Question

If x>3, which of the following is equivalent to
`(1)/((1)/(x+2)+(1)/(x+3))`?

A)
`(2x+5)/(x^2+5x+6)`
B)
`(x^2+5x+6)/(2x+5)`
C)
`2x+5`
D)
`x^2+5x+6`

Answer 1

B.
`(x^2+5x+6)/(2x+5)`

Explanation:

`\displaystyle (1)/((1)/(x+2)+(1)/(x+3))`

Add the fractions in the denominator.

`(1(x+3))/(\left(x+2\right)\left(x+3\right))+(1(x+2))/(\left(x+2\right)\left(x+3\right))`

Denominators are equal, so combine.

`(x+3+x+2)/(\left(x+2\right)\left(x+3\right))`

Combine like terms.

`(2x+5)/(\left(x+2\right)\left(x+3\right))`

Back to the problem.

`\displaystyle(1)/((2x+5)/(\left(x+2\right)\left(x+3\right)))`

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

`(\left(x+2\right)\left(x+3\right))/(2x+5)`

Expand the brackets in the numerator.

`(x^2+5x+6)/(2x+5)`

Answer 2

x^2 +5x+6

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

2x+5

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