Question

Question #bf791

Answer 1

The answer is (B) `(x^2 + 5x + 6)/(2x+5)`

Explanation:

The idea here is that you need to take your starting eexpressions

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

and do some algebraic manipulations to get it to math one of the four expressions given.

Focus on the denominator of the original fraction

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

In order to be able to add these two fractions, you need them to have the same denominator. This means that you're going to have to multiply the first one by `1 = (x+3)/(x+3)` and the second one by `1 = (x+2)/(x+2)` to get their common denominator, `(x+2)(x+3)`

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

`(x+3)/((x+2)(x+3)) + (x+2)/((x+2)(x+3))`

Now you can add these fractions by adding their numerators

`( x+3 + x+2)/((x+2)(x+3)) = (2x + 5)/((x+2)(x+3))`

The original fraction can thus be written as

`1/((2x+5)/((x+2)(x+3))`

Now, you know that dividing a number by a fraction is equivalent to multiplying the number by the inverse of the fraction

`color(blue)(a/(b/c) = a * c/b)`

In your case, `a = 1`, `b = (2x+5)` and `c = (x+2)(x+3)`. This means that you have

`1/((2x+5)/((x+2)(x+3))) = 1 * ((x+2)(x+3))/(2x+5) = ((x+2)(x+3))/(2x+5)`

Finally, you can expand the parantheses of the numerator to get

`(x+2)(x+3) = x^2 + 2x + 3x + 6 = x^2 + 5x + 6`

This means that the expression will be equal to

`((x+2)(x+3))/(2x+5) = (x^2 + 5x + 6)/(2x+5)`

Therefore, you have

`1/(1/(x+2) + 1/(x+3)) = color(green)((x^2 + 5x + 6)/(2x+5))`