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Robert McKeown: Hi everyone, Robert j McKeown

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here and welcome to another ALEKS walkthrough

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video This one is number six. And we're going to

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look at rational expressions, which I think are

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very intelligent, smart sounding kind of

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expression. Now, the specific what a what a

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rational expression is, I'll tell you in a few

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moments when we get into the slides. As an

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economist, we're often trying to rearrange an

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expression to get a certain result or to solve

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for an unknown variable. Like for example, we

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might want to solve for the price of a product,

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like gasoline or smartphones, or software

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products, or we might want to solve for

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quantities produced similar items. And so it's

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really, really difficult to get away and not use

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mathematics, and economics. And I really like

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today's topic, because it gets into the real

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nitty gritty of what you're going to be doing.

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As an economics undergraduate major, you're

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going to be rearranging expressions, you're

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going to be trying to show certain results. And

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that's what today's practice is all about. So

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becoming a master of this topic is going to pay

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off for you in every one of your core economics

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classes, from here to the end of your

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undergraduate career, and then into grad school,

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as well. And being good with math is a great job

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skill, a very employable skill to have. Now,

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again, what do I need from you, I need you to

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have your pencil, I need you to have your paper.

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Or if you're very sophisticated, and you have

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the money, invest in a tablet that you can

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write, and work through the problems with me.

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For this topic, and through this video, remember

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that the video doesn't cover every important

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topic in ALEKS, topic six, or every concept in

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topic six. I'm just trying to get you enough

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information to make sure that you get started

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and help you through maybe some of the more

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difficult problems. So let's get started. Topic

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six is all about rational expressions. So what

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is a rational expression? Well, it's when one

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polynomial, such as X plus five is divided by

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another polynomial, such as x squared minus

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nine. And remember, we seen a lot of quadratic

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equations quadratics quadratic expressions are

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also a type of polynomial. Now every question in

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this section involves rational expressions. To

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start off, why don't we talk about the domain of

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rational expression? So in an earlier topic, I

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told you that we must never ever ever divide by

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zero. And if you look here, it says the domain

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of any rational expression is all real numbers,

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except for those that make the denominator equal

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to zero. Now, why is this important? Let me show

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you an example. Here, we have x plus five

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divided by x squared minus nine, I can rewrite

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that using the difference of squares. Whenever I

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say something squared minus something else

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that's squared, I could I can use the difference

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of squares. For example, I'll rewrite this as x

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squared minus three squared, three squared is

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equal to nine. So I'm going to write this as

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three squared. I know from this rule, that X

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plus five is divided by x minus three,

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multiplied by x plus three. And you can prove to

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yourself at home that x minus three times x plus

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three is equal to x squared minus three squared

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or x squared minus nine. Now we know that all

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real numbers are in the domain, except for those

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real numbers that make the denominator equal to

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zero. So using this right here, it's easy to see

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that if x is equal to three

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Then we have three plus five, divided by three

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minus three, multiplied by three plus three,

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which is eight, over zero over six, which is

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equal to eight over zero. And I told you that we

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should never divide by zero, because we never

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observe a real number like eight divided by

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zero, so we just don't really know what that is.

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And so we say it's undefined. We don't know what

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it is. We can't define what eight divided by

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zero is. Similarly, if x is equal to negative

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three, what happens? Well, we get negative

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three, plus five over negative three minus three

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times negative three plus three.

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And this time we get negative two, or excuse me

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to over zero. So both x is equal to three, and x

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equal negative three are not in the domain.

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Here's a ALEKS question. And I've copied and

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pasted from ALEKS and tells us that find all the

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values of x that are not in the domain of g. So

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I can take a look at this thing here, I can do

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some factoring. So for these questions, the

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numerator never matters. It doesn't influence

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the domain. But I could look at this and I could

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say, well, I noticed that eight times nine is

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equal to 72. And negative eight minus nine is

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equal to 17. And when I recognize that, I can

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factor the numerator. So looking at the

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denominator, I see my favorite x squared minus

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two squared is my difference of squares. A very

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useful formula. It comes up a number of times,

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including finance. And I can write this as x

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plus two, and x minus two. Now, I didn't need to

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do that I, you could probably just look at the

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original expression and answer this question.

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But you'll see later questions, we need to be

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able to do these operations. So I'm going to

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show you now. And it's pretty obvious here that

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x equals two and x equals negative two are not

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in the domain, because when x is equal to two,

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or x is equal to negative two, the denominator

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is zero. Here we are an ALEKS. I'm going to put

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in our answers, we had negative two. And then

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I'm going to follow the instructions, put in a

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comma and a two. And I'll click on the check

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button. And fortunately, we got the right

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answer. Here's a great question. We have a

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complex fraction and rational expression but a

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complex fraction. And when I see a problem like

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this, I want to start off as my first step to

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find a lowest common denominator for the

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numerator. And then I want to do the same thing

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for the denominator. And after I do that, I'll

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think about what my next steps are. I don't like

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to see non common denominators. So that's

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something that I like to jump on, and fix right

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away. So starting with the numerator, I've got x

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squared on one side and 49 on the other. So for

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this first term, eight over 49, I'm going to

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multiply it by x squared over x squared, or I'm

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gonna multiply both the numerator and

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denominator by x squared. That way, I'm

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following the rules of algebra x squared.

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divided by x squared is just equal to one, I'm

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not changing, I'm not changing the expression in

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any way when I do this, that's the whole idea

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behind lowest common denominator. Now looking at

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the second term in the numerator, I need a 49.

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I'm going to multiply the top by 49 and the

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bottom by 49. And now I've got a common a common

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denominator, I could rewrite that as x squared

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minus 49. All over 49 x squared. Now, what about

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the denominator? Let's take a look at the

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denominator, one over seven, well, I can just

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multiply that by x. So I'm going to have x over

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seven x plus one over x, I can multiply the

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numerator by seven and the denominator by seven.

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And now I've got x plus seven divided by

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seven x. Now what's the rule when we're dividing

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two fractions?

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Well, the rule the operator is to flip the

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denominators, numerator and denominator and then

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multiply it by the numerator, what's the top so

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I can rewrite this as eight x squared minus 49

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multiplied by, well, maybe I'll write it out

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like this, I'll just rewrite it like that. And

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this whole thing is going to be multiplied by

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seven x over x plus seven. That's the rule. We

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saw that in an earlier video. That's the catch

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how we deal with division. I like the way that

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I've written it here, I can clearly see what's

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being multiplied and what's being divided. And I

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want to pay particular attention to the division

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because the division is going to what we say

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cancel things out. I can make a simplification

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here. And I can rewrite this expression as eight

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x squared minus 49 over seven x squared

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multiplied by seven x over x plus seven. And we

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can cancel out seven x divided by seven x. And

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I'm going to be left with eight x squared minus

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49 over seven x, x plus seven. So it was this

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exponent that was cancelled out previously. Not

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all the seven x's. I went ahead and I input our

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answer into ALEKS. And you can see that it was

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the simplest form possible. There wasn't

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anything more we could really do. There's a 49

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there, which is seven squared. But if we rewrote

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it that way, there's we can't simplify any

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further because that eight, the eight in front

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of the x squared is stopping us from doing any

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more factoring.

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Our next problem is asking us to solve for w. So

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we've got a rational expression. And there's

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three separate terms. Like before, I want to get

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started by finding a lowest common denominator.

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So notice that if I call that a and I call that

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B, A is equal to two B's. So if I multiply the

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numerator and denominator of this term here by

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to have the same denominator and I won't be

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altering the value of that term.

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I'm just feeling this in so you can see two w

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minus 12. Now I've got this negative one term

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here. Well, if I want to give it the same

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denominator, As the other two terms, two w minus

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12, I can multiply it, and its numerator

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denominator, because of course, negative one is

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equal to negative one over one, I can multiply

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its numerator and denominator by two w minus 12.

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So if I do that, I'm gonna get

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to w minus five over two w minus 12, you can see

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very clearly that that's got to be equal to one.

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And I don't want to forget my negative sign in

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front. The last term on the left is unchanged,

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it's still three over two w minus 12. There's a

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big minus right there. Now the great part is,

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since the denominator of every single term is

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the same, I could multiply both sides of this

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equation by two w minus 12. And if I multiply

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both sides of this equation by two w minus 12,

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I'm going to be left with negative three minus

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two w minus 12 is equal to 14. And you want to

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be really careful that you don't make a mistake

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with your signs. Very, very easy mistake to make

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one that I make fairly regularly. not unusual

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that I would forget a sign here or there. Given

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that reality in which I operate, I'm very

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careful to keep my sign negative on the outside

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of this bracket, because now I want to bring

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that negative into the bracket. So I'm going to

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rewrite this as minus three minus two w plus 12

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is equal to 14. Now notice that I've just got a

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W and a bunch of numbers. And so this is now

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linear. It's a linear equation or a linear

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expression. And that means I'm going to get one

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solution. Now I made a small mistake, I forgot

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that I forgot a negative sign. I forgot this

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negative sign partly because I didn't draw it.

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Clearly, I should have a negative 14 there. Now

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I want to finish this off and solve for w. So

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I'm going to have negative two w is equal to

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negative 14 minus 12 plus three, and I get w is

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equal to 23 over two