c) \( \frac{x}{2}-\frac{x}{3}=\frac{1}{6}\left(x-\frac{3}{2}\right)+x \) d) \( \frac{2 x}{3}-\frac{1}{2}=\frac{1}{3}\left(x-\frac{7}{3}\right) \)

Let's tackle the equation in part c) first: To simplify \( \frac{x}{2}-\frac{x}{3}=\frac{1}{6}\left(x-\frac{3}{2}\right)+x \), we want to combine like terms. The left side requires a common denominator, which would be 6, giving us \( \frac{3x}{6} - \frac{2x}{6} = \frac{x}{6} \). The right side simplifies to \( \frac{1}{6}(x - \frac{3}{2}) + x \), which can be rewritten as \( \frac{x}{6} - \frac{1}{4} + x \). After combining the terms and solving for \( x \), we get our solution. Now moving on to part d): We start by simplifying \( \frac{2 x}{3}-\frac{1}{2}=\frac{1}{3}\left(x-\frac{7}{3}\right) \). On the left side, we keep the term as is, and on the right side, we distribute \( \frac{1}{3} \) to both \( x \) and \( -\frac{7}{3} \) to get \( \frac{x}{3} - \frac{7}{9} \). Setting both sides equal to each other and providing a common denominator leads us to isolate \( x \) and solve for it accordingly. In both cases, careful arithmetic and maintaining the balance of the equations will lead you to the correct solutions.