I'm super confused as to why my solution path to a comparison problem didn't work. Any help would be appreciated.
Problem 41, Chapter 8 (Algebra), GRE 5 lb. book of problems
Here are the constraints:
X > Y
XY =/= 0
Here is the problem:
A: B:
X2/ (Y + 1/Y) Y2/ (X + 1/X)
My strategy was to cross-multiply to make this more manageable when using test cases.
So:
A: B:
X2 (X + 1/X) Y2/ (Y + 1/Y
X3 + X Y3 + Y
(I could have factored out, but didn't feel a need to as far as helping me calculate test cases.)
Because X > Y, the test cases I used, respectively, were +/+, +/-, -/-, choosing simple numbers.
(+/+) X=2, Y=1
A: 10 B: 2
(+/-) X= 1, Y= -1
A: 2 B; -2
(-/- ) X= -1, Y= -2
A: -2 B: -10
In all test cases, A is bigger using the equation I derived using cross-multiplication.
The book the answer is "D." Their strategy was not to try and simplify like I did, but instead, directly use test cases off the original formulas. So either, my derivation of the simpler formula was wrong, or my calculation of test cases was wrong. I don't think my use of test cases was incorrect, so where did I go wrong in cross-multiplying? I thought that such an operation was perfectly valid in examples like these?
I'm super confused. Any help would be greatly apprecaited. Thank you, in advance!