Bunuel wrote:
Tough and Tricky questions: Inequalities.
Is \(x \gt y\)?
(1) \(6x \gt 5y\)
(2) \(xy \lt 0\)
Kudos for a correct solution. Target question: Is x > y? Statement 1: 6x > 5y This statement doesn't FEEL sufficient, so I'll TEST some values.
There are several values of x and y that satisfy statement 1. Here are two:
Case a: x = 1 and y = -1. Notice that 6x > 5y becomes 6 > -5, which is true. In this case, the answer to the target question is
YES, x IS greater than yCase b: x = 9 and y = 10. Notice that 6x > 5y becomes 54 > 50, which is true. In this case, the answer to the target question is
NO, x is NOT greater than ySince we cannot answer the
target question with certainty, statement 1 is NOT SUFFICIENT
Statement 2: xy < 0Let' s TEST some more numbers.
There are several values of x and y that satisfy statement 2. Here are two:
Case a: x = 1 and y = -1. In this case, the answer to the target question is
YES, x IS greater than yCase b: x = -1 and y = 1. In this case, the answer to the target question is
NO, x is NOT greater than ySince we cannot answer the
target question with certainty, statement 2 is NOT SUFFICIENT
Statements 1 and 2 combined Statement 2 tells us that xy < 0 (i.e., xy is NEGATIVE)
If the product xy is NEGATIVE, then there are
two possible cases:
Case a:
x is POSITIVE and y is NEGATIVE .
Notice that this case ALSO satisfies statement 1, which says 6x > 5y. When we replace x and y, we get: 6(POSITIVE) > 5(NEGATIVE), which is always true.
In this case, the answer to the target question is
YES, x IS greater than yCase b:
x is NEGATIVE and y is POSITIVE. Notice that this case DOES NOT satisfy statement 1, which says 6x > 5y. When we replace x and y, we get: 6(NEGATIVE) > 5(POSITIVE), which is NEVER true.
So, it cannot be the case that x is NEGATIVE and y is POSITIVE.
In other words,
only case a can be true, which means the answer to the target question must be
YES, x IS greater than ySince we can answer the
target question with certainty, the combined statements are SUFFICIENT
Answer: C
Cheers,
Brent
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