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Advanced Math / Equivalent expressions Difficulty: Hard

Which expression is equivalent to $\frac{4}{4 x - 5} - \frac{1}{x + 1}$ ?

$\frac{1}{(x + 1) (4 x - 5)}$

$\frac{3}{3 x - 6}$

$- \frac{1}{(x + 1) (4 x - 5)}$

$\frac{9}{(x + 1) (4 x - 5)}$

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Explanation

Choice D is correct. The expression $\frac{4}{4 x - 5} - \frac{1}{x + 1}$ can be rewritten as $\frac{4}{4 x - 5} + \frac{(- 1)}{x + 1}$ . To add the two terms of this expression, the terms can be rewritten with a common denominator. Since $\frac{x + 1}{x + 1} = 1$ , the expression $\frac{4}{4 x - 5}$ can be rewritten as $\frac{(x + 1) (4)}{(x + 1) (4 x - 5)}$ . Since $\frac{4 x - 5}{4 x - 5} = 1$ , the expression $\frac{- 1}{x + 1}$ can be rewritten as $\frac{(4 x - 5) (- 1)}{(4 x - 5) (x + 1)}$ . Therefore, the expression $\frac{4}{4 x - 5} + \frac{(- 1)}{x + 1}$ can be rewritten as $\frac{(x + 1) (4)}{(x + 1) (4 x - 5)} + \frac{(4 x - 5) (- 1)}{(4 x - 5) (x + 1)}$ , which is equivalent to $\frac{(x + 1) (4) + (4 x - 5) (- 1)}{(x + 1) (4 x - 5)}$ . Applying the distributive property to each term of the numerator yields $\frac{(4 x + 4) + (- 4 x + 5)}{(x + 1) (4 x - 5)}$ , or $\frac{(4 x + (- 4 x)) + (4 + 5)}{(x + 1) (4 x - 5)}$ . Adding like terms in the numerator yields $\frac{9}{(x + 1) (4 x - 5)}$ .

Choice A is incorrect and may result from conceptual or calculation errors.

Choice B is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.