To solve this equation, we need to find the value of x.

First, let's simplify the left side of the equation:

(1/10 - 1/15 + 1/21 + ... + 1/120).x = 3/8

To simplify the sum of fractions, we need to find a common denominator. The least common multiple of 10, 15, 21, ..., 120 is 840. So, we can rewrite the left side of the equation as:

[(84/840) - (56/840) + (40/840) + ... + (7/840)].x = 3/8

Now, let's combine the fractions:

[(84 - 56 + 40 + ... + 7)/840].x = 3/8

To find the sum of the fractions, we can use the formula for the sum of an arithmetic series:

Sum = (n/2)(first term + last term)

In this case, the first term is 84, the last term is 7, and the common difference is -14. We can find the number of terms, n, by dividing the difference between the first and last term by the common difference and adding 1:

n = (7 - 84)/(-14) + 1 = 6

Now, let's find the sum:

Sum = (6/2)(84 + 7) = 6(91) = 546

Substituting this back into the equation:

[(546/840)].x = 3/8

To simplify further, we can divide both sides of the equation by the greatest common divisor of 546 and 840, which is 42:

[(546/42)(1/20)].x = (3/8)(1/42)

(13/20).x = 1/56

Now, let's isolate x by multiplying both sides of the equation by the reciprocal of (13/20):

x = (1/56)(20/13)

x = 20/728

Simplifying the fraction:

x = 5/182

Therefore, the value of x is 5/182.