Question

|x+5|=2x+1?? respostassss?

Answer 1

$\large\begin{array}{l} \textsf{Resolver a equa\c{c}\~ao modular}\\\\ \mathsf{|x+5|=2x+1\qquad(i)} \end{array}$


$\large\begin{array}{l} \bullet~~\textsf{Condi\c{c}\~oes de exist\^encia:}\\\\ \textsf{Perceba que a express\~ao do lado direito \'e igual ao m\'odulo}\\\textsf{de um n\'umero real. Como m\'odulo n\~ao pode ser negativo,}\\\textsf{devemos ter}\\\\ \mathsf{2x+1\ge 0}\\\\ \mathsf{2x\ge -1}\\\\ \mathsf{x\ge -\,\dfrac{1}{2}\qquad(ii)} \end{array}$


$\large\begin{array}{l} \bullet~~\textsf{Voltando \`a equa\c{c}\~ao (i), devemos ter}\\\\ \mathsf{|x+5|=2x+1}\\\\ \mathsf{x+5=\pm(2x+1)}\\\\ \begin{array}{rcl} \mathsf{x+5=2x+1}&~\textsf{ ou }~&\mathsf{x+5=-(2x+1)}\\\\ \mathsf{x+5=2x+1}&~\textsf{ ou }~&\mathsf{x+5=-2x-1}\\\\ \mathsf{x-2x=1-5}&~\textsf{ ou }~&\mathsf{x+2x=-1-5}\\\\ \mathsf{-x=-4}&~\textsf{ ou }~&\mathsf{3x=-6}\\\\ \mathsf{x=4}&~\textsf{ ou }~&\mathsf{x=-2\quad}\textsf{(n\~ao serve, pois }\mathsf{x\ge -\frac{1}{2}}\textsf{)} \end{array} \end{array}$


$\large\begin{array}{l} \textsf{Portanto, a \'unica solu\c{c}\~ao \'e}\\\\ \boxed{\begin{array}{c}\mathsf{x=4}\end{array}}\\\\\\ \textsf{Conjunto solu\c{c}\~ao: }\mathsf{S=\{4\}.} \end{array}$


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$\large\begin{array}{l} \textsf{D\'uvidas? Comente.}\\\\\\ \textsf{Bons estudos! :-)} \end{array}$


Tags: resolver equação modular condições existência testar solução