Question 239608
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*[tex \LARGE \ \ \ \ \ \ \ \ \ \ \frac{1}{6}|4x\ -\ 8|\ =\ 6x]


Multiply both sides by 6:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ |4x\ -\ 8|\ =\ 36x]


Apply the definition of the absolute value function:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ |x|\ =\ \left\{\ \ x\text{ if }x\ \geq\ 0\cr-x\text{ if }x\ <\ \,0\right]


So:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ 4x\ -\ 8\ =\ 36x]


or


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ 4x\ -\ 8\ =\ -36x]


Solve both equations for *[tex \Large x].  One value will be negative which would make the RHS of the original equation negative.  Since the absolute value function cannot have a negative output, the negative root of this equation must be excluded.


John
*[tex \LARGE e^{i\pi} + 1 = 0]
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