Solve the inequality (24x+4)^-1< 0

I think you meant  (24x+4)^(-1)<0.

This is equivalent to  1 / [4(6x+1)] < 0.

We must determine the set of values of x for which 6x+1 is not equal to zero and 6x+1 is less than 0 (because  1 / [4(6x+1)] < 0  for such values).

Solve 6x+1 < 0.    6x+1<0 becomes    6x < -1,    or   x < -1/6

This divides the number line into two halves:  (- infinity, -1/6) and (-1/6, infinity).

From each half, choose an x value not equal to -1/6.  If the original inequality is then true, you have found the interval that solves it.  If false, choose the other interval to represent your solution.