This changes the problem a tiny bit, but the approach is still the same as with the other inequality. 4/x < -1/3 means that 4/x + 1/3 < 0. Again, bring the fractions to their common denominator to get:

(12 + x)/3x < 0.

We're looking for the intervals for which 12 + x and x don't share the sign. Again, we split the number line into three cases:

a. x is smaller than -12: in this case, 12 + x and 3x will both be negative. Since they do have the same sign, this interval is out.

b. is is between -12 and 0: in this case, 12 + x will be positive, but 3x will be negative (since -12 < x < 0 and therefore x is negative). This one is part of the answer.

c. x is greater than zero means that both 3x and 12 + x will be positive, so they share the sign. This case is out.