We are given the equation:

$\frac{1}{2}(x + 1) + \frac{1}{3}(x + 1) = \frac{5}{12}(x - 2)$

Let's solve this step by step.

Step 1: Eliminate the fractions

To eliminate the fractions, find the least common denominator (LCD) of 2, 3, and 12, which is 12. Multiply the entire equation by 12 to get rid of the denominators:

$12 \times \left[\frac{1}{2}(x + 1) + \frac{1}{3}(x + 1)\right] = 12 \times \frac{5}{12}(x - 2)$

This simplifies to:

$6(x + 1) + 4(x + 1) = 5(x - 2)$

Step 2: Expand both sides

Now, expand both sides of the equation:

$6(x + 1) = 6x + 6$ $4(x + 1) = 4x + 4$ $5(x - 2) = 5x - 10$

So the equation becomes:

$(6x + 6) + (4x + 4) = 5x - 10$

Step 3: Combine like terms

On the left-hand side, combine the $x$-terms and the constants:

$10x + 10 = 5x - 10$

Step 4: Solve for $x$

Now, isolate $x$ by subtracting $5x$ from both sides:

$10x - 5x + 10 = -10$

Simplify:

$5x + 10 = -10$

Subtract 10 from both sides:

$5x = -20$

Finally, divide by 5:

$x = -4$

Solution:

The value of $x$ is $\boxed{-4}$.

Would you like further explanation on any step, or details about this type of equation?

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Here are 5 related questions for further practice:

1. Solve $\frac{1}{3}(x + 2) + \frac{1}{4}(x - 1) = \frac{5}{6}(x - 3)$.
2. Solve $\frac{1}{4}(x + 3) + \frac{1}{2}(x + 2) = \frac{7}{12}(x - 4)$.
3. Solve $3(x + 5) + 2(x - 2) = 4(x + 1)$.
4. Solve $\frac{2}{5}(x + 4) + \frac{3}{10}(x - 2) = \frac{4}{15}(x + 6)$.
5. Solve $\frac{3}{8}(x + 1) + \frac{1}{2}(x - 2) = \frac{7}{24}(x + 4)$.

Tip: Always start by clearing the denominators when solving equations with fractions; it simplifies the equation quickly.