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Answer 1 · 2016-09-24T00:06:02

#x in [2, oo)#

The absolute value function is defined as

#|a| = {(a" if "a>=0), (-a" if "a<0):}#

Thus, we consider two cases:

Case 1: #x-2 >= 0#

Then #|x-2| = x-2#, so

#3(x-2) = 3x-6#

#=> 3x-6 = 3x-6#

As this is a tautology, the equation is true for any #x# satisfying #x-2 >= 0#... that is, for any #x >= 2#

Case 2: #x-2 < 0#

Then #|x-2| = -(x-2) = 2-x#, so

#3(2-x) = 3x-6#

#=> 6-3x = 3x-6#

#=> 6x = 12#

#=> x = 2#

As we began this case with the restriction #x-2 < 0#, that is, #x < 2#, there are no solutions within the interval #(-oo, 2)#.

Taken together, we can see that the equation holds true if and only if #x >= 2#, so we have the solution set #x in [2, oo)#.