Question

How do you add `\frac { 1} { x + 3} + \frac { 1} { x + 4}`?

Answer 1

I found: #(2x+7)/((x+3)(x+4))

Explanation:

You need both fractions with the same denominator (the part below) to be able to add; to do that we can "choose" a common denominator!

We can choose `(x+3)(x+4)`
in fact, this is common to both and will make the two fractions with the same denominator...the problem is now the numerator!

You changed the denominator so now you need to "adapt" the numerator as well: no problem, we simply multiply the numerator by the bit we used in the denominator and was not present at the start!

`(1*(color(red)(x+4)))/((x+3)(x+4))+(1*(color(red)(x+3)))/((x+3)(x+4))=`

the red bits are the "corrections" we introduced to adapt the numerator.

Now we can add because the denominators are the same:
`=((x+3)+(x+4))/((x+3)(x+4))=(x+3+x+4)/((x+3)(x+4))=(2x+7)/((x+3)(x+4))`

Answer 2

In any adding of fractions, whether arithmetic or algebraic, you need to find the common denominator. (LCD)

`1/(x+3) + 1/(x+4)" "larr LCD = (x+3)(x+4)`

Make equivalent fractions.

=`1/(x+3) xx(x+4)/(x+4) + 1/(x+4) xx (x+3)/(x+3)`

=`((x+4) + (x+3))/((x+3)(x+4))`

=`(2x+7)/((x+3)(x+4))`