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

To solve this equation, first, combine the terms on the left side. Start by finding a common denominator for the fractions, which is 6: \[ \frac{2x + 1}{2} + \frac{1}{3} = \frac{3(2x + 1)}{6} + \frac{2}{6} = \frac{6x + 3 + 2}{6} = \frac{6x + 5}{6} \] Setting this equal to \(x\), we have: \[ \frac{6x + 5}{6} = x \] Next, multiply both sides by 6 to eliminate the fraction: \[ 6x + 5 = 6x \] Subtract \(6x\) from both sides: \[ 5 = 0 \] This contradiction tells us that there are no solutions to the equation, as \(5\) cannot equal \(0\). Thus, the equation is inconsistent.