X2y<0

The inequality \( x^2 y < 0 \) involves both the variable \( x \) and the variable \( y \). To understand the condition under which this inequality holds, we can analyze the components involved:

1. Understanding \( x^2 \):

   • The term \( x^2 \) is always non-negative, meaning \( x^2 \geq 0 \) for all real numbers \( x \).
   • Specifically, \( x^2 = 0 \) when \( x = 0 \).
   • Otherwise, \( x^2 > 0 \) when \( x \neq 0\).

2. Analyzing the product \( x^2 y \):

   • For the product \( x^2 y \) to be less than 0 (i.e., \( x^2 y < 0 \)), it is necessary that \( y < 0 \) because \( x^2 \) is non-negative.
   • If \( y < 0\), then since \( x^2 \) is positive (when \( x \neq 0\)), the product \( x^2 y \) will indeed be negative.

3. Conclusion:

   • The inequality \( x^2 y < 0 \) holds true under the condition that \( y < 0 \) and \( x \neq 0 \).
   • If \( y = 0 \) or \( x = 0 \), then \( x^2 y = 0 \), which does not satisfy the inequality.

In summary:

• \( x^2 y < 0 \) if and only if \( y < 0 \) and \( x \neq 0 \).