Solve for \( x \) and write your answer in simplest form. \[ 3\left(-\frac{1}{3} x+\frac{1}{2}\right)+2 x=-1-\left(6 x-\frac{1}{2}\right) \]

First, let's simplify the equation step by step. Starting with the equation: \[ 3\left(-\frac{1}{3} x+\frac{1}{2}\right)+2 x=-1-\left(6 x-\frac{1}{2}\right) \] We can distribute the \( 3 \) on the left side: \[ 3 \cdot -\frac{1}{3} x + 3 \cdot \frac{1}{2} + 2x = -1 - (6x - \frac{1}{2}) \] This simplifies to: \[ -x + \frac{3}{2} + 2x = -1 - 6x + \frac{1}{2} \] Now, combine the \( x \) terms on the left side: \[ (-x + 2x) + \frac{3}{2} = -1 - 6x + \frac{1}{2} \] This results in: \[ x + \frac{3}{2} = -1 - 6x + \frac{1}{2} \] Next, let's combine the constant terms on the right side: \[ x + \frac{3}{2} = -1 + \frac{1}{2} - 6x \] Calculating the right side gives: \[ x + \frac{3}{2} = -\frac{2}{2} + \frac{1}{2} - 6x \] Thus: \[ x + \frac{3}{2} = -\frac{1}{2} - 6x \] Now, add \( 6x \) to both sides: \[ x + 6x + \frac{3}{2} = -\frac{1}{2} \] This simplifies to: \[ 7x + \frac{3}{2} = -\frac{1}{2} \] Next, subtract \( \frac{3}{2} \) from both sides: \[ 7x = -\frac{1}{2} - \frac{3}{2} \] This results in: \[ 7x = -\frac{4}{2} = -2 \] Finally, divide both sides by \( 7 \): \[ x = -\frac{2}{7} \] Thus, the solution is: \[ \boxed{-\frac{2}{7}} \]