An easy True-False problem!
If x , y are real numbers, is it always true that
2 x + y ≥ x y
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3 solutions
@Mahdi Raza , @Vinayak Srivastava , @Alak Bhattacharya .
When at least one of x and y be negative, then the inequality will not hold. For the inequality to hold true, both of them must be positive.
Yes Sir, exactly! I am shocked at how only 46% of people stopped for a while to think that it is not the same as AM-GM inequality!
If x , y both are negative, the statement fails. A counterexample: x = − 1 , y = − 4 ⟹ 2 x + y = 2 − 5 x y = − 2 ⟹ 2 x + y < x y
Why i am thinking that you are wrong? Product will be -2. @Vinayak Srivastava .
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Why? x × y = − 1 × − 4 = 4 = ± 2
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Only -2 will be the answer.
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@A Former Brilliant Member – See here
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@A Former Brilliant Member – Ok, I'll put -2 only. Thanks!
@A Former Brilliant Member – @Kriti Kamal - Your link is only for the case where 2 roots are being multiplied, but in this problem, the variables will get multiplied first as x y is a number that has been written as a product of two of its variable factors.
Oh, − 2 is also a square root!
But both ways, it is greater, so it is correct.
@Kriti Kamal , I am still confused. We directly needed x y then why are we doing x × y ?
@Vinayak Srivastava - x y is ± 2 , not − 2 as @Kriti Kamal stated.
When one of the x and y becomes positive and other become negative,then the square root of their product will imaginary and there is no meaning of equality or inequality between purely imaginary and real number.