Let's solve the equation step by step:

The given equation is:

\[
\frac{X}{2} + \frac{x+2}{3} = (x + 1) + 5
\]

### Step 1: Simplify the right-hand side

The right-hand side simplifies as follows:

\[
(x + 1) + 5 = x + 6
\]

So, the equation becomes:

\[
\frac{X}{2} + \frac{x+2}{3} = x + 6
\]

### Step 2: Eliminate the fractions by finding a common denominator

The least common denominator (LCD) between 2 and 3 is 6. Multiply the entire equation by 6 to eliminate the fractions:

\[
6 \times \left(\frac{X}{2} + \frac{x+2}{3}\right) = 6 \times (x + 6)
\]

This simplifies to:

\[
3X + 2(x + 2) = 6(x + 6)
\]

### Step 3: Expand both sides

Expand both sides of the equation:

\[
3X + 2x + 4 = 6x + 36
\]

### Step 4: Simplify the equation

Now, combine like terms:

\[
3X + 2x + 4 = 6x + 36
\]

Move all the terms involving \(x\) to one side, and constant terms to the other side:

\[
3X + 2x - 6x = 36 - 4
\]

This simplifies to:

\[
3X - 4x = 32
\]

### Step 5: Factor out \(x\)

Factor \(x\) on the left-hand side:

\[
x(3 - 4) = 32
\]

This simplifies to:

\[
-x = 32
\]

### Step 6: Solve for \(x\)

To solve for \(x\), multiply both sides by -1:

\[
x = -32
\]

### Final Answer:

The solution to the equation is:

\[
x = -32
\]

Do you want further details or have any questions about this solution?

Here are some related questions:
1. What is the difference between linear and quadratic equations?
2. How do you solve equations with variables on both sides?
3. How can you solve equations involving fractions?
4. What is the least common denominator, and how is it used in solving equations?
5. How do you check if your solution to an equation is correct?

**Tip:** Always simplify both sides of an equation before solving—it makes the process much easier!

Explore how to solve a linear equation with fractions: X ÷ 2 + (x + 2) ÷ 3 = (x + 1) + 5. This example includes a step-by-step solution covering algebraic concepts, finding a common denominator, and solving for the variable. Suitable for students in Grades 8-10.

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