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Joined: 20 Feb 2017
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The operation is defined for all nonzero numbers a and b by a b =
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18 Sep 2022, 21:46
18 Sep 2022, 21:46
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The operation ⊗ is defined for all nonzero numbers a and b by a ⊗ b = a/b – 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
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
Re: The operation is defined for all nonzero numbers a and b by a b =
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21 Sep 2022, 05:07
21 Sep 2022, 05:07
1
Expert Reply
Given: \(a@b=\frac{a}{b}-\frac{b}{a}\), for all nonzero numbers \(a\) and \(b\).
I. \(x@(xy) = x(1@y)\): \(LHS=x@(xy)=\frac{x}{xy}-\frac{xy}{x}=\frac{1}{y}-y\) and \(RHSx(1@y)=x(\frac{1}{y}-y)\) as you see LHS doen't equal to RHS;
II. \(x@y = -(y@x)\): \(LHS=x@y=\frac{x}{y}-\frac{y}{x}\) and \(RHS=-(y@x)=-(\frac{y}{x}-\frac{x}{y})=\frac{x}{y}-\frac{y}{x}\) --> LHS=RHS;
III. \((\frac{1}{x})@(\frac{1}{y}) = y@x\): \(LHS=(\frac{1}{x})@(\frac{1}{y})=\frac{y}{x}-\frac{x}{y}\) and \(RHS=y@x=\frac{y}{x}-\frac{x}{y}\) --> LHS=RHS.
Answer: E (II and III).
I. \(x@(xy) = x(1@y)\): \(LHS=x@(xy)=\frac{x}{xy}-\frac{xy}{x}=\frac{1}{y}-y\) and \(RHSx(1@y)=x(\frac{1}{y}-y)\) as you see LHS doen't equal to RHS;
II. \(x@y = -(y@x)\): \(LHS=x@y=\frac{x}{y}-\frac{y}{x}\) and \(RHS=-(y@x)=-(\frac{y}{x}-\frac{x}{y})=\frac{x}{y}-\frac{y}{x}\) --> LHS=RHS;
III. \((\frac{1}{x})@(\frac{1}{y}) = y@x\): \(LHS=(\frac{1}{x})@(\frac{1}{y})=\frac{y}{x}-\frac{x}{y}\) and \(RHS=y@x=\frac{y}{x}-\frac{x}{y}\) --> LHS=RHS.
Answer: E (II and III).
gmatclubot
Re: The operation is defined for all nonzero numbers a and b by a b = [#permalink]
21 Sep 2022, 05:07
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