sarahfiqbal wrote:
ScottTargetTestPrepCan you help explain this please?
My approach:
I picked x=2,-2 and y= 3, -3
When I make the calculations, I am getting I, II and III as correct answers.
The question is asking for the statements which "must" be true, meaning the statement should be true for ANY value of x and y satisfying |x/y| < 1. As far as I understood, you found values which make statements I, II and III hold, but the question is not asking for that.
For statement I, if you take x = 2 and y = -3, then |(x + 1)/(y + 1)| becomes |3/-2|, which is greater than 1. That single example is enough to show that it is not true that statement I MUST be true. There can be values of x and y where statement I is true, but even one counter example is enough to show that it is not always true.
Similarly, for statement III, if you take x = -2 and y = 3, then |(x - 1)/(y - 1)| becomes |-3/2|, which is greater than 1. For the same reasons as above, we conclude that it's not true that statement III must be true either.
While we actually know the answer at this point (there is only one answer choice which does not involve statement I or III), let's show that statement II must be true. Notice that you can't just show that |(x^2 + 1)/(y^2 + 1)| < 1 by picking values for x and y; you can find a million pairs of x and y which satisfy statement II, but it does not prove that the statement is true for all x and y with |x/y| < 1. Instead, you proceed as follows:
If x/y > 0, then |x/y| = x/y. In this case, we have 0 < x/y < 1. Since the square of a number between 0 and 1 is also between 0 and 1,
we have x^2/y^2 < 1 as well. Then, x^2 < y^2. Adding 1 to each side, we get x^2 + 1 < y^2 + 1; thus (x^2 + 1)/(y^2 + 1) < 1. Since (x^2 + 1)/(y^2 + 1) is positive, |(x^2 + 1)/(y^2 + 1)| < 1.
If x/y < 0, then |x/y| = -x/y. We have 0 < -x/y < 1. As above, the square of a number between 0 and 1 is again between 0 and 1; thus 0 < x^2/y^2 < 1. We follow the same steps as above to conclude that |(x^2 + 1)/(y^2 + 1)| < 1 in this case as well.
Just as a side note, if the question was "Which of the following can be true?", then your approach and your answer would have been correct.
I just had concern regarding the highlighted part above where denominator from LHS can't go to RHS unless it is equal correct?
And only '+' & '-' signs can be shuffled from LHS to RHS or vice versa.