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Advanced Math / Equivalent expressions Difficulty: Hard

$the fraction with numerator 2, and denominator x minus 2, end fraction, plus, the fraction with numerator 3, and denominator x plus 5, end fraction, equals, the fraction with numerator r x plus t, and denominator, open parenthesis, x minus 2, close parenthesis, times, open parenthesis, x plus 5, close parenthesis, end fraction$

The equation above is true for all $x greater than 2$ , where r and t are positive constants. What is the value of rt ?

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Explanation

Choice C is correct. To express the sum of the two rational expressions on the left-hand side of the equation as the single rational expression on the right-hand side of the equation, the expressions on the left-hand side must have the same denominator. Multiplying the first expression by $the fraction with numerator x plus 5, and denominator x minus 5, end fraction$ results in $the fraction with numerator 2 times, open parenthesis, x plus 5, close parenthesis, and denominator, open parenthesis, x minus 2, close parenthesis, times, open parenthesis, x plus 5, close parenthesis, end fraction$ , and multiplying the second expression by $the fraction with numerator x minus 2, and denominator x minus 2, end fraction$ results in $the fraction with numerator 3 times, open parenthesis, x minus 2, close parenthesis, and denominator, open parenthesis, x minus 2, close parenthesis, times, open parenthesis, x plus 5, close parenthesis, end fraction$ , so the given equation can be rewritten as $the fraction with numerator 2 times, open parenthesis, x plus 5, close parenthesis, and denominator, open parenthesis, x minus 2, close parenthesis, times, open parenthesis, x plus 5, close parenthesis, end fraction, plus, the fraction with numerator 3 times, open parenthesis, x minus 2, close parenthesis, and denominator, open parenthesis, x minus 2, close parenthesis, times, open parenthesis, x plus 5, close parenthesis, end fraction, equals, the fraction with numerator r x plus t, and denominator, open parenthesis, x minus 2, close parenthesis, times, open parenthesis, x plus 5, close parenthesis, end fraction$ , or $the fraction with numerator 2 x plus 10, and denominator, open parenthesis, x minus 2, close parenthesis, times, open parenthesis, x plus 5, close parenthesis, end fraction, plus, the fraction with numerator 3 x minus 6, and denominator, open parenthesis, x minus 2, close parenthesis, times, open parenthesis, x plus 5, close parenthesis, end fraction, equals, the fraction with numerator r x plus t, and denominator, open parenthesis, x minus 2, close parenthesis, times, open parenthesis, x plus 5, close parenthesis, end fraction$ . Since the two rational expressions on the left-hand side of the equation have the same denominator as the rational expression on the right-hand side of the equation, it follows that $open parenthesis, 2 x plus 10, close parenthesis, plus, open parenthesis, 3 x minus 6, close parenthesis, equals, r x plus t$ . Combining like terms on the left-hand side yields $5 x plus 4, equals, r x plus t$ , so it follows that $r equals 5$ and $t equals 4$ . Therefore, the value of $r t$ is $5 times 4, which equals 20$ .

Choice A is incorrect and may result from an error when determining the sign of either r or t. Choice B is incorrect and may result from not distributing the 2 and 3 to their respective terms in $the fraction with numerator 2 times, open parenthesis, x plus 5, close parenthesis, and denominator, open parenthesis, x minus 2, close parenthesis, times, open parenthesis, x plus 5, close parenthesis, end fraction, plus, the fraction with numerator 3 times, open parenthesis, x minus 2, close parenthesis, and denominator, open parenthesis, x minus 2, close parenthesis, times, open parenthesis, x plus 5, close parenthesis, end fraction, equals, the fraction with numerator r x plus t, and denominator, open parenthesis, x minus 2, close parenthesis, times, open parenthesis, x plus 5, close parenthesis, end fraction$ . Choice D is incorrect and may result from a calculation error.