2m}{3}\cdot 6 \\\\3m-8\ \textgreater \ \frac{12m}{3}\\\\3m-8\ \textgreater \ 4m~~\texttt{(fa\c{c}a~a~subtra\c{c}\~ao~de~(3m)~de~ambos~os~lados)}$

$-3\hspace{-8}\diagup m\hspace{-8}\diagup +3\hspace{-7}\diagup m\hspace{-7}\diagup-8\ \textgreater \ 4m-3m\\\\-8\ \textgreater \ m\\\\ \fbox{\fbox{ m\ \textless \ -8 }}\\\\ \fbox{\fbox{]-\infty,~-8[~=~m\in\mathbb{R}~|~m\ \textless \ -8 }}$