Simplify the complex fraction. ( 6 x 2 − 11 x − 10 7 x 2 + 7 x ) ( 2 x − 5 7 x + 1 ) \frac { \left( \frac { 6 x ^ { 2 } - 11 x - 10 } { 7 x ^ { 2 } + 7 x } ight) } { \left( \frac { 2 x - 5 } { 7 x + 1 } ight) } ( 7 x + 1 2 x − 5 ) ( 7 x 2 + 7 x 6 x 2 − 11 x − 10 ) A) 3 x + 2 x , x ≠ − 5 2 \frac { 3 x + 2 } { x } , x
eq - \frac { 5 } { 2 } x 3 x + 2 , x  = − 2 5 B) 7 ( x + 1 ) x ( 3 x + 2 ) , x ≠ − 5 2 \frac { 7 ( x + 1 ) } { x ( 3 x + 2 ) } , x
eq - \frac { 5 } { 2 } x ( 3 x + 2 ) 7 ( x + 1 ) , x  = − 2 5 C) ( 3 x + 2 ) ( 7 x + 1 ) 7 x ( x + 1 ) , x ≠ 5 2 \frac { ( 3 x + 2 ) ( 7 x + 1 ) } { 7 x ( x + 1 ) } , x
eq \frac { 5 } { 2 } 7 x ( x + 1 ) ( 3 x + 2 ) ( 7 x + 1 ) , x  = 2 5 D) ( 3 x + 2 ) ( x + 1 ) x ( 7 x + 1 ) , x ≠ 5 2 \frac { ( 3 x + 2 ) ( x + 1 ) } { x ( 7 x + 1 ) } , x
eq \frac { 5 } { 2 } x ( 7 x + 1 ) ( 3 x + 2 ) ( x + 1 ) , x  = 2 5 E) 7 x + 1 7 x , x ≠ 5 2 \frac { 7 x + 1 } { 7 x } , x
eq \frac { 5 } { 2 } 7 x 7 x + 1 , x  = 2 5

Question

Simplify the complex fraction.   ( 6 x 2 − 11 x − 10 7 x 2 + 7 x )  ( 2 x − 5 7 x + 1 )  \frac { \left( \frac { 6 x ^ { 2 } - 11 x - 10 } { 7 x ^ { 2 } + 7 x } ight)  } { \left( \frac { 2 x - 5 } { 7 x + 1 } ight)  } ( 7 x + 1 2 x − 5 ​ )  ( 7 x 2 + 7 x 6 x 2 − 11 x − 10 ​ )  ​ A)    3 x + 2 x , x ≠ − 5 2 \frac { 3 x + 2 } { x } , x 
eq - \frac { 5 } { 2 } x 3 x + 2 ​ , x  = − 2 5 ​ B)    7 ( x + 1 )  x ( 3 x + 2 )  , x ≠ − 5 2 \frac { 7 ( x + 1 )  } { x ( 3 x + 2 )  } , x 
eq - \frac { 5 } { 2 } x ( 3 x + 2 )  7 ( x + 1 )  ​ , x  = − 2 5 ​ C)    ( 3 x + 2 )  ( 7 x + 1 )  7 x ( x + 1 )  , x ≠ 5 2 \frac { ( 3 x + 2 )  ( 7 x + 1 )  } { 7 x ( x + 1 )  } , x 
eq \frac { 5 } { 2 } 7 x ( x + 1 )  ( 3 x + 2 )  ( 7 x + 1 )  ​ , x  = 2 5 ​ D)    ( 3 x + 2 )  ( x + 1 )  x ( 7 x + 1 )  , x ≠ 5 2 \frac { ( 3 x + 2 )  ( x + 1 )  } { x ( 7 x + 1 )  } , x 
eq \frac { 5 } { 2 } x ( 7 x + 1 )  ( 3 x + 2 )  ( x + 1 )  ​ , x  = 2 5 ​ E)    7 x + 1 7 x , x ≠ 5 2 \frac { 7 x + 1 } { 7 x } , x 
eq \frac { 5 } { 2 } 7 x 7 x + 1 ​ , x  = 2 5 ​

Victoria Correia · Accepted Answer

Final Answer : C, Explanation :   To simplify the given complex fraction, we first factorize the numerators and denominators where possible. The numerator of the complex fraction is   6 x 2 − 11 x − 10 7 x 2 + 7 x \frac{6x^2 - 11x - 10}{7x^2 + 7x} 7 x 2 + 7 x 6 x 2 − 11 x − 10 ​  , and the denominator of the complex fraction is   2 x − 5 7 x + 1 \frac{2x - 5}{7x + 1} 7 x + 1 2 x − 5 ​  .1. Factorize   6 x 2 − 11 x − 10 6x^2 - 11x - 10 6 x 2 − 11 x − 10  : This can be factored into   ( 3 x + 2 ) ( 2 x − 5 ) (3x + 2)(2x - 5) ( 3 x + 2 ) ( 2 x − 5 )  .2. Factorize   7 x 2 + 7 x 7x^2 + 7x 7 x 2 + 7 x  : This can be factored into   7 x ( x + 1 ) 7x(x + 1) 7 x ( x + 1 )  .3. The fraction   2 x − 5 7 x + 1 \frac{2x - 5}{7x + 1} 7 x + 1 2 x − 5 ​   remains as is since it cannot be further simplified.The complex fraction becomes:  ( 3 x + 2 ) ( 2 x − 5 ) 7 x ( x + 1 ) 2 x − 5 7 x + 1 = ( 3 x + 2 ) ( 2 x − 5 ) 7 x ( x + 1 ) × 7 x + 1 2 x − 5 \frac{\frac{(3x + 2)(2x - 5)}{7x(x + 1)}}{\frac{2x - 5}{7x + 1}} = \frac{(3x + 2)(2x - 5)}{7x(x + 1)} 	imes \frac{7x + 1}{2x - 5} 7 x + 1 2 x − 5 ​ 7 x ( x + 1 ) ( 3 x + 2 ) ( 2 x − 5 ) ​ ​ = 7 x ( x + 1 ) ( 3 x + 2 ) ( 2 x − 5 ) ​ × 2 x − 5 7 x + 1 ​  Cancel out the   ( 2 x − 5 ) (2x - 5) ( 2 x − 5 )   terms:  ( 3 x + 2 ) 7 x ( x + 1 ) × ( 7 x + 1 ) \frac{(3x + 2)}{7x(x + 1)} 	imes (7x + 1) 7 x ( x + 1 ) ( 3 x + 2 ) ​ × ( 7 x + 1 )  This simplifies to:  ( 3 x + 2 ) ( 7 x + 1 ) 7 x ( x + 1 ) \frac{(3x + 2)(7x + 1)}{7x(x + 1)} 7 x ( x + 1 ) ( 3 x + 2 ) ( 7 x + 1 ) ​  Therefore, the correct answer is   ( 3 x + 2 ) ( 7 x + 1 ) 7 x ( x + 1 ) , x ≠ − 5 2 \frac{(3x + 2)(7x + 1)}{7x(x + 1)}, x 
eq -\frac{5}{2} 7 x ( x + 1 ) ( 3 x + 2 ) ( 7 x + 1 ) ​ , x  = − 2 5 ​  , which matches choice C.

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