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# The operation  is defined for all nonzero numbers a

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Manager
Joined: 10 Oct 2005
Posts: 113
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The operation  is defined for all nonzero numbers a [#permalink]  10 Oct 2005, 05:38
The operation  is defined for all nonzero numbers a and b by a  b = a/b â€“ b/a. If x and y are nonzero numbers, which of the following statements must be true?

I. x  xy = x(1  y)
II. x  y = -(y  x)
III. 1/x  1/y = y  x

A. I only
B. II only
C. III only
D. I and II
E. II and III
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Director
Joined: 09 Jul 2005
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E is the only correct answer. If I am correct I will demostrate.
Manager
Joined: 15 Jul 2005
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yep, (E) it is.

I) LHS = x  xy = x/xy-xy/x = (1/y)-y.
RHS = x(1y) = x(1/y-y/1) = x/y-xy.

LHS is not equal to RHS.

II) LHS = xy= x/y-y/x
RHS = -(yx) = -(y/x-x/y) = -y/x+x/y = x/y-y/x

LHS = RHS

III) LHS = 1/x1/y = (1/x)/(1/y)-(1/y)/(1/x) = y/x-x/y
RHS = yx=y/x-x/y

LHS = RHS.
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