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We're going to do an introduction to rational expressions.
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Now, a rational expression is really just a fraction.
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So, rational fraction basically means the same thing.
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To find the value of a rational expression when x equals negative 3, for instance,
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you would plug in negative 3 for x and simplify.
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So, here's some rational expressions.
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And we want to find the value when x equals negative 3.
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So, you put this on pause and then come back and see if you get the same answer
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simplifying by plugging x equal negative 3 that I do.
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All right, so we're going to plug in negative 3.
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All right, so for the first one, we're going to plug in negative 3 for x.
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So, we're going to replace x with negative 3.
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Every place there's an x, you put a negative 3.
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Oops, that was a minus 2.
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Better catch it.
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So, that's going to be 6 over negative 5, which is just negative 6 fifths.
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You can leave your answer like that.
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It's reduced.
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If you want, you could write negative 1 and 1 5th or negative 1.2.
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But those are all the same.
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All right, next one.
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x squared plus 1 over 2x.
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All right, x squared means x times x.
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2x means 2 times x.
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So, since x is negative 3, I have negative 3 times negative 3 plus 1 over 2 times negative 3,
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which is 9 plus 1 over negative 6.
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It's 10 over negative 6, which is negative 5 thirds.
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Again, you can leave your answer in that format, negative 5 thirds, as long as it's reduced,
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or you could put negative 1 and 2 thirds.
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All right, and the last one, we're putting in negative 3 for x.
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So, the denominator of a 0.
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And what is 5 divided by 0?
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That's undefined.
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It doesn't reduce.
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You can't say what number it is.
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It's undefined.
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Anytime you have a 0 in the denominator, a fraction is undefined.
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So, the question is, how do you know which numbers make a rational expression undefined?
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And the answer is, anytime the denominator of a fraction is 0, that fraction is undefined.
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That's the one time that a fraction is undefined.
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If there is a 0 in the denominator.
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So let's say you were looking at 5 over x minus 2, this would be defined for any value of
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x except when the denominator is 0.
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So this is undefined only in one time.
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It's undefined when x minus 2, which is the denominator, equals 0.
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So that means x equals 2 is what makes it undefined.
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If x is 2, it makes that fraction undefined.
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Okay.
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Undefined when x equals 2.
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Alright, let's do another one.
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x plus 5 over 3x.
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Right now, when is it that rational expression is undefined?
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It's when the denominator equals 0.
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Notice it has absolutely nothing to do with the numerator.
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We don't care about the numerator.
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Nothing to do at all with being an undefined rational expression.
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So again, this is only undefined when the denominator is 0.
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Okay, now the denominator is 3x.
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So to say when, 3x is equal to 0.
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Now we could solve that if you divide both sides by 3 when x equals 0.
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So it's undefined when x equals 0.
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Alright, how about x squared plus 1 over 7?
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When is this undefined?
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x squared plus 1 over 7?
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Can the denominator ever be 0?
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That's the question.
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Well, the denominator 7, 7, can never all of a sudden become 0.
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So x squared plus 1 over 7 is never undefined.
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Which means it is always defined.
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So basically, for the first one, 5 over x minus 2, it's undefined when x equals 2.
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What that means is, every other value of x makes this fraction defined.
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x plus 5 over 3x is undefined only when x is 0.
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That means for all other values of x, when you plug it in, you will be able to define
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it.
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Which means you're going to get some number.
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And x squared plus 1 over 7 is never undefined.
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So no matter what you put in for x, it will always be defined.
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Okay.
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How about x squared, I mean x plus 5 over x squared minus 16?
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Is this ever undefined?
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Well, again, it's undefined when the denominator equals 0.
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So it's undefined when x squared minus 16 equals 0.
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Hey, does that look familiar?
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Hmm.
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Well, we've got an x squared term.
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We're going to have to factor.
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So we have x plus 4 times x minus 4 equals 0.
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x plus 4 equals 0, or x minus 4 equals 0.
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So for this fraction, there's two numbers where it's undefined.
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So x plus 5 over x squared minus 16 is undefined when x equals negative 4, or when x equals positive
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4.
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So really, that's your basic introduction to rational expressions.
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And I'm going to go over y.
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You can't have a 0 in the denominator now.
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I'm sure you're really curious.
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Okay.
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y is a fraction undefined if 0 is in the denominator.
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There are a few ways of explaining this.
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But let's just take an example.
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8 over 0.
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What does that mean?
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That means 8 divided by 0.
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Now let's say there was an answer over here.
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The answer, division's really the opposite.
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The inverse of multiplication.
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It means whatever's in the box, when you multiply it by 0, you get 8.
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Well, what could you put in the box so that when you multiply it by 0, you get 8.
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Nothing.
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Nothing works.
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Nothing works to make 8 equals 0 times something in that box.
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So therefore, the answer to this is it's undefined.
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So let's say what if you were trying to do 16 divided by 8?
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Okay, well this means 16 divided by 8, right?
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And you're saying what could you put in the box so that 16 is 8 times whatever's in
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the box.
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Yeah, too.
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That's why 16 over 8 is defined.
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But 8 divided by 0 or any number divided by 0, 0 will be undefined.
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The one that's tricky is if they're both 0.
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0 divided by 0 is also undefined.
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The reason is whatever goes in the box has to just be a single number.
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You can't have more than one choice.
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So this one's the trickiest of all.
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You could say well, heck.
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So divided by 0 means 0 divided by 0, right?
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And the question is what could you put in the box?
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Well, heck, to make that work, anything could go in the box.
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There's too many choices for what could go in the box.
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Oh, you could put a 5 in here, 5 times 0 is 0, 10 times 0, negative 1 times 0, 1 half
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times 0, they're all going to give you 0.
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So 0 divided by 0 is also undefined.
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Okay.
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So that's the introduction to rational expressions.