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Question

The operation \otimes is defined for all nonzero numbers a and b by a \otimes b = \frac{a}{b} - \frac{b}{a} . If x and y are nonzero numbers, which of the following statements must be true? I. x \otimes x y = x (1 \otimes y) II. x \otimes y = - (y \otimes x) III. \frac{1}{x} \otimes \frac{1}{y} = y \otimes x

Answer Choices

A. $I only$
B. $II only$
C. $III only$
D. $I and II$
E. $II and III$

Steps

Explanation	Calculations	Help
$We recall the definition of the operation.$	$a \otimes b = \frac{a}{b} - \frac{b}{a}$
$We evaluate x \otimes (x y) by substituting a = x and b = x y into the definition.$	$x \otimes (x y) = \frac{x}{x y} - \frac{x y}{x}$
$We simplify the expression for x \otimes (x y) .$	$x \otimes (x y) = \frac{x}{x y} - \frac{x y}{x} = \frac{1}{y} - y$	Theory & Tactics Method Card ALG5-D Click to view full details Theory & Tactics Method Card FDPR2-A Click to view full details
$We evaluate 1 \otimes y by substituting a = 1 and b = y into the definition.$	$1 \otimes y = \frac{1}{y} - \frac{y}{1}$
$We simplify the expression for 1 \otimes y .$	$1 \otimes y = \frac{1}{y} - y$
$We multiply the simplified value by x .$	$x \times (1 \otimes y) = x \times (\frac{1}{y} - y) = \frac{x}{y} - x \times y$
$We compare the results of x \otimes (x y) and x \times (1 \otimes y) to check Statement I.$	$x \otimes (x y) = \frac{1}{y} - y x \times (1 \otimes y) = \frac{x}{y} - x \times y$
$We evaluate x \otimes y .$	$x \otimes y = \frac{x}{y} - \frac{y}{x}$
$We evaluate y \otimes x .$	$y \otimes x = \frac{y}{x} - \frac{x}{y}$
$We find the negative of y \otimes x and then simplify it.$	$- (y \otimes x) = - (\frac{y}{x} - \frac{x}{y}) = - \frac{y}{x} + \frac{x}{y} = \frac{x}{y} - \frac{y}{x}$
$We observe that this matches x \otimes y, so Statement II is always true.$	$\frac{x}{y} - \frac{y}{x} = x \otimes y$
$We evaluate \frac{1}{x} \otimes \frac{1}{y} by substituting a = \frac{1}{x} and b = \frac{1}{y} into the definition.$	$\frac{1}{x} \otimes \frac{1}{y} = \frac{\frac{1}{x}}{\frac{1}{y}} - \frac{\frac{1}{y}}{\frac{1}{x}} = \frac{y}{x} - \frac{x}{y}$
$We compare this result with y \otimes x and see they are the same, so Statement III is always true.$	$\frac{y}{x} - \frac{x}{y} = y \otimes x$

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Final Answer

E

Question

The operation \otimes is defined for all nonzero numbers a and b by a \otimes b = \frac{a}{b} - \frac{b}{a} . If x and y are nonzero numbers, which of the following statements must be true? I. x \otimes x y = x (1 \otimes y) II. x \otimes y = - (y \otimes x) III. \frac{1}{x} \otimes \frac{1}{y} = y \otimes x

Answer Choices

A. $I only$
B. $II only$
C. $III only$
D. $I and II$
E. $II and III$

Steps

Explanation	Calculations	Help
$We recall the definition of the operation.$	$a \otimes b = \frac{a}{b} - \frac{b}{a}$
$We evaluate x \otimes (x y) by substituting a = x and b = x y into the definition.$	$x \otimes (x y) = \frac{x}{x y} - \frac{x y}{x}$
$We simplify the expression for x \otimes (x y) .$	$x \otimes (x y) = \frac{x}{x y} - \frac{x y}{x} = \frac{1}{y} - y$	Theory & Tactics Method Card ALG5-D Click to view full details Theory & Tactics Method Card FDPR2-A Click to view full details
$We evaluate 1 \otimes y by substituting a = 1 and b = y into the definition.$	$1 \otimes y = \frac{1}{y} - \frac{y}{1}$
$We simplify the expression for 1 \otimes y .$	$1 \otimes y = \frac{1}{y} - y$
$We multiply the simplified value by x .$	$x \times (1 \otimes y) = x \times (\frac{1}{y} - y) = \frac{x}{y} - x \times y$
$We compare the results of x \otimes (x y) and x \times (1 \otimes y) to check Statement I.$	$x \otimes (x y) = \frac{1}{y} - y x \times (1 \otimes y) = \frac{x}{y} - x \times y$
$We evaluate x \otimes y .$	$x \otimes y = \frac{x}{y} - \frac{y}{x}$
$We evaluate y \otimes x .$	$y \otimes x = \frac{y}{x} - \frac{x}{y}$
$We find the negative of y \otimes x and then simplify it.$	$- (y \otimes x) = - (\frac{y}{x} - \frac{x}{y}) = - \frac{y}{x} + \frac{x}{y} = \frac{x}{y} - \frac{y}{x}$
$We observe that this matches x \otimes y, so Statement II is always true.$	$\frac{x}{y} - \frac{y}{x} = x \otimes y$
$We evaluate \frac{1}{x} \otimes \frac{1}{y} by substituting a = \frac{1}{x} and b = \frac{1}{y} into the definition.$	$\frac{1}{x} \otimes \frac{1}{y} = \frac{\frac{1}{x}}{\frac{1}{y}} - \frac{\frac{1}{y}}{\frac{1}{x}} = \frac{y}{x} - \frac{x}{y}$
$We compare this result with y \otimes x and see they are the same, so Statement III is always true.$	$\frac{y}{x} - \frac{x}{y} = y \otimes x$

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Final Answer

E