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2x-1/x>1 multiply by x
2x-1 > x subtract x
x -1 >0 add 1
x>1

my text gives your answer and also x<0 as the solution set.

so x<0 or x>1

what I do not get if if you plug a number less >0 the inequality does not hold true.

yes it does!!
try -1
-2-1/-1 =-3/-1 = 3 which is >1 ;)