https://file.aiquickdraw.com/mathbot/file/1726888778271k50r8omg.jpg

The equation provided is:

\[
\frac{3w - 6}{7} = 4 - \frac{w + 6}{2}
\]

Let's solve this step by step.

### Step 1: Eliminate the denominators
To get rid of the fractions, we multiply the entire equation by the least common denominator (LCD) of 7 and 2, which is 14.

\[
14 \times \frac{3w - 6}{7} = 14 \times \left(4 - \frac{w + 6}{2}\right)
\]

This simplifies to:

\[
2(3w - 6) = 14 \times 4 - 7(w + 6)
\]

Simplifying further:

\[
6w - 12 = 56 - 7w - 42
\]

### Step 2: Simplify both sides
Combine like terms on both sides:

\[
6w - 12 = 14 - 7w
\]

### Step 3: Move all terms involving \( w \) to one side
Add \( 7w \) to both sides:

\[
6w + 7w - 12 = 14
\]

This simplifies to:

\[
13w - 12 = 14
\]

### Step 4: Solve for \( w \)
Add 12 to both sides:

\[
13w = 26
\]

Now, divide both sides by 13:

\[
w = 2
\]

So, the solution is:

\[
w = 2
\]

Would you like any further details or clarifications? 

Here are five related questions for deeper understanding:
1. How would you solve if the equation had a different common denominator?
2. What happens if \( w \) is negative?
3. How can we verify this solution by substituting \( w = 2 \) back into the original equation?
4. How would you handle this if the equation was in the form of an inequality?
5. Can you derive a general formula for similar types of equations?

**Tip:** Always check your solution by plugging it back into the original equation!

Learn how to solve the linear equation involving fractions: (3w - 6)/7 = 4 - (w + 6)/2. This step-by-step solution helps you eliminate fractions, simplify, and solve for w. Suitable for students in Grades 8-10.

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