log4 (4x+3) < log4 (5x-3/2x-3)

I think you meant:



Think I'll put the more complicated side on the left
and reverse the inequality:









4 raised to the power of both sides will preserve the inequality
since log is an increasing function.













It might help to draw a graph of y = the left side and see where the
graph is positive:



It has vertical asymptotes at x=3/2 and x=-3/4
We need the x-intercepts which are the zeros of the numerator.






So the graph is positive (above the x-axis) when x is between
the left horizontal asymptote and the left-most x-intercept,

That's 
And, the graph is positive again when x is between the right-most
asymptote and the right-most x-intercept.

That's 

Answer:   and 

Edwin