\( \frac{3}{}(2x-\frac{x}{2})=\frac{x+1}{3} \)

To solve the equation \( \frac{3}{}(2x-\frac{x}{2})=\frac{x+1}{3} \), let's first simplify the left-hand side. We combine the terms inside the parentheses: \[ 2x - \frac{x}{2} = \frac{4x}{2} - \frac{x}{2} = \frac{3x}{2} \] Now the equation looks like: \[ 3 \cdot \frac{3x}{2} = \frac{x+1}{3} \] This simplifies to: \[ \frac{9x}{2} = \frac{x+1}{3} \] Next, we cross-multiply to eliminate the fractions: \[ 9x \cdot 3 = (x + 1) \cdot 2 \] This expands to: \[ 27x = 2x + 2 \] To isolate \(x\), subtract \(2x\) from both sides: \[ 27x - 2x = 2 \] \[ 25x = 2 \] Now divide both sides by 25: \[ x = \frac{2}{25} \] So the solution to the equation is: \[ x = \frac{2}{25} \]