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If x>3, which of the following is equivalent to [tex]\frac{1}{\frac{1}{x+2}+\frac{1}{x+3}}[/tex]?
A) [tex]\frac{2x+5}{x^2+5x+6}[/tex]
B) [tex]\frac{x^2+5x+6}{2x+5}[/tex]
C) [tex]2x+5[/tex]
D) [tex]x^2+5x+6[/tex]

Respuesta :

wegnerkolmp2741o

Answer:

x^2 +5x+6

----------------------

2x+5

Step-by-step explanation:

1

-----------------

1/(x+2) + 1/(x+3)

Multiply by ( x+2) * (x+3) in the numerator and denominator

1 * ( x+2) * (x+3)

-----------------

(1/(x+2) + 1/(x+3)) *( x+2) * (x+3)

Distribute

( x+2) * (x+3)

-----------------

((x+3) + (x+2))

Combine like terms

( x+2) * (x+3)

-----------------

2x+5

Foil the numerator

x^2 +2x+3x+6

---------------------

2x+5

Combine like terms

x^2 +5x+6

----------------------

2x+5

09pqr4sT

Answer:

B. [tex]\frac{x^2+5x+6}{2x+5}[/tex]

Step-by-step explanation:

[tex]\displaystyle \frac{1}{\frac{1}{x+2}+\frac{1}{x+3}}[/tex]

Add the fractions in the denominator.

[tex]\frac{1(x+3)}{\left(x+2\right)\left(x+3\right)}+\frac{1(x+2)}{\left(x+2\right)\left(x+3\right)}[/tex]

Denominators are equal, so combine.

[tex]\frac{x+3+x+2}{\left(x+2\right)\left(x+3\right)}[/tex]

Combine like terms.

[tex]\frac{2x+5}{\left(x+2\right)\left(x+3\right)}[/tex]

Back to the problem.

[tex]\displaystyle\frac{1}{\frac{2x+5}{\left(x+2\right)\left(x+3\right)}}[/tex]

Apply fraction rule 1/b/c = c/b

[tex]\frac{\left(x+2\right)\left(x+3\right)}{2x+5}[/tex]

Expand the brackets in the numerator.

[tex]\frac{x^2+5x+6}{2x+5}[/tex]

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