Your staring inequality is

    log(x+6) < log(3-2x).


First, determine the domain, i.e. the set of real numbers, where this inequality makes sense.


Logarithm must have positive arguments:  x+6 > 0  and  3-2x > 0.

First inequality gives  x > - 6;  the second inequality gives  3 > 2x,  or   x < 1.5.


So, the domain is this set  -6 < x < 1.5.     (1)



Next, logarithm is monotonic function; therefore, from the given inequality we have

    x + 6 < 3 - 2x.


Simplify it

    x + 2x < 3 - 6

      3x   <   -3

       x   <   -1.


Comparing the domain (1) with the last inequality, we see that the solution to the problem is this set

    -6 < x < -1.