The operation is defined for all nonzero numbers a and b by...

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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 x y=x(1 y) II. x y=-(y x) III. 1/x 1/y=y x

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E We recall the definition of the operation. a b = a/b - b/a We evaluate x (xy) by substituting a = x and b = xy into the definition. x (xy) = x/xy - xy/x We simplify the expression for x (xy). x (xy) = xxy - xyx = 1/y - y We evaluate 1 y by substituting a = 1 and b = y into the definition. 1 y = 1/y - y/1 We simplify the expression for 1 y. 1 y = 1/y - y We multiply the simplified value by x. x (1 y) = x (1/y - y) = x/y - x y We compare the results of x (xy) and x (1 y) to check Statement I. x (xy) = 1/y - y x (1 y) = x/y - x y We evaluate x y. x y = x/y - y/x We evaluate y x. y x = y/x - x/y We find the negative of y x and then simplify it. -(y x) = -(y/x - x/y) = -y/x + x/y = x/y - y/x We observe that this matches x y, so Statement II is always true. x/y - y/x = x y We evaluate 1/x 1/y by substituting a = 1/x and b = 1/y into the definition. 1/x 1/y = 1/x1/y - 1/y1/x = y/x - x/y We compare this result with y x and see they are the same, so Statement III is always true. y/x - x/y = y x

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

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