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.

X<-1/6

(24x+4) ^-1<0

1 / (24x+4) <0

1 / (4 (6x+1)) <0 - denominator needs to be less than 0

4 (6x+1) <0

6x+1<0

6x<-1

x<-1/6