Let's solve the equation \( \frac{x}{2} + 1 = \frac{2x}{3} - 3 \).

### Step 1: Eliminate the fractions
To eliminate the fractions, find a common denominator for the terms with \( x \). The least common denominator (LCD) of 2 and 3 is 6.

Multiply every term in the equation by 6:

\[
6 \times \left(\frac{x}{2}\right) + 6 \times 1 = 6 \times \left(\frac{2x}{3}\right) - 6 \times 3
\]

This simplifies to:

\[
3x + 6 = 4x - 18
\]

### Step 2: Rearrange the equation
To solve for \( x \), first move the \( x \) terms to one side of the equation by subtracting \( 3x \) from both sides:

\[
6 = 4x - 3x - 18
\]

Simplify:

\[
6 = x - 18
\]

### Step 3: Solve for \( x \)
Add 18 to both sides to isolate \( x \):

\[
x = 24
\]

### Final Answer:
The solution to the equation is \( x = 24 \).

Would you like further details or have any questions? Here are some related questions to consider:

1. How can we verify the solution to the equation \( \frac{x}{2} + 1 = \frac{2x}{3} - 3 \)?
2. What is the significance of finding a common denominator when solving equations with fractions?
3. How can this approach be applied to solve equations with more complex fractions?
4. What are some common mistakes to avoid when solving equations involving fractions?
5. How would the solution process change if the equation included additional variables?

**Tip:** Always double-check your solution by substituting it back into the original equation to ensure both sides are equal.

Learn how to solve the equation x/2 + 1 = 2x/3 - 3 step-by-step. This problem involves eliminating fractions, rearranging equations, and solving for x. Suitable for students in Grades 7-9.

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