Manbehindthecurtain wrote:
Are x and y both positive?
(1) \(2x - 2y = 1\)
(2) \(\frac{x}{y} > 1\)
Target question: Are x and y both positive? Statement 1: 2x - 2y = 1 There are several pairs of numbers that satisfy this condition. Here are two:
Case a: x = 1 and y = 0.5, in which case
x and y are both positiveCase b: x = -0.5 and y = -1, in which case
x and y are not both positiveSince we cannot answer the
target question with certainty, statement 1 is NOT SUFFICIENT
Statement 2: x/y > 1 This tells us that x/y is positive. This means that either x and y are both positive or x and y are both negative. Here are two possible cases:
Case a: x = 4 and y = 2, in which case
x and y are both positiveCase b: x = -4 and y = -2, in which case
x and y are not both positiveSince we cannot answer the
target question with certainty, statement 2 is NOT SUFFICIENT
Statements 1 and 2Statement 1 tells us that 2x - 2y = 1.
Divide both sides by 2 to get: x - y = 1/2
Solve for x to get x = y + 1/2
Now take the statement 2 inequality (x/y > 1) and replace x with y + 1/2 to get:
(y + 1/2)/y > 1
Rewrite as: y/y + (1/2)/y > 1
Simplify: 1 + 1/(2y) > 1
Subtract 1 from both sides: 1/(2y) > 0
If 1/(2y) is positive, then
y must be positive.
Statement 2 tells us that either x and y are both positive or x and y are both negative.
Now that we know that
y is positive, it
must be the case that
x and y are both positiveSince we can now answer the
target question with certainty, the combined statements are SUFFICIENT
Answer: C
Cheers,
Brent
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