Question 198871


{{{abs(3x-6)=3}}} Start with the given equation



Break up the absolute value (remember, if you have {{{abs(x)=a}}}, then {{{x=-a}}} or {{{x=a}}})


{{{3x-6=-3}}} or {{{3x-6=3}}} Set the expression {{{3x-6}}} equal to the original value 3 and it's opposite -3





Now lets focus on the first equation  {{{3x-6=-3}}}



{{{3x=-3+6}}}Add 6 to both sides



{{{3x=3}}} Combine like terms on the right side



{{{x=(3)/(3)}}} Divide both sides by 3 to isolate x




{{{x=1}}} Divide





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Now lets focus on the second equation {{{3x-6=3}}}




{{{3x=3+6}}}Add 6 to both sides



{{{3x=9}}} Combine like terms on the right side



{{{x=(9)/(3)}}} Divide both sides by 3 to isolate x




{{{x=3}}} Divide






So the solutions to {{{abs(3x-6)=3}}} are:


{{{x=1}}} and {{{x=3}}}




Notice if we graph  {{{y=abs(3x-6)}}} and {{{y=3}}} (just set each side equal to y and graph), we get



{{{graph(500,500,-1,5,-10,10,abs(3x-6),3)}}}  Graph of {{{y=abs(3x-6)}}} (red) and {{{y=3}}}(green)


and we can see the two graphs intersect at {{{x=1}}} and {{{x=3}}}. So this verifies our answer.