SOLUTION: If x is an integer greater than 1 and if y = x + (1/x), which of the following must be true ? I y ≠ x II y is an integer III xy

SOLUTION: If x is an integer greater than 1 and if y = x + (1/x), which of the following must be true ? I y ≠ x II y is an integer III xy > x^2 (A) I

Question 296658: If x is an integer greater than 1 and if y = x + (1/x), which of the following must be true ?
I y ≠ x
II y is an integer
III xy > x^2
(A) I only
(B) III only
(C) I and II only
(D) I and III only
(E) I , II ,and III only

Answer by jim_thompson5910(35256) (Show Source): You can put this solution on YOUR website!
First, we need to see if statement I is true or not. Assume that , if this is the case, then . Subtract 'x' from both sides to get . Since this equation is NEVER true for any 'x' values (let alone positive 'x' values), we can say that is false where . So this means that and that statement I is true.
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If , then is NOT an integer. So this consequently means that is also NOT an integer. So statement II is false meaning that choices C and E can be ignored.
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Finally, because 'x' is an integer and 'x' is positive, we also know that or that (do a simple substitution here). Multiply both sides by 'x' to get (note: the sign doesn't flip because 'x' is positive).

So we've shown that if 'x' is an integer and 'x' is positive, then where . So statement III is true.

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So because statements I and III are both true, we can say that the answer is choice D.
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SOLUTION: If x is an integer greater than 1 and if y = x + (1/x), which of the following must be true ? I y &#8800; x II y is an integer III xy > x^2 (A) I

SOLUTION: If x is an integer greater than 1 and if y = x + (1/x), which of the following must be true ? I y ≠ x II y is an integer III xy > x^2 (A) I