Expert: Robi Bhattacharjee - 9/20/2008

Question
If x is an integer greater than 1 and if y=x+(1/x), which of the following must be true?
I. y≠ x (y does NOT equal x)
II. y is an integer
III. xy>x^2

The correct answer is that I and III are true. I know why II isn't true but why are I and III true?

Thanks in advance

Answer
Hello Emily,

Great problems!

For this question, the easiest way to approach it is to try to prove each thing wrong.

So lets see if we can find a case where I is violated.

If y = x. Then x = x + 1/x since y = x + 1/x. This means that 1/x = 0

so 1 = 0x and x = 1/0. This means that there is no x for which 1/x is 0 since 1/0 does not exist. So y does NOT equal x for all x.

This means that I is true.

For II, y is not always an integer. Let x = 2 for example. y = x + 1/x

so y = 2 + 1/2 = 2 and 1/2 which is not an integer. This means that II is false.

For III, we will just simplify an equation.

y = x + 1/x

xy = x^2 + 1.

Since x is > 1, y cannot be negative since y is > x. This is because 1/x will always be the difference between x and y so since it is positive, y > x.

This means that both y and x are positive. This means that xy is positive.

So xy - 1 = x^2

so xy must be > x^2 since x^2 is one less than xy. The reason I had to prove that xy is positive is since if it was negative, it would be smaller than x^2 + 1.

So basically, xy > x^2 since y > x.

So this means that III is true.

So I and III are true.


I hope this helped,

Please ask if you have any questions.


Robi