Which value of \( x \) satisfies the equation \( \frac{5}{2}\left(x+\frac{1}{3}\right)=\frac{65}{6} ? \)

To solve the equation \( \frac{5}{2}\left(x+\frac{1}{3}\right)=\frac{65}{6} \), we first eliminate the fraction by multiplying both sides by 2: \[ 5\left(x+\frac{1}{3}\right) = \frac{65}{3} \] Next, we can expand the left side: \[ 5x + \frac{5}{3} = \frac{65}{3} \] Now, to isolate \( 5x \), we subtract \( \frac{5}{3} \) from both sides: \[ 5x = \frac{65}{3} - \frac{5}{3} \] Simplifying the right side gives: \[ 5x = \frac{65 - 5}{3} = \frac{60}{3} = 20 \] Now, we divide both sides by 5 to solve for \( x \): \[ x = \frac{20}{5} = 4 \] Thus, the value of \( x \) that satisfies the equation is \[ \boxed{4} \]