Question

Resolva em |R as equações: A) |x-8|=3 ; B)|3x-1|= -4 ; C) |2x+3|=3x-6 ;

Answer 1

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$\large\begin{array}{l} \textsf{Resolver as equa\c{c}\~oes modulares:}\\\\ \textsf{Tenha sempre em mente que m\'odulo nunca pode ser igual}\\\textsf{a um n\'umero negativo.}\\\\\\ \mathsf{A)~\big|x-8\big|=3}\\\\ \mathsf{x-8=\pm 3} \end{array}$

$\large\begin{array}{l}\begin{array}{rcl} \mathsf{x-8=-3}&~\textsf{ ou }~&\mathsf{x-8=3}\\\\ \mathsf{x=-3+8}&~\textsf{ ou }~&\mathsf{x=3+8}\\\\ \mathsf{x=5}&~\textsf{ ou }~&\mathsf{x=11} \end{array}\\\\\\ \textsf{Conjunto solu\c{c}\~ao: }\mathsf{S=\{5,\,11\}.} \end{array}$

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$\large\begin{array}{l} \mathsf{B)~\left|3x-1\right|=-4}\\\\ \textsf{Como }\mathsf{-4<0}\textsf{ (n\'umero negativo), a equa\c{c}\~ao acima n\~ao}\\\textsf{tem solu\c{c}\~ao.}\\\\\\ \textsf{Conjunto solu\c{c}\~ao: }\mathsf{S=\varnothing}\\\\\textsf{(conjunto vazio).} \end{array}$

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$\large\begin{array}{l} \mathsf{C)~\left|2x+3\right|=3x-6}\\\\ \textsf{Como o m\'odulo nunca \'e negativo, devemos ter como}\\\textsf{condi\c{c}\~ao de exist\^encia}\\\\ \mathsf{3x-6\ge 0}\\\\ \mathsf{3x\ge 6}\\\\ \mathsf{x\ge \dfrac{6}{3}}\\\\ \mathsf{x\ge 2\qquad\quad(i)} \end{array}$


$\large\begin{array}{l} \textsf{Resolvendo a equa\c{c}\~ao:}\\\\ \mathsf{\left|2x+3\right|=3x-6}\\\\ \mathsf{2x+3=\pm\,(3x-6)}\\\\ \begin{array}{rcl} \mathsf{2x+3=3x-6}&~\textsf{ ou }~&\mathsf{2x+3=-(3x-6)}\\\\ \mathsf{2x+3=3x-6}&~\textsf{ ou }~&\mathsf{2x+3=-3x+6}\\\\ \mathsf{2x-3x=-6-3}&~\textsf{ ou }~&\mathsf{2x+3x=6-3} \end{array} \end{array}$

                    $\large\begin{array}{rcl} \mathsf{-x=-9}&~\textsf{ ou }~&\mathsf{5x=3}\\\\ \mathsf{x=9}&~\textsf{ ou }~&\mathsf{x=\dfrac{3}{5}}\quad\left(\textsf{n\~ao serve, pois }\mathsf{\dfrac{3}{5}<2}\right) \end{array}$


$\large\textsf{Conjunto solu\c{c}\~ao: }\mathsf{S=\{9\}.}$

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


Tags: equação modular condição de existência solução resolver álgebra