Adding and Subtracting Rational Expressions
Find the Excluded Values
Objective
Alright, so what about our lesson today?
Well, to be honest, we still need to review before we actually get into it.
But what do we need to review?
Well, a little more 3rd grade math, which is adding fractions!
So, how do we add fractions?
Well, if we have the same denominator, we just add the numerators together.
Like so:
But what if it isn’t easy?
What if we don’t have the same denominators?
Well, that’s when we need to find the least common multiple!
Finding the Least Common Multiple
So here’s how we add fractions.
Let’s start off with an example:
We can’t just add these right?
We have no idea how to add them.
However, what if we changed them to something we can add?
But, we can’t change the equation right?
So, how do we change the numbers, without changing the amount?
Well, we can try to multiply the fractions by 1, but what kind of one?
This is where the Least Common Multiple comes in.
So, to start, let’s see which multiple the two denominators share.
To find it, let’s list out the multiples of 2:
2 4 6 8 10 12 14 16 18 20
Now, let’s list out the multiples of 3:
3 6 9 12 15 18 21 24 27 30
Now, which number do they both share?
6
6
Now we have to find out how to make 3 into 6.
We multiply it by 2!
Now we have to find out how to make 2 into 6.
We multiply it by 3!
THE CHEATING WAY
So, there is actually another way of creating a common denominator instead of finding the least common multiple
But, it’s not the best way.
However, it will work every single time.
We multiply by opposite denominators.
Sounds weird, but let’s try it:
So, what we do, is look at:
( )
= 35
Now, we multiply each side by the opposite number to get 35:
Again, this works every time, but it can take more work.
SO…….WHAT DOES THIS HAVE TO DO WITH ALGEBRA 2?
Again, this is not to insult your intelligence.
The truth is, you probably haven’t needed to find the least common denominator for years.
So, a little refresher on how to do it isn’t that bad of a thing.
But why do we need it?
Well, mainly because we are going to be adding fractions…….
With variables.
So, for example:
EXAMPLE 1
Add the following:
Alright, so this one seems a little weirder than what we’ve done before.
However, we can still do this.
First things first, we can’t add them together the way they are.
Mainly because we need a least common denominator.
Well, what we have so far is just x and 2.
Is there a possible way for us to maybe use the cheat way to find the LCD?
Well sure!
So, to use the cheat way, let’s first multiply the left side by 2/2:
And the right side by x/x:
Now, if we multiply the left side, we get:
And if we multiply the right side, we get:
Now we can add them together!
So, adding them together we get:
THAT WASN’T SO BAD, WAS IT?
Pretty easy right?
Well, it does get a little more complicated.
However, we’re usually using the cheat method to find the LCD.
So, let’s try a few more to make sure you got it.
EXAMPLE 2:
Add the following:
Again, we can’t seem to add these together the way they are.
Mainly because we need a least common denominator.
Well, what we have so far is x + 1 and x.
Is there a possible way for us to maybe use the cheat way to find the LCD?
Well sure!
However, it gets a little complicated.
First, we need to multiply the left side by x/x:
And the right side by x+1/x+1:
Now, if we multiply the left side, we get:
And if we multiply the right side, we get:
Now we can add them together!
So, adding them together we get:
And that’s our answer!
Now let’s try a little harder one:
Example 3:
Add the following:
Again, we can’t seem to add these together the way they are.
Mainly because we need a least common denominator.
Well, what we have so far is x + 1 and x - 2.
Is there a possible way for us to maybe use the cheat way to find the LCD?
Well sure!
However, it gets a little complicated.
First, we need to multiply the left side by x - 2/x - 2:
And the right side by x + 1/x + 1:
Now, if we multiply the left side, we get:
And if we multiply the right side, we get:
Now we can add them together!
So, adding them together we get:
And that’s our answer!
So, now, what if there is something harder?
But we don’t really want to multiply by a binomial?
Something like:
Example 4:
Add the following:
Again, we can’t seem to add these together the way they are.
Mainly because we need a least common denominator.
However, do we really want to multiply this monster by anything to make it even bigger?
Or, maybe there’s a way to make it smaller?
Well, can we factor this?
In fact we can!
When we factor the left side, we see that:
And when we factor the right side, we see:
Now that we see we have some common factors, we can see how easy we can get a LCD.
All we need to do is multiply the right side by -4/-4, so we can get:�12(x + 13) on both sides.
So:
And now we add, and we get:
Example 5:
Add the following:
Again, we can’t seem to add these together the way they are.
Mainly because we need a least common denominator.
However, do we really want to multiply this monster by anything to make it even bigger?
Or, maybe there’s a way to make it smaller?
Well, can we factor this?
In fact we can!
When we factor the left side, we see that:
And when we factor the right side, we see:
Now that we see we have some common factors, we can simplify some.
And how much easier was that?
So, let’s continue doing this:
And what we’re left with is:
Adding what’s left, we get:
So then what are excluded values?
Exactly how they sound.
Values of x, that need to be excluded.
So how does this work with what we’ve learned?
Well, it’s basically what values can x not equal in the expression we have in front of us.
It seems really complicated at first, but this is another time where it’s easier to show than explain.
EXAMPLE 1
Add the following:
Alright, so this one seems a little weirder than what we’ve done before.
However, we can still do this.
First things first, we can’t add them together the way they are.
Mainly because we need a least common denominator.
Well, what we have so far is just x and 2.
Is there a possible way for us to maybe use the cheat way to find the LCD?
Well sure!
So, to use the cheat way, let’s first multiply the left side by 2/2:
And the right side by x/x:
Now, if we multiply the left side, we get:
And if we multiply the right side, we get:
Now we can add them together!
So, adding them together we get:
Now, what value can x not be?
Well, to find it, we need to set the denominator to 0.
So:
And finally, the excluded value is 0.
SO THAT’S ALL?
That’s it!
So, all we need to do is simplify the expression.
Then find out what x cannot equal.
So, let’s look at some more examples:
EXAMPLE 2:
Add the following:
Again, we can’t seem to add these together the way they are.
Mainly because we need a least common denominator.
Well, what we have so far is x + 1 and x.
Is there a possible way for us to maybe use the cheat way to find the LCD?
Well sure!
However, it gets a little complicated.
First, we need to multiply the left side by x/x:
And the right side by x+1/x+1:
Now, if we multiply the left side, we get:
And if we multiply the right side, we get:
Now we can add them together!
So, adding them together we get:
Now, what value can x not be?
Well, again, to find it, we need to set the denominator to 0.
So:
And finally, the excluded value is -1 and 0.
EXAMPLE 3:
Add the following:
Again, we can’t seem to add these together the way they are.
Mainly because we need a least common denominator.
Well, what we have so far is x + 1 and x - 2.
Is there a possible way for us to maybe use the cheat way to find the LCD?
Well sure!
However, it gets a little complicated.
First, we need to multiply the left side by x - 2/x - 2:
And the right side by x + 1/x + 1:
Now, if we multiply the left side, we get:
And if we multiply the right side, we get:
Now we can add them together!
So, adding them together we get:
Now, what value can x not be?
Well, again, to find it, we need to set the denominator to 0.
So:
But wait, we’ve already seen the factors of this.
It’s:
And finally, the excluded value is -1 and 2.
EXAMPLE 4:
Add the following:
Again, we can’t seem to add these together the way they are.
Mainly because we need a least common denominator.
However, do we really want to multiply this monster by anything to make it even bigger?
Or, maybe there’s a way to make it smaller?
Well, can we factor this?
In fact we can!
When we factor the left side, we see that:
And when we factor the right side, we see:
Now that we see we have some common factors, we can see how easy we can get a LCD.
All we need to do is multiply the right side by -4/-4, so we can get:�12(x + 13) on both sides.
So:
And now we add, and we get:
Now, what value can x not be?
Well, again, to find it, we need to set the denominator to 0.
So:
And finally, the excluded value is -13.
EXAMPLE 5:
Add the following:
Again, we can’t seem to add these together the way they are.
Mainly because we need a least common denominator.
However, do we really want to multiply this monster by anything to make it even bigger?
Or, maybe there’s a way to make it smaller?
Well, can we factor this?
In fact we can!
When we factor the left side, we see that:
And when we factor the right side, we see:
Now that we see we have some common factors, we can simplify some.
And what we’re left with is:
Adding what’s left, we get:
So what are the excluded values?
Well, there are none!
There is no x in the denominator, so we don’t need to worry about it!