`{:("Column A",3x + 7y > 7x + 3y,"Column B),(x + 2y,,y + 2x):}`

Options

A. If column A is larger
B. If column B is larger
C. If the columns are equal
D. If there is not enough information to decide

Answer

To solve the inequality and compare the two columns, we will follow these steps:

### Step 1: Simplify the given inequality
We start with the inequality:
\[ 3x + 7y > 7x + 3y \]

### Step 2: Rearrange the inequality
We will move all terms involving \( x \) to one side and all terms involving \( y \) to the other side:
\[ 3x + 7y - 7x - 3y > 0 \]
This simplifies to:
\[ 7y - 3y > 7x - 3x \]
So, we have:
\[ 4y > 4x \]

### Step 3: Divide by 4
Now, we can divide both sides of the inequality by 4 (since 4 is positive, the direction of the inequality remains the same):
\[ y > x \]

### Step 4: Analyze Columns A and B
Now we need to compare the expressions in Column A and Column B:
- Column A: \( x + 2y \)
- Column B: \( y + 2x \)

### Step 5: Substitute \( y \) in terms of \( x \)
Since we know \( y > x \), we can analyze the expressions. We can substitute some values to see which column is larger.

### Step 6: Test with specific values
Let's take \( y = 3 \) and \( x = 2 \):
- For Column A:
\[ x + 2y = 2 + 2(3) = 2 + 6 = 8 \]
- For Column B:
\[ y + 2x = 3 + 2(2) = 3 + 4 = 7 \]

Here, Column A (8) is greater than Column B (7).

### Step 7: Test with negative values
Now, let's test with negative values, say \( y = -2 \) and \( x = -4 \):
- For Column A:
\[ x + 2y = -4 + 2(-2) = -4 - 4 = -8 \]
- For Column B:
\[ y + 2x = -2 + 2(-4) = -2 - 8 = -10 \]

Again, Column A (-8) is greater than Column B (-10).

### Conclusion
In both cases, we find that Column A is larger than Column B. Therefore, the answer is:
**Column A is larger.**