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