>> This is part one of solving rational equations.
Now, a rational equation is an equation
that has fractions in it.
And, remember, a fraction is undefined
if zero is in the denominator.
So we're going to have to keep that in mind
as we solve these types of equations.
Keep in mind that we're solving equations.
We're trying to find out what number you could
in for the variable to make the equation true.
There are several methods.
I'm going to go over the most common one to begin with.
So let's say we wanted to solve x over 6 plus 4x
over 3 equals x over 18.
One thing you might notice is
that there is no zero in the denominator.
So that's a good sign.
OK. So in previous videos, when we were first solving equations,
and we had some fractions, what we did was we multiplied
by the least common denominator to clear the denominators.
That means get rid of all the fractions.
So we're going to look at all the denominators
and determine the least common denominator, and I've gone
over ahead to get the least common denominator
in other videos, and for this problem,
the least common denominator is going to be 18, because 6, 3,
and 18 all go into 18.
So what we're going to do is multiply,
since this is an equation, we can multiply both sides
of the equation by the least common denominator.
So I'm going to multiply each term on each side
of the equation by 18.
So this is what we get.
We do that.
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>> And the reason we're doing this is
if there is a denominator, and we're multiplying that term
by the least common denominator, it will cancel into it.
So we'll be able to clear the fractions.
So now we're going to do that.
So the first term, 18 times x over 6,
we can do 6 into 18, and that's 3.
Second term, we've got 18 times 4x over 3.
The 3 goes into the 18, and you get 6, and on the other side
of the equals sign, 18's cancel, and you get 1.
You can write 1 or leave it, and, of course,
all of the denominators cancel to 1,
and that's exactly what we wanted.
We didn't want to have any denominators
when we did, do this.
So what do we have?
Three times x, 3x plus 6 times 4x is 24x equals,
and this will just give you x.
and now what we have is an equation that has no fractions.
So we have 27x equals x. Alright.
Now, remember, we're solving an equation with just one variable.
So what we want to do is put the variables on one side
and constants on the other.
Lot of people get stuck right here.
Try to divide by x, and you can't do that.
You have to subtract x from both sides.
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>> So we're going to subtract one x from both sides.
And that gives us 26x equals 0,
and we can divide both sides by 26.
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>> And we finally found a solution.
X equals 0.
Now, remember, it's extremely important to check your answers,
especially when you're working with rational equations.
So let's see if x equals 0 really is the correct solution.
So here's my original up here at the top.
So if we put in zero.
Let's just check it.
If we put in 0 for x, we'll get 0 over 6 plus 4 times 0
over 3 is equal to 0 over 18,
and I hope you can convince yourself that's going to be true
because you'll end up with 0 plus 0 equals 0.
So, actually, it does check.
So the solution to this problem is 0.
And when you check, make sure there's no way you end
up with a 0 in the denominator, which, of course,
they were all numbers to begin with, 6, 3, and 18.
So that didn't happen.
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>> Here's the same problem again.
I'm going to show you another method some people use.
What they do is they use the jealous method
to determine the least common denominator
of all the fractions, and this is only going to work
when you have an equation, and we're going to write all
of the fractions with that least common denominator.
So the 6 looks over and decides it needs a 3,
and actually 6 times 3 is the least common denominator.
So the first fraction is multiplied by 3 over 3.
The second fraction, then, to get 18, you'd have to multiply
by 6 over 6, and the third fraction already has 18.
Now, here's the idea.
So you really have 3x over 18 plus 24x
over 18 equals x over 18.
If all of the denominators are the same,
then the numerator should be equal.
And another way of thinking about that is
if you multiply both sides by 18,
that would actually eliminate all of the denominators.
So if you make all the denominators the same,
then you can pretty much ignore the denominators
because when you multiply by 18, they would all cancel.
So I get 3x plus 24x equals x. If you want
to see why that's really true algebraically,
you're really just multiplying everything by 18 over 1.
So all of the denominators cancel with this 18.
OK. And then from here, everything else is the same.
You would have 27 equals x, 26x equals 0, x equals 0,
and when you checked it, of course,
it would also be the same.
So this is an alternative way to do it.
Where people get confused is trying to do
that when it's not an equation.
So you can only eliminate, be careful,
you can only eliminate denominators -
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>> By clearing fractions -
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>> If you're solving equations.
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>> OK. So if you are just adding fractions, you're still going
to have the denominator.
You can't just make the denominator disappear.
Alright. Let's go onto a new problem.
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>> Here's another problem.
Three minus 12 over x equals 7.
First thing I want you to do is note that there's only one thing
in the denominator, an x, and it cannot be 0.
So remember, before you even start the problem,
look at the denominator, and make sure
that you write whatever's in the denominator can't be 0.
I didn't do that when it was a number
like 3 because, obviously, 3's not 0.
But do keep in mind, if you get x equals 0,
it's not really going to be a solution.
OK. There's a couple of ways to begin this problem.
If you want, you can subtract, subtract 3 from both sides
to get, you know, the constants on the same side,
or you can just start by multiplying both sides
by the least common denominator, x,
or you can make all the denominators x. I'm going
to begin by just going ahead and multiplying both sides
by the least common denominator of x. So I'm going
to multiply each term by the least common denominator,
and the least common denominator's x.
So we have 3 times x, right.
Minus, they're going to have 12 over x times x because you have
to multiply each term by the least common denominator
and then seven times x. So, remember, why did I multiply it
by x. Because the least common denominator,
there was only one denominator
so the least common denominator is x. Alright.
So let's see what we get.
We have 3 times x is x. Now, for this term,
that was the only one that had a fraction.
Those are going to cancel.
So I have minus 12 equals 7x.
Now we get an equation without fractions,
which is easier to solve.
So I'm going to subtract 3x from both sides.
Be careful of my signs.
This gives me negative 12 equals 4x, and if I divide both sides
by 4, this will [inaudible] all review for you this part.
Once you get rid of the fractions, it's easy.
So we're going to get x equals negative 3.
Now, notice up here at the beginning, I say, said,
said x cannot be 0, and I didn't get 0.
So that's hopefully the correct answer.
And we would just want to check that in the original.
So let's check it.
In the original, we had 3 minus 12 over x equals 7,
and we're going to put a negative 3 for x.
So I have 3 minus 12 times negative 3 on the left-hand side
of the equation, and so that becomes 3 minus negative 4,
which will be plus 4, which is 7.
On the right side, it's already simplified.
We've got 7.
So x equals negative 3 is the correct solution, and if I put
that [inaudible] braces, we get negative 3.
I don't have time on this video
to do it the other two ways I mentioned, but you could do this
by subtracting 3 from both sides first, and then multiplying by x
from the original problem, or you could do it
by making all the denom, all the fractions
or all the terms having a denominator of x
and then multiplying all sides by x,
which means you'll have the same equation, 3x minus 12 equals 7x.
We'll do a lot more problems on part two.
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