Let's solve the equation step by step:

$\frac{1}{3}(x - 2) - \frac{1}{2}(x + 1) = \frac{5}{6}(x + 2)$

Step 1: Distribute the fractions

Distribute the fractions on both sides:

$\frac{1}{3}(x - 2) = \frac{1}{3}x - \frac{2}{3}$ $\frac{1}{2}(x + 1) = \frac{1}{2}x + \frac{1}{2}$ $\frac{5}{6}(x + 2) = \frac{5}{6}x + \frac{10}{6}$

Substituting these into the original equation:

$\frac{1}{3}x - \frac{2}{3} - \left( \frac{1}{2}x + \frac{1}{2} \right) = \frac{5}{6}x + \frac{10}{6}$

Step 2: Simplify both sides

Now, simplify the left-hand side by distributing the negative sign:

$\frac{1}{3}x - \frac{2}{3} - \frac{1}{2}x - \frac{1}{2} = \frac{5}{6}x + \frac{10}{6}$

Step 3: Get a common denominator

To combine the terms, get a common denominator for the fractions involving $x$. The least common denominator (LCD) of 3 and 2 is 6.

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

So the equation becomes:

$\left( \frac{2}{6}x - \frac{3}{6}x \right) - \frac{2}{3} - \frac{1}{2} = \frac{5}{6}x + \frac{10}{6}$

Simplifying the $x$-terms:

$-\frac{1}{6}x - \frac{2}{3} - \frac{1}{2} = \frac{5}{6}x + \frac{10}{6}$

Step 4: Simplify constants

Now simplify the constants by getting a common denominator. The LCD of 3 and 2 is 6:

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

Thus:

$-\frac{1}{6}x - \frac{4}{6} - \frac{3}{6} = \frac{5}{6}x + \frac{10}{6}$

Combine the constants on the left-hand side:

$-\frac{1}{6}x - \frac{7}{6} = \frac{5}{6}x + \frac{10}{6}$

Step 5: Move all $x$-terms to one side

Add $\frac{1}{6}x$ to both sides:

$-\frac{7}{6} = \frac{6}{6}x + \frac{10}{6}$

Step 6: Move constants to the other side

Subtract $\frac{10}{6}$ from both sides:

$-\frac{7}{6} - \frac{10}{6} = x$

Simplify:

$x = -\frac{17}{6}$

Final Answer:

$x = -\frac{17}{6}$

Would you like a more detailed breakdown or have any questions?

Here are 5 related questions to explore further:

1. How do you solve equations with variables on both sides?
2. What are common strategies for handling fractions in algebraic equations?
3. How do you find the least common denominator (LCD)?
4. What are the steps to isolate the variable in an equation?
5. How can you check the solution to an algebraic equation?

Tip: Always simplify the equation step by step to avoid mistakes, especially when working with fractions!