ajit257 wrote:
If x and y are nonzero integers, is \((x^{-1}+y^{-1})^{-1}> (x^{-1}*y^{-1})^{-1}\)?
(1) x = 2y
(2) x + y > 0
Target question: Is \((x^{-1}+y^{-1})^{-1}> (x^{-1}*y^{-1})^{-1}\)This is a good candidate for
rephrasing the target question. Take: \((x^{-1}+y^{-1})^{-1}> (x^{-1}*y^{-1})^{-1}\)
Rewrite as: \((\frac{1}{x}+\frac{1}{y})^{-1}> (\frac{1}{x}*\frac{1}{y})^{-1}\)
Rewrite as: \((\frac{y}{xy}+\frac{x}{xy})^{-1}> (\frac{1}{xy})^{-1}\)
Simplify to get: \((\frac{x+y}{xy})^{-1}> (\frac{1}{xy})^{-1}\)
Apply exponents to get: \(\frac{xy}{x+y}> xy\)
REPHRASED target question: Is \(\frac{xy}{x+y}> xy\)?Aside: the video below has tips on rephrasing the target question Statement 1: \(x = 2y\) Replace \(x\) with \(2y\) to get:
Is \(\frac{(2y)y}{(2y)+y}> (2y)y\)?Simplify to get:
Is \(\frac{2y^2}{3y}> 2y^2\)?Since \(2y^2\) must be POSITIVE, we can safely divide both sides of the inequality by \(2y^2\) to get:
Is \(\frac{1}{3y}> 1\)?IMPORTANT: Since y is an INTEGER,
1/3y will ALWAYS be less than 1.
For example, if y = 1, we get: 1/3 < 1
If y = 2, we get: 1/6 < 1
If y = 3, we get: 1/9 < 1
etc.
Likewise, if y is NEGATIVE, then 1/3y will always be NEGATIVE, and a negative value is always less than 1.
So, the answer to the REPHRASED target question is
NO, \(\frac{xy}{x+y}\) is NOT greater than \(xy\)Since we can answer the
target question with certainty, statement 1 is SUFFICIENT
Statement 2: x + y > 0 Let's TEST some values.
There are several values of x and y that satisfy statement 2. Here are two:
Case a: x = 2 and y = 1. Plug into REPHRASED target question to get: \(\frac{(2)(1)}{2+1}> (2)(1)\).
Simplify: \(\frac{2}{3}> 2\) In this case, the answer to the REPHRASED target question is
NO, \(\frac{xy}{x+y}\) is NOT greater than \(xy\)Case b: x = 3 and y = -1. Plug into REPHRASED target question to get: \(\frac{(3)(-1)}{3+(-1)}> (3)(-1)\).
Simplify: \(\frac{-3}{2}> -3\) In this case, the answer to the REPHRASED target question is
YES, \(\frac{xy}{x+y}\) IS greater than \(xy\)Since we cannot answer the
target question with certainty, statement 2 is NOT SUFFICIENT
Answer: A
Cheers,
Brent
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