Vertical asymptote when 2x-1=0, x=½.
When x is very large we get the horizontal asymptote:
3/(2x-1)<-1 needs to be converted into a graph equation:
Let y=3/(2x-1)+1, then y=1 is the horizontal asymptote, because 3/(2x-1)→0.
The y intercept (when x=0) is -3+1=-2.
The x intercept (when y=0) is 3=-2x+1, 2x=1-3=-2, x=-1 is the x-intercept.
The inequality is solved by first assuming 2x-1>0 then:
3<-2x+1, 2x<1-3, 2x<-2, x<-1, but when x<-1, 2x-1=-3 and -3<0 so 2x-1 must be negative and 3>-2x+1, 2x>-2, x>-1 and x<½ (solution of inequality), which is usually written -1<x<½. It’s also the solution of y<0.
The graph shows y=3/(2x-1)+1 in red, the vertical asymptote in green and the horizontal asymptote in orange. The blue shaded area shows the inequality.