Solve and check the equation. Begin your work by rewriting the equation without fractions. \( \frac{x}{3}+\frac{x}{2}=\frac{5}{6} \) Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. The solution set is \( \} \). (Type an integerfor a simplified fraction.) B. The solution set is \( \{x \mid x \) is a real number\}. C. The solution set is \( \varnothing \).

To begin with, let's eliminate the fractions by finding a common denominator for the terms. The least common multiple of 3, 2, and 6 is 6. We can multiply the entire equation by 6: \[ 6 \left(\frac{x}{3}\right) + 6 \left(\frac{x}{2}\right) = 6 \left(\frac{5}{6}\right) \] This simplifies to: \[ 2x + 3x = 5 \] Combining like terms gives us: \[ 5x = 5 \] Now, divide both sides by 5 to solve for \(x\): \[ x = 1 \] Checking the solution, we substitute \(x=1\) back into the original equation: \[ \frac{1}{3} + \frac{1}{2} = \frac{5}{6} \] Finding a common denominator for the left side (which is 6): \[ \frac{2}{6} + \frac{3}{6} = \frac{5}{6} \] So, the left side equals the right side, confirming that \(x = 1\) is indeed the solution. Thus, the correct choice is A. The solution set is \( \{1\} \).