Up next in 10
How To Find The LCD & Simplify More Complex Rational Expressions
263 views
Feb 20, 2025
Today we worked on using the lowest common denominator of a rational expression to simplify it. During our lessons , we also learned how to solve equations involving fractions. For more lecture visit http://tayibs.com/get-math-help/
View Video Transcript
0:00
so a couple of weeks ago we worked on
0:01
like simplifying fractions and rational
0:03
expressions and now it's getting a
0:05
little bit more complicated and today
0:08
what we have is this expression here as
0:12
a fraction and on top of another
0:15
fraction and we have to learn to
0:16
simplify them now the first thing we
0:19
want to do is find the lowest common
0:20
denominator and then when we find the
0:23
lowest common denominator we can use the
0:25
quotient rule and just flip the bottom
0:27
one and factor it right so let's let's
0:30
write down the equation so is 1 over X
0:34
plus 2 minus 2 okay
0:39
the whole thing over 4 over X plus 2
0:42
plus 2 I'm going to split this into two
0:46
I'm gonna take the numerator and I'm
0:48
gonna take the denominator apart and I'm
0:51
gonna work on this together right so the
0:53
first thing we have is 1 over X plus 2
1:00
minus 2 and I want to find the lowest
1:02
common denominator all right
1:04
so this fraction is like over 1 right
1:08
yeah so the thing is this what is very
1:10
simple this fraction the X plus 2 I want
1:12
to make this college this denominator is
1:14
the same right so it's gonna be since
1:17
they don't have anything in common so
1:19
the lowest common denominator LCD
1:21
something why is with 1 times X plus 2
1:25
right I'm gonna put in parentheses
1:27
because you wanna you want to make sure
1:29
these two fractions have the same
1:30
denominator you cannot add any fractions
1:33
or subtract any fraction if they don't
1:35
have the same denominator okay so now
1:38
here's what I'm gonna do here I'm gonna
1:41
transform this one into X plus 2 and I'm
1:43
gonna multiply this one by one
1:44
right which is doesn't do much to it
1:47
they stay the same and I'm gonna
1:48
multiply this one by X plus 2 I'm gonna
1:50
put a parenthesis to make it easier but
1:53
here's the thing when I do something to
1:54
the denominator I have to do the same
1:56
thing on the numerator - the numerator
1:59
right so there's gonna be X plus 2 so we
2:02
end up having 1 over X plus 2 minus 2 X
2:11
plus 2
2:12
over X plus 2 and then we can put these
2:15
two fractions together since they are
2:16
the same denominator we can just put
2:18
them into one single denominator which
2:20
is X plus 2 or once you get them all
2:22
like all the same yes put him on you
2:25
don't have to if it's both the same
2:26
thing you just pull on one first that's
2:30
the whole point is to put him together
2:31
right over 2 X plus 2 right and then
2:35
we're gonna do this with an X 1 with
2:37
this expression here alright same thing
2:41
so we're just gonna jump to the you know
2:44
burn all those steps and we're not gonna
2:46
wish our time because we already know
2:47
what to do there so the denominator is
2:49
gonna be Express to steal right and I'm
2:51
gonna multiply the 2 times X plus 2
2:54
right because we did the same step here
2:56
it's the same denominator and he's like
3:00
a similar fraction so we're not gonna
3:01
waste time trying to elaborate on every
3:03
single piece all right now we can put
3:05
them back together so now we have the
3:07
first fraction which is 1 minus 2x plus
3:13
2 over X plus 2 right over 4 plus 2x
3:22
plus 2 over X plus 2x plus 2 correct it
3:32
looks but it's not complicated so I'm
3:33
gonna take this expression and this is
3:35
the thing I'm going to keep this the
3:37
same and I'm gonna flip this one right
3:38
because that's how your quotient rule
3:40
I'm gonna recall the question will a
3:43
over B divided by C over D is equal to a
3:50
times D right over a times e over BC
3:56
okay so this is the same process here
3:59
I'm gonna be times B times C is being
4:01
the same alright so we're gonna do the
4:05
same thing here so we're gonna take 1
4:08
minus 2 X plus 2 right and I know it
4:13
looks really complicated but it's not
4:14
okay you're gonna see how I wanted to
4:17
explain something to you first and then
4:19
we're gonna multiply it by the
4:20
reciprocal right which is X plus 2
4:24
over 4 right + 2 X plus 2 all right and
4:32
then look what can I do with these two
4:33
here
4:34
cross and cross them up right so I'm
4:36
left with 1 minus 2x plus 2 over 4 plus
4:42
2x plus 2 now the last step would be
4:45
sorry
4:46
look at Ben so I was fresh nothing too
4:48
much the last thing would just be to
4:51
split this to work on it at the top one
4:54
right we're gonna multiply the h1 101
4:57
minus 2 X minus 2 right / 4 + 2 X + 4
5:03
okay this is crazy to read this over 1
5:08
minus 2 X but my - follow me I can't
5:12
multiply guys forgive me for that now
5:15
it's gonna be negative 3 minus 2x over 8
5:22
plus 2 X we combining like terms
5:25
okay and that should be the final result
5:27
only here goes you just you just can the
5:31
second problem in this case we have a
5:33
second-degree expression and we have to
5:39
turn around simplify this so I actually
5:41
wrote out the steps and I'm gonna just
5:44
go through them one at a time so the
5:46
first thing we want to do is just single
5:48
out on the numerator and that's what we
5:50
need over here and then we want to find
5:53
the lowest common denominator now to
5:54
find the lowest common denominator we
5:56
have to factor on this expression
5:59
alright expression and I want two
6:02
numbers whose product is negative 6 and
6:04
Psalms 5 so we have 6 and 1 and since
6:07
this is negative and this is positive so
6:09
it's gonna be 6 and then X plus 6 and X
6:11
minus 1
6:12
so we split that and then we do the same
6:15
thing over here now we want to find the
6:16
LCD because this is X minus 1 they
6:18
already have X minus 1 in common so all
6:21
I need to add is X plus 6 which I did on
6:23
both sides like because if you do
6:25
something in the denominator you have to
6:27
do the same thing on the numerator and
6:29
then now we have the same expression
6:31
here now we put them into one single
6:33
fraction denominator okay and then the
6:36
next thing is we
6:37
do the same thing with our within
6:39
denominator which is right here since
6:41
they have the same exact denominators I
6:45
just went ahead and just put this on top
6:49
of that okay so I hope you understand
6:52
what I'm doing here and then the next
6:54
thing is just to flip this fraction
6:57
because we're using the quotient rule
6:59
and our division rule or whatever and
7:02
we're gonna just simplify the like terms
7:05
which are cut out and then we're gonna
7:07
do it use the distribution rule here and
7:09
we end up with this which is simplifying
7:12
combining like terms so you get up to
7:14
negative 5 minus 6 over 10 plus X 9 DX
7:17
hopefully you get what I'm saying
7:19
because I had to just write out except I
7:21
didn't wanna do them you know I was I
7:23
was working so this is clear enough all
7:26
right if you have any questions be sure
7:28
to just send me a email all right we
7:30
went from basically simplifying
7:34
fractions to not solving equations are
7:37
you sleepy so we're gonna be doing the
7:42
same thing here now before we solve it
7:43
we have to find a common denominator for
7:45
these two fractions okay now one is
7:48
people say that some fraction one is one
7:50
over one basically right so again one
7:54
over one I keep on breaking order spread
7:57
because you sharpen them too much 1 over
7:59
1 right 1 over 1 and the common
8:03
denominator will be n right now here's
8:05
the key a lot of people like to like
8:08
complicate things but what I do is this
8:10
I know that if I multiply every single
8:14
entry by n I'm gonna simplify this and
8:16
make the problem a lot easier so I'm
8:18
going to put a renters here I multiply
8:20
this by n and n because every single
8:23
entry here has can be if I if I was
8:28
trying to find a common denominator
8:29
right for each of this fraction ub n for
8:32
this one and for this one and i have to
8:34
turn this into 1 times n right so if I
8:37
do that I'll make my problem 7 because I
8:40
multiply everything by n guess what's
8:41
gonna happen this squad cancel this one
8:43
right it's gonna cancel this one and I'm
8:45
gonna be left with 3 and square
8:48
Plus 10 and - train right - N equals to
8:54
2 because if you multiply n times these
8:56
it gives you n right and then now you
9:00
can solve this problem cause it's a lot
9:01
easier to work with now you see what I'm
9:03
saying yeah so now what we have is we
9:05
can put all on one side so we're gonna
9:07
do 3n squared plus 10 and minus 10 minus
9:12
n minus 2 equals to 0 okay and that
9:17
gives you 3 n squared plus 10 N and then
9:22
negative 10 minus 2 is negative 12 but
9:25
we're gonna put it in order 10 n minus n
9:28
is 9 in so I should make it easier I
9:31
don't want me I'll see y'all come behind
9:32
ya combine like terms right minus 12 is
9:36
equal to 0 now look it's a lot easier
9:38
now they all have white in common and
9:40
and but at the same time is the second
9:43
degree right in black and Pilate 3
9:45
correct travellightly 300-watt n square
9:48
plus 3 n minus 4 equals to zero and I
9:53
can solve this because now I want two
9:56
numbers whose product is 4 negative 4
9:58
and some Street right 4 & 1 oh you just
10:02
turn it into a second factor Buffon
10:04
right so I've got my 3 here so then we
10:06
end and and she's a negative so it's
10:10
gonna be plus 4 minus 1 and all I have
10:13
to do is set each one of these equal to
10:16
0 solve for n I can let you do that
10:19
alright so hopefully it makes sense okay