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

Let's simplify the left-hand side first! We have \(2x - \frac{x}{2}\). To combine, think of it as \(2x - 0.5x\), which gives us \(1.5x\) or \(\frac{3x}{2}\). So now the equation looks like: \[ \frac{3}{2}\cdot\frac{3x}{2} = \frac{x+1}{3} \] Next, we can cross-multiply to eliminate the fractions: \[ 3 \cdot \frac{3x}{2} \cdot 3 = 1 \cdot (x + 1) \implies \frac{9x}{2} = \frac{x + 1}{3} \] Now, multiply both sides by \(6\) to clear the denominators: \[ 9x \cdot 3 = (x + 1) \cdot 2 \implies 27x = 2x + 2 \] Subtract \(2x\) from both sides: \[ 25x = 2 \] Now divide by \(25\): \[ x = \frac{2}{25} \] So the final answer is \(x = \frac{2}{25}\)!