Carcass wrote:
If \(x < y\), which of the following must be true?
A. \(2x < y\)
B. \(2x > y\)
C. \(x^2 < y^2\)
D. \(2x-y < y\)
E. \(2x-y < 2xy\)
When I scan the answer choices, I see that D looks promising.
D) 2x − y < yAdd y the both sides of the inequality:
2x < 2yDivide both sides by 2 to get:
x < ySo, D is equivalent to the original inequality x < y
Answer: DAlternate approach: Test values
If x < y, then it could be the case that x = 1 and y = 2.
Plug these values into the five answer choices to get:
A. 2(1) < 2, which simplifies to 2 < 2. Not true. Eliminate.
B. 2(1) > 2, which simplifies to 2 > 2. Not true. Eliminate.
C. 1² < 2², which simplifies to 1 < 2. TRUE. Keep.
D. 2(1) − 2 < 2, which simplifies to 0 < 2. TRUE. Keep.
E. 2(1) − 2 < 2(1)(2), which simplifies to 0 < 4. TRUE. Keep.
We're down to options C, D and E.
Strategy: We first tested 2 positive values. Now that's test 1 negative and 1 positive. We'll test x = -3 and y = 2.
Plug these values into the remaining answer choices to get:
C. (-3)² < 2², which simplifies to 9 < 4. Not true. Eliminate.
D. 2(-4) − 2 < 2, which simplifies to -10 < 2. TRUE. Keep.
E. 2(-3) − 2 < 2(-3)(2), which simplifies to -8 < -12. Not true. Eliminate.
By the process of elimination, the
answer is D