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In $3-20$ , perform the indicated additions or subtractions and write the result in simplest form. In each case, list any values of the variables for which the fractions are not defined.$$\frac{3}{x+2}+\frac{x-2}{x}$$

$\frac{x^{2}+3 x-4}{x(x+2)}, x=0$ and $x=-2$

Algebra

Chapter 2

THE RATIONAL NUMBERS

Section 4

Adding and Subtracting Rational Expressions

Fractions and Mixed Numbers

Decimals

Equations and Inequalities

Oregon State University

Harvey Mudd College

University of Michigan - Ann Arbor

Lectures

01:32

In mathematics, the absolu…

01:11

03:39

In $3-20$ , perform the in…

01:36

04:14

04:40

03:04

00:41

01:30

03:30

03:13

00:53

we're being asked to add these two fractions together, whether we're adding or subtracting. The one thing we have to have first is a common denominator, which we do not have here. So our first step is gonna be in try and get these to a common denominator. One denominator is X plus two. The other denominator is X X plus two is a quantity. Unfortunately, it's X plus two, not x times two. If we had just had two x and X, this would be pretty easy. We could just multiply the second fraction. By two, we'd have denominators of two X on both them, and we could add and be done. But that's not the case. This is X Plus two. So there isn't any way that I can simply get the second fraction too easily equal the same denominator as the first. You can't add to top him to the numerator and denominator. That's not a thing, right? You have to multiply by something. So unfortunately, there's really nothing in common here. Yes, they both have an X, but in one of them the X is bound with the two, and then the other one is just a standalone X. So they don't really have anything in common when you have fractions and they're denominators have absolutely nothing in common. The only way that you can really go about trying to solve it out is you multiply each fraction by the others denominator. Meaning? I'm going to take this first fraction of three over X plus two and I'm gonna multiply it by my second fractions denominator. Namely this X right here. So if I multiply by X and the job dinner, I have to multiply the numerator by X as well. Okay, three times X is three X. I'm not actually going to multiply out this denominator yet because it's possible I may be able to simplify stuff later on. I'm just going to write it as literally x times the quantity of X plus two. You don't have to do that, but remember, we've talked about with the multiplying and dividing rational functions in the section before this. If you can simplify rational expressions, what you have to do is factor the numerator factor the denominator and see if things canceled. So I might as well leave my denominator factored until I know that I can go ahead and multiply back together. Okay, so I've got that for my first fraction three x over x times the quantity of explosives to again. If you don't like that, just go ahead and distribute the X. And there it would give you X squared plus two X. You can totally have that if you want. For my second fraction, I've got X minus two over X, and I am going to multiply that by the other fractions denominator, which in this case would mean my first fractions denominator of X plus two. So that's what we're gonna multiply by on both top and bottom numerator and denominator in my numerator, that means I'm taking the quantity of X minus two times the quantity of X plus two. So that's going to be foiling. Right? First term times, first term X times X is X squared outside terms negative too. Times are how do we first outer Inter, lest so outside terms would be the X and the two doesn't really matter. But I'm gonna try and do it by full just cause I imagine that's how most you have learned it. So I figure it'll help so X Times two would be two x inner two terms would be negative two times x, so that would be negative. Two X and then last two terms would be negative. Two times two. So negative four in my denominator, just like last time. Instead of actually distributing that X, I'm simply going to write it as X times the quantity of X plus two. Now, when I write that up here, I'm going to go ahead and clean up the middle here because two X minus two X would cancel. Right. So I'm gonna rewrite it up here as X squared minus four over X times the quantity of X plus two because we can see two x minus two X would cancel out. All right, you can see that we have now got ourselves a common denominator. So since we've got ourselves our common denominator, let's go ahead and combine. Remember, when we add fractions, we do not combine the denominator. We just rewrite that exactly as it was. We only combine the numerator that we can only combined with like terms, and I don't see any here. Remember, an X squared is not alike with Justin X so I can combine these into one fraction. But I can't actually combine them together, meaning it's just X squared plus three X minus four. That's our answer. You could factor the numerator. And if you factor the numerator, what you will see is that nothing cancels. So since nothing cancels, that is our answer, we do still need to determine any values that would give us an undefined problem. So let's look back at the original problem. I have a denominator of X plus two in a denominator of X. Remember, what gives us undefined is having a zero in the denominator. So since we can't have a zero in the denominator, I'm going to literally write that algebraic lee and say, X plus two cannot be zero and X cannot be zero, and then we'll just solve them both out. 1st 1 we would need to subtract two to the other side. Giving us that X looks like it can not equal negative, too. For the 2nd 1 it's kind of already solved out. So our answer is X squared plus three x minus four over X Times X plus two. Assuming that X does not equal negative too, and X does not equal zero

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