Question

A soma das raízes da equação |2x-3|=x-1

Answer 1

$\texttt{Pelo conceito de m\'odulo:}\\\\\\|x|=\left\{\begin{matrix}x,~se~x\geq0&\\-x,~se~x~<0&\end{matrix}\right$


$\texttt{Portanto temos o seguinte:}\\\\\\|2x-3|=\left\{\begin{matrix} 2x-3 & \\ -2x+3 & \end{matrix}\right.\\\\\\\texttt{Teremos duas solu\c{c}\~oes para x, observe:}$


$\texttt{Solu\c{c}\~ao 1:}\\\\\\2x-3=x-1\\\\x-3=-1\\\\ \fbox{ x=2 }\\\\\\\texttt{Solu\c{c}\~ao 2:}\\\\\\-2x+3=x-1\\\\-2x=x-4\\\\-3x=-4\\\\\\x=\dfrac{-4}{-3}\\\\\\\fbox{ x=\dfrac{4}{3} }$



$\texttt{Soma da ra\'izes (solu\c{c}\~oes):}\\\\\\2+\dfrac{4}{3}~\Leftrightarrow~ \dfrac{2}{1}+\dfrac{4}{3}~\Leftrightarrow~ \dfrac{6}{3}+\dfrac{4}{3}= \large\fbox{\fbox{ \dfrac{10}{3} }}$