When you have inequalities involving moduluses you need to separate it into the greater than and less than 0 cases as follows

first do the case1) |2x+1|<4x-2

We want the intersection of and 2x+1<4x-2

2x+1<4x-2

3<2x

x>3/2

So the intersection is x > 3/2

Next we look at 2x+1 < 0

So we want the intersection of 2x+1<0 and -(2x+1)<4x-2

solving this gives x<-1/2 and x>1/6. This has no intersection so the final answer is simply x > 3/2.

If the intersection did exist, we would take the union of it and the first solution to get the final result.