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If \(x > 3\), which of the following is equivalent to \(\frac{1}{\frac{1}{x+2}+\frac{1}{x+3}}\)?...


Question

If \(x > 3\), which of the following is equivalent to \(\frac{1}{\frac{1}{x+2}+\frac{1}{x+3}}\)?

Options

A)
\(\frac{2x+5}{x^2+5x+6}\)
B)
\( \frac{x^2+5x+6}{2x+5} \)
C)
\(2x + 5 \)
D)
\(x^2 + 5x + 6\)

The correct answer is B.

Explanation:

Choice B is correct. To rewrite \(\frac{1}{\frac{1}{x+2}+\frac{1}{x+3}}\), multiply by \(\frac{(x+2)(x+3)}{(x+2)(x+3)}\). This results in the expression \(\frac{(x+2)(x+3)}{(x+2)(x+3)}\), which is equivalent to the expression in choice B.
Choices A,C, and D are incorrect and could be the result of common algebraic errors that arise while manipulating a complex fraction.


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