You cannot take the log of 0 or negative numbers.
(3x-1)/x-4 >0
Find critical points: x-4 can't be 0, x=4 is a critical point. x=4 is also excluded from the domain because it creates division by 0.
3x-1 = 0
x = 1/3 is another critical point.
< 1/3, try 0.
(3(0)-1)/(0-4) = -1/-4 = 1/4 which is greater than 0. This works.
1/3 is excluded, because 3(1/3) - 1 = 0.
1/3 to 4, try 2.
(3(2)-1)/(2-4) = 5/-2 = -5/2 is negative. This interval doesn't work.
>4 try 10.
(3(10)-1)/(10-4) = 29/6 is positive. This works.
The domain is (-inf, 1/3) U (4,inf)
The domain is x<1/3 or x>4
