Rational Equations
Summary
TLDRThis instructional video script focuses on solving rational equations, which are equations involving fractions with variables. The approach emphasized is to gather all fractions on one side with a common denominator before proceeding with the solution. The script explains the importance of avoiding division by zero and provides step-by-step examples to illustrate the process of finding a common denominator and solving for the variable. It also highlights the significance of excluding values that would result in a zero denominator. The examples gradually increase in complexity, demonstrating how to handle different scenarios, including when denominators are factored or when there are no additions or subtractions involved. The goal is not only to solve the equations but also to prepare for tackling rational inequalities and graphing rational functions.
Takeaways
- 🧩 Rational equations involve fractions with variables, also known as rational expressions.
- ❌ Division by zero is undefined in mathematics, which is a crucial concept when solving rational equations.
- 🔍 If the numerator of a fraction is zero and the denominator is not zero, the expression evaluates to zero.
- 🚫 Exclude values that would make the denominator zero, as these cannot be solutions to the equation.
- 📚 Finding a common denominator is essential for combining fractions and solving rational equations.
- 📉 Rational equations can be solved by moving all fractions to one side of the equation and setting the sum equal to zero.
- 🔢 Factoring can simplify the process of finding a common denominator, especially when denominators can be factored.
- 📌 Multiplying the numerator and denominator of a fraction by the same non-zero expression does not change its value.
- 📝 When subtracting fractions, ensure that the entire numerator of the fraction being subtracted is accounted for.
- 🔑 Solutions to rational equations must not result in division by zero, so check the solutions against the excluded values.
- 📉 In cases where the solutions do not satisfy the original equation, it indicates there is no solution.
Q & A
What is a rational equation?
-A rational equation is an equation that contains rational expressions, which are fractions with variables in the numerator and/or denominator.
Why should we avoid dividing by zero when solving rational equations?
-Division by zero is undefined in mathematics. If the denominator of a rational expression is zero, the expression itself becomes undefined, which means we cannot have a solution that results in division by zero.
How does the value of zero affect the numerator and denominator of a fraction?
-If the numerator of a fraction is zero, the entire expression evaluates to zero, regardless of the denominator (as long as it's not zero). However, if the denominator is zero, the expression is undefined.
What is the purpose of finding a common denominator when solving rational equations?
-Finding a common denominator allows us to combine all the fractions in the equation into a single fraction, which simplifies the process of solving the equation by making it easier to isolate the variable.
What technique does the video suggest for solving rational equations, and why?
-The video suggests getting all the fractions together on one side with a common denominator. This approach is used because it is also applicable to solving rational inequalities and graphing rational functions, making it a versatile technique.
What are the excluded values for the potential solution in the first example equation x - 4 / (2ar * 3x + 1) over (2x to the 3 power * 5x - 4) equals 0?
-The excluded values are x = 0 and x = 4/5 because these values would make the denominator equal to zero, which is not allowed.
How do we solve the equation 5/x + 1 = 3x/(x + 1)?
-We get all the fractions on one side by subtracting 3x/(x + 1) from both sides, combine the fractions since they have a common denominator, and then solve for x by setting the numerator equal to zero and ensuring the denominator is not zero.
What is the solution to the equation 6/x = -2/(x - 4)?
-The solution is x = 3, as it is the value that makes the numerator zero without making the denominator zero.
How does the video approach solving rational inequalities?
-The video mentions that the same technique used for solving rational equations will be applied to rational inequalities, although it does not go into detail on how this is done.
What is a shortcut mentioned in the video for finding a common denominator?
-A shortcut mentioned is to factor the denominators that can be factored, which can simplify the process of finding a common denominator.
What is the solution to the equation 6/x = 11/(3x + 8)?
-The solution is x = 7/24, after finding a common denominator and solving for x by setting the numerator equal to zero.
Outlines
📚 Introduction to Solving Rational Equations
This paragraph introduces the concept of solving rational equations, which are equations containing fractions with variables, known as rational expressions. The approach discussed involves moving all fractions to one side of the equation with a common denominator. The instructor emphasizes not to multiply both sides by the least common denominator to cancel out the denominators, a method that might be quicker but not the focus of this video. The importance of understanding that division by zero is undefined and zero divided by any non-zero number is zero is highlighted, as these are critical concepts in solving rational equations. The first example provided demonstrates how to solve a basic rational equation by setting the numerator to zero while ensuring the denominator is not zero.
🧩 Simplifying Rational Equations with Common Denominators
The second paragraph delves into solving more complex rational equations by combining fractions with common denominators. The process involves moving all terms to one side of the equation and then combining them into a single fraction. The example given shows how to subtract fractions with the same denominator and solve for the variable by setting the numerator to zero. The importance of excluding values that would result in a zero denominator is reiterated, as these values are not valid solutions. The paragraph also covers how to handle equations with different denominators by multiplying the numerator and denominator by appropriate factors to achieve a common denominator.
🔍 Advanced Techniques for Rational Equations with Multiple Expressions
This paragraph presents a more advanced example involving three rational expressions. The solution process includes finding a common denominator by factoring the denominators where possible, which simplifies the task. The instructor demonstrates how to multiply each term's numerator and denominator by the necessary factors to match the common denominator. After combining all terms over the common denominator, the numerator is simplified, and the equation is solved by setting the numerator equal to zero and solving for the variable. The paragraph also discusses how to handle excluded values that would result in division by zero.
📘 Special Cases in Solving Rational Equations
The final paragraph addresses special cases in solving rational equations, such as when there are no additions or subtractions in the denominators, making the process of finding a common denominator straightforward. It also covers how to handle whole numbers by converting them into fractions and then adjusting the equation to achieve a common denominator. The paragraph concludes with an example that demonstrates these concepts, resulting in a solution for the variable. The instructor advises students to write out all steps until they are comfortable with the process before attempting to simplify steps, ensuring a solid understanding of the material.
Mindmap
Keywords
💡Rational Equations
💡Common Denominator
💡Undefined
💡Numerator
💡Denominator
💡Factored
💡Excluded Values
💡Rational Inequalities
💡Rational Functions
💡Least Common Denominator (LCD)
Highlights
Introduction to solving rational equations with a focus on using common denominators.
Explanation of rational expressions as fractions with variables.
Approach to solving rational equations without multiplying by the least common denominator.
Importance of avoiding division by zero in rational expressions.
Zero numerator results in an expression evaluating to zero, unless the denominator is zero.
Guide on finding a common denominator for fractions.
First example solving a basic rational equation with zero on one side.
Exclusion of values that make the denominator zero from potential solutions.
Identification of valid solutions by setting the numerator to zero.
Second example involving combining fractions with the same denominator.
Third example with different denominators requiring adjustment for a common denominator.
Technique of multiplying by one to adjust fractions for a common denominator.
Fourth example with rational expressions and an inequality.
Method for solving rational equations with three rational expressions.
Factoring denominators to find a common denominator.
Handling of an equation with no addition or subtraction in the denominators.
Conversion of whole numbers into fractions to facilitate common denominators.
Transcripts
today we're going to look at solving
rational equations meaning equations
with rational
expressions rational expressions are
just fractions but they have variables
in them they have polinomial and the
approach we're going to take here is
getting all the fractions together on
one side with a common denominator and
going from there the reason we're going
to do that is because we're going to use
the same technique to solve rational
inequalities
and we're going to use similar
techniques when we graph rational
functions so even though there might be
a faster way to solve rational equations
and I'll mention it uh you multiply both
sides by the least common denominator to
cancel all the denominators that's not
how I'll be approaching it in this video
so let's go ahead and get
started so again a rational equation has
fractions and the in mathematics means
fraction because it has the word
ratio a couple things you need to know
about fractions is if you
got anything divided by
zero then this is undefined you cannot
divide by zero there's no number that
you get if you try to do this but if you
divide Zero by something as long as it's
something not equal to zero well that's
always zero so if your numerator is zero
the the expression evaluates to
zero if your denominator is zero the
expression is undefined that's going to
be key as we go through solving these
equations so the other thing you need to
know is how to get a common denominator
so if you're not as familiar with that
you might want to slow down there in
that part of the video in the examples
and and see if you can catch on so let's
do our first example let's try to solve
this equation
how about x -
4^
2ar * 3x +
1 over
2x to the 3 power * 5x - 4 equal
0 now our first example here is going to
be pretty basic and just hit home at the
concepts we were just talking about the
idea here here is the right side's
already zero that's good for us and the
left side all we have to do to solve
this equation is figure out what makes
the numerator equals
zero and doesn't make the denominator
equal zero so I'm going to make a note
here in this step I don't want my
denominator to be zero and the two
values that would do that is if x were
equal to
zero or if five so we don't want X to be
zero and we don't want 5x - 4 to be zero
you can check that if either one of
those are zero and you multiply this
denominator out you'll get
zero so X can't be zero and 5x - 4 can't
be zero that means that X can't be if
you add four and divide by five x also
can't be four fths so these are these
are values we want to exclude from our
potential solution we don't want those
however if we want to figure out what
makes the expression equal to zero we
look at the numerator right we want the
numerator to be zero well in that case x
- 4 could be 0er or 3x + 1 could be
zero now the reason this is so easy is
because we have zero on one side and
this side everything's factored for us
that's super
nice having it factored makes it simple
to identify what makes it equal zero so
X could be 4 or X could be
1/3 and those are both valid Solutions
because they're not either one of our
excluded values from above so that's a
pretty basic one let's try something a
little more
involved let's try to solve 5/x + 1
equal uh how about 3x overx +
1 so this is set up a bit differently
than the last one what we want to do is
get all the fractions together on one
side so um we could subtract the 3x
overx plus one fraction over or we could
subtract the 5 over X+ one fraction over
it doesn't really matter um let's
subtract the 3x over X+1
fraction so 5 overx + 1 minus 3x overx +
1 equal 0er we like the zero over here
because that means all of our terms are
together on the left side we just have
to combine these now these two fractions
the good news is they have the same
denominator so all we have to do is put
them together as one fraction over that
common denominator and subtract the two
numerators 5 -
3x and now this is set up like our last
example a fraction equals z so we can
see that our denominator we don't want
that to be zero so X can't
be1 our numerator equals 0o will give us
our solution so let's subtract five and
divide by
-3 and negative divided negative is
positive and 5/3 is our solution because
it is not our excluded
value so that not not too bad right they
had the same denominator so that made it
pretty simple let's try
another let's try something like uh how
about 6
overx = uh -2 over x -
4 so again we want all of our fractions
on the same side so I'm going to add the
fraction from the right over to the left
to give us 6X + 2x - 4 =
Z and now the two fractions on the left
side do not have the same denominator so
this is what we're going to do I'm going
to take my 6 overx fraction and I'm
going to multiply the top and the Bottom
by X+ 4 or Sorry xus 4 you'll see why
shortly one thing I want to note is by
multiplying by x - 4 over x - 4 that's
the same thing as multiplying by one
right cuz x - 4 x - 4 is just 1 and we
know that multiplying by one doesn't
change the value of what we have so this
is perfectly fine to do you're allowed
to multiply by
one then we take our 2x - 4 fraction and
we want the denominators to match well
so what I'm going to do is multiply the
top and the bottom of this fraction by
X and we can see that the denominators
now are x * x - 4 in both cases
so now what we get is we get we're going
to put these together over one
denominator x * x - 4 and on top I'll go
ahead and combine like terms so I get 6x
-
24 +
2x that equals zero and then let's just
uh combine like terms on top we get 8x -
24 over x * x - 4
equal
0 well we can see from our denominator
that X can't be 0 or
4 and in our
numerator if we want our numerator to be
zero then add 24 and divide by 8 we can
see that X is three so three is the
solution because it's not either of our
excluded
values so again the key to getting a
common denominator is multiplying top
and the Bottom by something so that the
denominators in all the fractions match
and that that can take some
practice okay let's try another one how
about 5 overx - 2 is greater than sorry
is equal to 3 over x -
3 so let's get our fractions on the same
side by
subtracting 3 overx - 3
then we need a common denominator so I'm
going to take a little bit of a shortcut
here I'm going to go ahead and multiply
the first fraction on top and bottom by
xus
3 I'm just going to put it in this step
but I'll use a different
color and over here I'm going multiply
top and bottom by xus
2 and so now our denominators match so
we can put both fractions
together into to a single fraction with
that common
denominator and on top I'm going to get
5x -
15 and this negative here distributes
with the three because I'm subtracting
this entire numerator so I get -3x + 2
so 5x -3x is
2x minus the 15 + the 6 is -
99 and our excluded values are 3 and two
and our numerator is zero if x =
9es okay so hopefully you're getting the
idea we can start skipping some some of
those smaller steps but I always
encourage students write as many steps
as it
needs a whole lot until you really get
the idea down before you start to skip
steps
okay now let's try one that's got three
rational
expressions X over x -
4 + 6 over x -
9 = 7 x - 48 over x^2 -
13x +
36 now we know we're going to try to put
all these fractions together with a
common denominator and one thing we
haven't seen yet is when trying to find
a common denominator it can be handy if
you factor the denominators that can be
factored so x^2 - 13x + 36 you could use
the AC method or whatever method you
like but it factors as x - 9 * x -
4 that's going to be handy uh when we
try to find a common denominator so I'm
going to subtract this fraction over x
overx - 4 and I can already see that
this will be my common denominator so
what I'm going to do is go ahead and
write
in on this fraction I'm multiply top and
bottom by x -
9 you know write more steps if you need
to but um hopefully this makes sense my
next fraction 6 overx - 9 I'll have to
multiply top and bottom by x -4
and then I'm going to subtract this
entire fraction over when I when I do
that I could put minus this whole
fraction actually let's go ah and do
that
minus 7 x - 48 over x - 9 * x
-4 equals 0 and again I just want to
remind you when you got a minus in front
of a fraction that minus applies to the
entire numerator
Okay the reason I say that is because in
the next step we're going to put all
these numerators together over the
common denominator x - 9 * x -
4 and so I get X squar that's the only X
squ term I'm going to have on top so
I'll write that down then I get -
9x and here I get plus 6X so that's -
3x -
7x is - 10 x okay
so we had - 99x + 6 x is -3 - 7 is -10
now we've used all the terms here we get
a -4 out of this guy and we get a
positive 48 out of that guy so we get
plus
24 all right so now our we've put
everything together our excluded values
are X can't be 9 or
4 and if we want to if we want the
numerator to be Z Z we we need to factor
this or use the quadratic formula well
luckily it factors very nicely as x - 6
* x -
4 and we can see we get two solutions x
- 6 could be zero meaning x = 6 or x - 4
could be 0 which means x =
4 but we already said X can't be four so
we throw that guy out and we only get
xal 6 if you plug 6 in for x to the
original equation you should get a true
statement and you can see if you try to
plug four in well it would cause you to
try to divide by zero which you can't
even
do there are cases where none of your
Solutions work and when that happens you
just say no
solution let's do one last example
solve the
equation 6 /x =
11 over
3x +
8 and this one's a little more a
little different than the others there's
no adding or subtracting in the
denominators that actually makes this
one a little bit easier but since it
doesn't look the same as the others it
might might throw you off a little bit
what I'm going to do this time is I'm
going to subtract the 6 overx to the
right because I already have two of my
terms over
here so I've got 11 over
3x and I've got this eight what I like
to do with an eight is make it into a
fraction so I'll call it plus 8 over 1
and then finally - 6
overx and I want the all three
denominators here to match so 3x is what
I can turn all of them into for example
on this fraction I can multiply the top
and the Bottom by a
three on this fraction I can multiply
top and bottom by
3x because 1 * 3x is
3x so we get
11 +
24x minus 18 I could have probably
combined the like terms there before and
that's all over the common denominator
of
3x and I can see at this step um my
excluded value is zero I don't want X to
be zero because I would make my
denominator zero and then on
top to make the fraction equals z to
solve the equation I'd have
24x 11 - 18 is
-7 and then I can add 7 and divide by
24 to get X by
itself and the solution we get is X =
724 so a couple new things here um when
you don't have addition or subtraction
in your denominators it's just
multiplication so no problem in building
a common denominator and when you have
an integer or a whole number turn it
into a fraction put it over one and then
you can multiply top and bottom by
whatever you need to get the same
denominator
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