Bunuel wrote:
If \(x\) and \(y\) are consecutive integers (\(x \gt y\)) and \(x^2-1 \gt y^2-4y+x-1\), then which of the following must be true?
A. \(y \le 0\)
B. \(y \gt 0\)
C. \(y \gt 1\)
D. \(y \gt 7\)
E. \(y \gt 8\)
Another way to approach the question is with
easy numbers and then test the answers to see what you can eliminate. As long as you pick
consecutive integers and the original inequality holds true, then you will have created a valid system of inputs. In this case, I arbitrarily chose y = 1 and x = 2
because I am lazy, and I would rather not do a bunch of time-consuming squaring and all that if I walk away with a better sense of
what must be true from such simple inputs. We need to test in the original inequality:
\((2)^2-1 \gt (1)^2-4(1)+(2)-1\)
\(4-1 \gt 1-4+2-1\)
\(3 \gt -2\)
This is true, 3 is greater than -2. Thus, we have created a valid system of inputs for our variables. Now check the answers against what we know to be true:
A. \(y \le 0\)
No, y is not less than 0. We chose 1, and the inequality held up.
Red light. B. \(y \gt 0\)
Yes, y is a positive integer, 1. Keep this for now.
Yellow light.C. \(y \gt 1\)
No, our input of 1 for y is
not greater than 1.
Red light.D. \(y \gt 7\)
This is clearly not going to work. We just proved that y can equal 1, and 1 is not greater than 7.
Red light.E. \(y \gt 8\)
This will not work either, for the same reason as in (D) above.
Red light.With just one answer choice that checked out, the winner must be (B). For you skeptics out there who might be wondering what if a different combination of inputs had been selected, you might simply have to refine your numbers until one answer emerged, but I always advise keeping such numbers simple for ease of access. Why choose a negative if a positive would do? (More people make mistakes with signs when dealing with negatives.) Would y = 0 and x = 1 work? No, but you would get 0 on both sides--0 = 0, NOT 0 > 0--which would reveal that you had reached a crucial point, and that the answer was just around the corner.
Good luck with your studies.
- Andrew
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