To solve this equation, we will first combine the fractions on the left side:

4 5/19 + x + 1 1/38 = 12 1/2

Converting the mixed numbers to improper fractions:

(419 + 5)/19 + x + (138 + 1)/38 = 25/2

(76 + 5)/19 + x + (38 + 1)/38 = 25/2

81/19 + x + 39/38 = 25/2

Now, find a common denominator to add the fractions on the left side:

(812)/(192) + x + 39/38 = 12 1/2

162/38 + x + 39/38 = 25/2

Now, combine the fractions:

(162 + 39)/38 + x = 25/2

201/38 + x = 25/2

Next, subtract 201/38 from both sides to isolate x:

x = 25/2 - 201/38

x = 190/38 - 201/38

x = -11/38

Therefore, the value of x is -11/38.