Question

\(x^2 y < 0\)
 

Column A	Column B
\(xy\)	0

Hint:

The inequality \(x^2>0\) holds for any non-zero real number \(x\), regardless of whether \(x\) is positive or negative. This is a crucial property to remember when working with inequalities involving squared variables.

Answer 1

Step 1: Understanding the Concept:
This question tests our understanding of inequalities and the properties of positive and negative numbers. We are given an inequality and must determine the sign of a related expression.
Step 2: Key Formula or Approach:
Analyze the given inequality \(x^2 y<0\) to determine the signs of the variables \(x\) and \(y\). Then use this information to determine the possible sign of the expression \(xy\).
Step 3: Detailed Explanation:
The inequality is \(x^2 y<0\).
The term \(x^2\) (the square of a real number) must be non-negative.
If \(x=0\), then \(0 \times y<0\), which simplifies to \(0<0\). This is false. Therefore, \(x \neq 0\).
Since \(x \neq 0\), \(x^2\) must be strictly positive.
So, our inequality has the form: \((\text{positive number}) \times y<0\).
For this product to be negative, \(y\) must be a negative number. So, we know for sure that \(y<0\).
However, we have no information about the sign of \(x\). \(x\) could be positive or negative.
Now let's evaluate Column A, which is \(xy\).
Case 1: \(x\) is positive.
If \(x>0\) and \(y<0\), then their product \(xy\) will be negative.
In this case, \(xy<0\). Column B is greater.
Case 2: \(x\) is negative.
If \(x<0\) and \(y<0\), then their product \(xy\) will be positive.
In this case, \(xy>0\). Column A is greater.
Step 4: Final Answer:
Since the value of \(xy\) can be either greater than or less than 0 depending on the sign of \(x\), the relationship between the two columns cannot be determined.