`{:("Column A",x//a > 4 and y//a < -6,"Column B),(x,(a^2=9 ,ab^2=-8),y):}`

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 problem, we need to analyze the given inequalities and conditions step by step.

### Step 1: Analyze the conditions
We have two inequalities:
1. \( \frac{x}{a} > 4 \)
2. \( \frac{y}{a} < -6 \)

And two equations:
1. \( a^2 = 9 \)
2. \( ab^2 = -8 \)

### Step 2: Solve for \( a \)
From the equation \( a^2 = 9 \), we can find the possible values for \( a \):
\[ a = 3 \quad \text{or} \quad a = -3 \]

### Step 3: Analyze each case for \( a \)

#### Case 1: \( a = 3 \)
Substituting \( a = 3 \) into the inequalities:
1. From \( \frac{x}{3} > 4 \):
\[ x > 12 \]
2. From \( \frac{y}{3} < -6 \):
\[ y < -18 \]

Now, we need to check the second condition \( ab^2 = -8 \):
\[ 3b^2 = -8 \implies b^2 = -\frac{8}{3} \]
This is not possible since \( b^2 \) cannot be negative.

#### Case 2: \( a = -3 \)
Substituting \( a = -3 \) into the inequalities:
1. From \( \frac{x}{-3} > 4 \):
\[ x < -12 \]
2. From \( \frac{y}{-3} < -6 \):
\[ y > 18 \]

Now, we check the second condition \( ab^2 = -8 \):
\[ -3b^2 = -8 \implies b^2 = \frac{8}{3} \]
This is valid since \( b^2 \) is positive.

### Step 4: Compare values of \( x \) and \( y \)
From our findings:
- For \( a = -3 \), we have \( x < -12 \) and \( y > 18 \).

### Conclusion
Since \( x < -12 \) and \( y > 18 \), it is clear that:
\[ y > 18 > x \]
Thus, Column B (which is \( y \)) is larger than Column A (which is \( x \)).

### Final Answer
Column B is larger than Column A.

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