First, the denominator of the fraction must be different then 0 that

3 - 5x = 0 follows that 3/5 can not be solution to the equation.

By the definition for absolute value we know that

|x| = x if and only if x >=0

|x| = -x if and only if x < 0

There for we have the following 2 cases:

1. 3-5x > 0 that is 3 > 5x or 3/5 > x

For x < 3/5 we have |3 - 5x| = 3 - 5x and we need to solve the following equation

1 / (3 - 5x) = 8

(3 - 5x) * 8 = 1

24 - 40x = 1

40 x = 24 - 1

x = 23 / 40

x = 0.575 < 0.6 (3/5 = 0.6) follows x = 0.575 is one solution for the equation 1/|3-5x|=8.

And indeed 1 / |3 - 5*0.575| = 1 / |0.125| = 1 / 0.125 = 8

2. 3 - 5x < 0 that is 3 < 5x or 3/5 < x

For x > 3/5 we have |3 - 5x| = - ( 3 - 5x) = 5x - 3 and we need to solve the following equation

1 / (5x - 3) = 8

(5x - 3) * 8 = 1

40x - 24 = 1

40 x = 24 + 1

x = 25 / 40

x = 0.625 > 0.6 (3/5 = 0.6) follows x = 0.625 is second solution for the equation 1/|3-5x|=8.

And indeed 1 / |3 - 5*0.625| = 1 / |- 0.125| = 1 / 0.125 = 8

Finally we can say that we found all the solution of the equation 1/|3-5x|=8 that is 0.575 and 0.625.