How do you solve absolute value inequality abs(2x-4)-1>0?

1 Answer
Alan P.
Apr 7, 2015

Split the absolute value sub-expression into its two cases
(2x-4)<0 rarr x<2
and
(2x-4)>=0 rarr x>=2

If (2x-4)<0
then
abs(2x-4)-1>0
is equivalent to
-2x+4-1 >0
-2x > -3
x < 2/3 remember multiplying by a negative reverses the inequality.
(Note that this is consistent with the requirement x<2)

If (2x-4)>=0
then
abs(2x-4) -1>0
is equivalent to
2x-4-1 >=0
2x>5
x>5/2
(Again, note that this is consistent with the requirement x>=2)

So the solution to the given inequality is all values of x such that
x<2/3 or x> 2 1/2

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