Question

If x>3, which of the following is equivalent to 2B%5Cfrac%7B1%7D%7Bx%2B3%7D%7D" id="TexFormula1" title="\frac{1}{\frac{1}{x+2}+\frac{1}{x+3}}" alt="\frac{1}{\frac{1}{x+2}+\frac{1}{x+3}}" align="absmiddle" class="latex-formula">?
A) $\frac{2x+5}{x^2+5x+6}$
B) $\frac{x^2+5x+6}{2x+5}$
C) $2x+5$
D) $x^2+5x+6$

Answer 1

Answer:

B. $\frac{x^2+5x+6}{2x+5}$

Step-by-step explanation:

$\displaystyle \frac{1}{\frac{1}{x+2}+\frac{1}{x+3}}$

Add the fractions in the denominator.

$\frac{1(x+3)}{\left(x+2\right)\left(x+3\right)}+\frac{1(x+2)}{\left(x+2\right)\left(x+3\right)}$

Denominators are equal, so combine.

$\frac{x+3+x+2}{\left(x+2\right)\left(x+3\right)}$

Combine like terms.

$\frac{2x+5}{\left(x+2\right)\left(x+3\right)}$

Back to the problem.

$\displaystyle\frac{1}{\frac{2x+5}{\left(x+2\right)\left(x+3\right)}}$

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

$\frac{\left(x+2\right)\left(x+3\right)}{2x+5}$

Expand the brackets in the numerator.

$\frac{x^2+5x+6}{2x+5}$

Answer 2

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