  # Adding and Subtracting Fractions with Unlike Denominators

In adding and subtracting fractions with unlike denominators, you have to get a least common denominator which is exactly like the least common multiple it just so happens to be in the denominator. But first, looking at each one of these denominators, we must make sure we have them completely factored. The x minus 2 and the x plus 2 are prime. However, this x squared minus 4 can actually be written as a product of two primes of x minus 2 times x plus 2. So, let's find our least common denominator. So, look at the first one, and we place x minus 2 in the least common denominator. So, we are done with the first one. Looking at the second one, we have this x plus 2 that's not in in the least common denominator, so we have to place that there as well. Notice, once we get to this last one over here, the x minus 2 and the x plus 2 are already in the least common denominator, therefore we don't have to place extra ones in. So, the least common denominator happens to be x minus 2 times x plus 2. Now, actually doing the calculations, we are well aware that our least common denominator must contain x minus 2 times x plus 2. So, for each of these fractions, we have to write an equivalent fraction. On this first one, we are missing the x plus 2. On the second one, we’re missing the x minus 2. And on this last one, we are not missing anything. But, if you feel like you should put something there, you can put 1 and that will be fine. So, now what we must do is we just have to multiply all of the numerators together to get these into equivalent fractions. So, we have 7 times x plus 2. Note the negative here. Make sure you place that down in the next step. We have negative 1 times x minus 2. We also have the symbol here, the plus, and then times 4. Now all we need to do from here is just simplify. And what I mean by simplify, is just using the distributive property. Distributing the 7 through the x plus 2, and the negative 1 through the x minus 2. So, we have, 7 times x plus 14 minus x plus 2, and then plus four, all over x minus 2 times x plus 2. We are just going to keep simplifying down. So, now all we have to do is mark our like terms and add those together. We have 7x and negative x, which gives us 6x. Then we have 14 plus 2 plus 4, which is basically 14 plus 6, which gives us 20, all divided by x minus 2 times x plus 2. Now, if we want, we could factor the 6x plus 20. The greatest common factor out of those two terms is actually going to be a 2. So, we have 2 times 3x plus 10 all divided by x minus 2 times x plus 2. If you feel like you don't have full understanding of the least common denominator, including the least common multiple, or you're having trouble factoring, please refer back to the other videos.

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