Adding and Subtracting Rational Expressions - Problem 1

Whenever we are adding or subtracting fractions we always need a common denominator and in order to do that when dealing with rational expressions I always find it easiest to factor out the denominators completely.

So looking at my first denominator I see I have a common factor 7, so I can take that out 7 leaving me with 1 plus 2x. Looking at my second denominator, I see that I have a 5 I can take out leaving me with 1 plus 2x as well.

So finding my LCD, my least common denominator, I know I need a 1 plus 2x and I also need a number that has a 7 and a 5. The only number that has both of those terms the smallest one is 7 times 5 35. So in order to obtain a 35 I need to multiply this first one by 5 over 5 and the second term by 7 over 7. So what I end up with then is 20 over 35(1 plus 2x) plus 14 over 35(1 plus 2x). Once our denominators are the same, we can just combine like terms in the numerator giving us 34 over 35(1 plus 2x).

In this case nothing can cancel sometimes we will be able to factor, cancel some things, in this case what we are left with is our final answer.