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Equivalent expressions SAT question

Math · Advanced Math · Hard · Answer it here, explanation included.

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

ExplanationShow

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.

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