Solve the following equations 1. \( \frac{x}{2}-\frac{x}{3}=2 \) 2. \( \frac{x+2}{4}-\frac{x-6}{3}=\frac{1}{2} \)

To solve the first equation \( \frac{x}{2} - \frac{x}{3} = 2 \), we first need a common denominator. The least common multiple of 2 and 3 is 6. We can rewrite the equation as: \[ \frac{3x}{6} - \frac{2x}{6} = 2 \] Combining the fractions gives: \[ \frac{3x - 2x}{6} = 2 \Rightarrow \frac{x}{6} = 2 \] Multiplying both sides by 6 yields: \[ x = 12 \] Now, for the second equation \( \frac{x+2}{4} - \frac{x-6}{3} = \frac{1}{2} \), we again find a common denominator, which is 12. Rewriting the equation gives us: \[ \frac{3(x+2)}{12} - \frac{4(x-6)}{12} = \frac{6}{12} \] This simplifies to: \[ \frac{3(x+2) - 4(x-6)}{12} = \frac{6}{12} \] Multiplying through by 12 leads to: \[ 3(x+2) - 4(x-6) = 6 \] Expanding the left side results in: \[ 3x + 6 - 4x + 24 = 6 \] Combining like terms: \[ -x + 30 = 6 \] Subtracting 30 from both sides gives: \[ -x = -24 \Rightarrow x = 24 \] Thus, the solutions are \( x = 12 \) for the first equation and \( x = 24 \) for the second equation.