Vincen wrote:Is |x - y| > |x + y|?
(1) x^2 - y^2 = 9
(2) x - y = 2
Is |x-y| > |x+y|?
When each side of an inequality is enclosed in absolute value symbols, we can square the inequality:
(x-y)² > (x+y)²
x² - 2xy + y² > x² + 2xy + y²
0 > 4xy
xy < 0.
Question stem, rephrased:
Do x and y have DIFFERENT SIGNS?
Statement 1: x² - y² = 9
Perfect squares: 1, 4, 9,
16, 25...
The difference between the two perfect squares in blue is 9, implying that one solution for Statement 1 is as follows:
x² = 25, with the result that x = ±5.
y² = 16, with the result that y = ±4.
Since x and y could have the same sign or different signs, INSUFFICIENT.
Statement 2: x - y = 2
If x=3 and y=1, then x and y have the same sign.
If x=1 and y=-1, then x and y have different signs.
INSUFFICIENT.
Statements combined:
Statement 1 implies the following:
(x+y)(x-y) = 9.
Substituting x-y=2 into (x+y)(x-y) = 9, we get:
(x+y)(2) = 9
x+y = 9/2.
Since we have two variables (x and y) and two distinct linear equations (x-y = 2 and x+y = 9/2), we can solve for both variables, allowing us to determine whether x and y have the same sign or different signs.
SUFFICIENT.
The correct answer is
C.
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