Solving the Equation
To solve the given equation 6x + 1/3 + 1 = x - 3/6, we need to isolate the variable x by simplifying the equation step by step.

Step 1: Combine Like Terms
First, let's combine the constants on both sides of the equation.
6x + 1/3 + 1 = x - 3/6
6x + 4/3 = x - 1/2

Step 2: Get rid of Fractions
To eliminate fractions, we can multiply the entire equation by the least common multiple (LCM) of the denominators. In this case, the LCM of 3 and 2 is 6.
6 * (6x + 4/3) = 6 * (x - 1/2)
36x + 8 = 6x - 3

Step 3: Solve for x
Now, we can simplify the equation further by expanding and combining like terms.
36x + 8 = 6x - 3
36x - 6x = -3 - 8
30x = -11
x = -11/30

Step 4: Check the Solution
Finally, we should substitute the value of x back into the original equation to verify if it satisfies the equation.
6(-11/30) + 1/3 + 1 = (-11/30) - 3/6
-11/5 + 1/3 + 1 = -11/30 - 1/2
-33/15 + 5/15 + 15/15 = -11/30 - 15/30
-13/15 = -26/30
-13/15 = -13/15
Therefore, the solution x = -11/30 is correct and satisfies the given equation.