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Determine whether the statement is true, false, or sometimes true.

If [tex]x[/tex] is positive and [tex]y = -x[/tex], then [tex]x^2 y \ \textgreater \ 0[/tex].

A. True
B. False
C. Sometimes true

Sagot :

GinnyAnsweridn
Let's analyze the given conditions and the inequality step by step:

1. Given Conditions:
- [tex]\( x \)[/tex] is positive.
- [tex]\( y = -x \)[/tex].

2. Substitute [tex]\( y \)[/tex] in the inequality [tex]\( x^2 y > 0 \)[/tex]:
- Since [tex]\( y = -x \)[/tex], we substitute to get [tex]\( x^2 (-x) > 0 \)[/tex].

3. Simplify the expression:
- [tex]\( x^2 (-x) = -x^3 \)[/tex].

4. Analyze the simplified expression:
- Since [tex]\( x \)[/tex] is positive, [tex]\( x^3 \)[/tex] (the cube of a positive number) is also positive.
- Multiplying a positive number ([tex]\( x^3 \)[/tex]) by -1 results in a negative number.
- Therefore, [tex]\( -x^3 \)[/tex] is negative.

5. Determine the truth value of the inequality:
- We end up with [tex]\( -x^3 > 0 \)[/tex].
- A negative number (which [tex]\( -x^3 \)[/tex] is) cannot be greater than zero.

Conclusion:
Given that [tex]\( x \)[/tex] is positive, and substituting [tex]\( y = -x \)[/tex], the inequality [tex]\( x^2 y > 0 \)[/tex] results in a false statement. Therefore, the answer to whether the statement is true, false, or sometimes true is:

False.
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