Let's solve the expression:

$\frac{1}{2} - \frac{1}{3} - \frac{5}{6} + \frac{2}{3}$

Step 1: Find a common denominator.

The denominators are 2, 3, and 6. The least common denominator (LCD) of 2, 3, and 6 is 6.

Step 2: Rewrite all fractions with the LCD of 6.

$\frac{1}{2} = \frac{3}{6}, \quad \frac{1}{3} = \frac{2}{6}, \quad \frac{5}{6} = \frac{5}{6}, \quad \frac{2}{3} = \frac{4}{6}$

Step 3: Substitute into the original expression.

$\frac{3}{6} - \frac{2}{6} - \frac{5}{6} + \frac{4}{6}$

Step 4: Combine the fractions.

First, combine all the numerators:

$(3 - 2 - 5 + 4) / 6 = 0 / 6$

Step 5: Simplify.

$0 / 6 = 0$

Thus, the value of the expression is $\boxed{0}$.

Would you like more details or have any questions? Here are some related questions you might consider:

1. How do you find the least common denominator of fractions?
2. What are the steps for adding and subtracting fractions with different denominators?
3. How do you simplify complex fractions?
4. What is the importance of equivalent fractions in arithmetic?
5. How can you handle fractions in algebraic expressions?

Tip: Always look for the least common denominator to make fraction operations easier.