To solve the equation |3x - 1/6| + (-5/12) = 2/7, we need to consider two cases: when the expression inside the absolute value is positive and when it is negative.

Case 1: 3x - 1/6 > 0
In this case, the absolute value can be removed, so the equation becomes:
3x - 1/6 - 5/12 = 2/7
To simplify, we can find a common denominator for the fractions on the right side:
3x - 1/6 - 10/12 = 2/7
Multiplying the fractions by 2/2 to get a common denominator:
3x - 1/6 - 20/12 = 2/7
Now, we can combine the fractions:
3x - 1/6 - 20/12 = 2/7
3x - 1/6 - 10/6 = 2/7
Combining the fractions on the left side:
3x - 11/6 = 2/7
To eliminate the fraction, we can multiply both sides by 42 (the least common multiple of 6 and 7):
42(3x - 11/6) = 42(2/7)
42(3x - 11/6) = 12
Distributing the 42 on the left side:
126x - 462/6 = 12
Simplifying the fraction on the left side:
126x - 77 = 12
Adding 77 to both sides:
126x = 89
Dividing both sides by 126:
x = 89/126
Simplifying the fraction:
x = 17/24

Case 2: 3x - 1/6 < 0
In this case, the expression inside the absolute value becomes negative, so we need to change its sign and solve the equation:
-(3x - 1/6) - 5/12 = 2/7
Distributing the negative sign:
-3x + 1/6 - 5/12 = 2/7
To simplify, we can find a common denominator for the fractions on the right side:
-3x + 1/6 - 10/12 = 2/7
Multiplying the fractions by 2/2 to get a common denominator:
-3x + 1/6 - 20/12 = 2/7
Now, we can combine the fractions:
-3x + 1/6 - 20/12 = 2/7
-3x + 1/6 - 10/6 = 2/7
Combining the fractions on the left side:
-3x - 59/6 = 2/7
To eliminate the fraction, we can multiply both sides by 42 (the least common multiple of 6 and 7):
42(-3x - 59/6) = 42(2/7)
42(-3x - 59/6) = 12
Distributing the 42 on the left side:
-126x - 2478/6 = 12
Simplifying the fraction on the left side:
-126x - 413 = 12
Adding 413 to both sides:
-126x = 425
Dividing both sides by -126:
x = 425/-126
Simplifying the fraction:
x = -17/5

Therefore, the solutions to the equation are x = 17/24 and x = -17/5.