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## Algebra (all content)

### Course: Algebra (all content) > Unit 13

Lesson 6: Solving rational equations- Rational equations intro
- Rational equations intro
- Equations with one rational expression (advanced)
- Rational equations (advanced)
- Equations with rational expressions
- Equations with rational expressions (example 2)
- Rational equations
- Equation with two rational expressions (old example)
- Equation with two rational expressions (old example 2)
- Equation with two rational expressions (old example 3)

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# Equation with two rational expressions (old example 2)

Sal solves the equation 5/(2x)-4/(3x)=7/18 by first finding the LCM (least common multiple) of 2x and 3x. Created by Sal Khan and Monterey Institute for Technology and Education.

## Want to join the conversation?

- How can I find out what x cannot be?(13 votes)
- It's a mathematical rule that you can't have a 0 in the denominator because any number divided by 0 is undefined.

In general that's the logic you have to follow, " is there any number in this equation that would make it undefined (and thus impossible to solve) ?

"if yes, what would x had to be to get that number?"

And that is your "X that can't be"

Example given in the previous video:

5/(x-1)= 3/(x+3)

You know that you can't have 0 in the denominator.

In the first fraction you have the denominator x-1. What would x had to be for the denominator be 0?

it had to be 1; 1-1=0. So X can't be 1.

By the same logic:

x+3

what number would make x+3=0.

-3 so -3+3=0

You can also rule out -3.

For this example you already know that X cannot be 1 or -3.(41 votes)

- At2:04the example is dividing x by x and 18 by 2.

I thought we were supposed to multiply the 18x by the denominator as we do in the

"Solving Rational Equations 1" example (can be seen at0:40onwards)?(5 votes)- He's basically doing just that. When you are multiplying 18x by 5 over 2x, the x can be "cancelled out", leaving a 2, and that 2 can also be cancelled out as well. Basically, cancelling out is just division, so he is using those interchangeably.

In other words, what you are supposed to do is multiply every term in both sides of the equation by the LCD (lowest common denominator) of the fractions, and that will always cancel out the denominators. (When you set it up as a fraction multiplication, you might notice that you'll be able to cancel factors out diagonally to simplify. That's where the cancelling out happens. )

It isn't specifically multiplying by the denominators, but it's actually dividing into the denominators, putting a one in the denominator afterwards, and taking the result and multiplying that by the numerator (You can think of it that way too). This always works because you are taking the LCD of all of the fractions in the rational equation, and the whole point in using the LCD is that it has factors that will divide into the factors of each of the denominators evenly.

Just don't forget to multiply every term in the equation, on both sides, by this LCD. That includes non-fractions on either side of the equation. That's why it's most practical to do this when your equation is mostly fractions, although you can do whatever you'd like to solve your equations.(5 votes)

- what happens if there is a variable in the numerator?

ex. x-3/(3x+2)=1/5(4 votes) - At2:06, 18 is divided by the denominator (2) and then the quotient (9) is multiplied by the numerator.

I'm pretty fuzzy on the logic behind this. If anyone can explain why this works I think I'd retain this a lot better than I am now.(3 votes)- The key for this problem (and many problems involving fractions) is to ELIMINATE the fractions:) If you can find a number into which all the denominators are divisible, all you need to do is multiply that number by each term, and then when you simply the fractions the denominators will cancel out.

The specific issue you seem to have is the order in which Sal is simplifying. It would be just as correct to multiply the 18x by 5 (the numerator) to get 90x, and then divide that by 2x (the denominator) to get 45. See? Same result. Sal did it in smaller steps to avoid having to multiply 18x by 5, which some people can't do in their heads. Remember the 18x is really 18x/1 so you are just multiplying fractions; Sal chose to do some cancellation/simplification before he multiplied. Did this make sense or did I make it worse?(4 votes)

- I want to know the answer and know how to check it. Here's my problem: 3/3x+4=2/5x-6(3 votes)
- Since you are adding fractions, the first thing you should do is convert everything to a common denominator. You didn't use parentheses so I can't tell if you mean

'3/(3x+4) = 2/(5x-6)' or '3/(3x) + 4 = 2/(5x) - 6'.

My guess is you meant '3/(3x+4) = 2/(5x-6)'. So multiple the left side by (5x-6)/(5x-6), and the right side by (3x+4)/(3x+4). You're really just multiplying both sides by 1, so this is perfectly valid. That gives you a numerator of 3*(5x-6) on the left side and 2*(3x+4) on the right side. The denominator on both sides is (3x+4)(5x-6). Now you only have to compare the numerators to solve for x.

3*(5x-6) = 2*(3x+4) ... 15x - 18 = 6x + 8 ... 9x = 26 ... x = 26/9. To check if this is correct, substitute the solution back into the original expression and verify that both sides are equal. Doing that I get 9/38 = 9/38.(2 votes)

- How do you solve a quadratic equation with multiple different denominators?(2 votes)
- Well first of how did you end up with a quadratic equation with multiple denominators? was a=1/2 or c=1/2? considering the denominator consists of 2ac?(3 votes)

- I was lost after the first 1 minute and 12 seconds. I'm in college remedial math, and I still don't understand equations.(3 votes)
- Its like saying 4=4. We know that is true... but there is a bunch of ways ways we can write that same thing....

ex. 2x2=4, 2+2=4,6-2=4 etc..... correct?

now, if I say, some number 'x' + 2 gives me 3, then we know that 'x' must be 1..... correct?

So writing that in a bit more mathematical fashion we can say

x + 2 = 3.

now if you go back to the 4=4 equation, if i subtract 1 from the right side, we would get 4=3... that's not true. but if i subtracted 1 from both sides, i get 3=3 which is true. so in general the equality is maintained as long as we do the same operation on both sides of the equation. now back to x + 2 = 3... subtract both sides by 2 you get x = 1..... its the same with division and multiplication.... if you have doubts mail me or add me as your coach... my email id is informjaka@gmail.com(2 votes)

- how do I solve the rational equation 3x=12+15/x(2 votes)
- It is essentially the same as any other equation. If a number does not have a denominator, it by definition technically has a denominator of one, so simply multiply all numbers by x, cancel out the denominator, then solve from there.(3 votes)

- 2a/3 + 1/5 = 7/6

How should I begin to solve this I'm confused.(2 votes)- Here what you got to do:

2a/3 = 7/6 - 1/5 ---> 20a/30 = 35/30 - 6/30 ---> 20a = 29 ---> a = 29/20

If you want the decimal form, you divide 29 by 20, then you will have:

1.45(2 votes)

- How do you solve x/3 -1/2 = x/4.(1 vote)
- Multiply the equation by the LCD which in this case is 12. Then isolate 𝑥.(2 votes)

## Video transcript

Solve the equation, 5 over
2x minus 4 over 3x is equal to 7 over 18. And they tell us that x can't to
be equal to 0, because that would make these two expressions
here undefined. Hopefully the answer here is not
0, and then this becomes-- this is kind of extra,
unnecessary information. So let's figure out
how to solve this. So a good place to start-- I
don't like having x's in my denominators. So let's multiply-- and in fact,
in general, I don't like having fractions in
my equations. So let's see if we can multiply
both sides of this equation by some things that
will get rid of the fractions. So let me just rewrite it
so we have some space. 5 over 2x, and then we have
minus 4 over 3x is equal to 7 over 18. Now, if we want to get rid of
the 2 in the denominator here, we could multiply
everything by 2. If we want to get rid of this 3
in the denominator, we could multiply everything by 3. If we want to get rid of this
18 in the denominator, we could multiply everything
by 18. And 18 also includes
a 2 and a 3. The prime factorization
of 18 is 2 times 9, which is 3 times 3. So when you're multiplying both
sides of the equation by 18, you're actually multiplying
it by a 2 and a 3 and another 3. So let's just multiply both
sides of this equation by 18. So I'll multiply this term
right here by 18. And then, this term
right here by 18. That'll get rid of all of
these numbers in the denominator. 18 divided by 3 is 6. 18 divided by 2 is 9. But we don't just have numbers
in the denominator, we also have these x's in
the denominator. So let's also multiply both
sides of the equation by x, so that we get rid of these. So we're essentially going to
multiply both sides of the equation by 18x. We're taking, essentially,
the least common multiple of 2x, 3x and 18. This is the smallest number that
is divisible by all three of these characters. Now, when we do that our
denominators will disappear. x divided by x is 1. 18 divided by 2 is 9. So this term becomes 9
times 5, which is 45. And then this term right here,
x divided by x is 1. 18 divided by 3 is 6. So you have 6 times 4 is 24,
but you have a subtraction sign here so minus 24 is equal
to-- let me do that in that yellow-- is equal to-- and then
you have this term, 7 over 18 times 18x. Well, the 18's cancel out and
you're just left with 7 times x is equal to 7x. And now this becomes a much,
much simpler equation. Now, what's 45 minus 24? Let's see, 45 minus 20
would be would be 25. Then you subtract
4 more, it's 21. So you get 21 is equal to 7x. Divide both sides by 7 and
you get x is equal to 3. And let's verify that
that works. So we have 5 over 2x. So that's the same thing as
5 over 2 times 3, minus 4 over 3 times 3. So this is 5/6 minus 4 over
18-- sorry, 4 over 9. We want to find a common
denominator. 18 is the least common multiple
of 6 and 9, so let's put it over 18. 5/6 is the same thing
as 15 over 18. Multiply the numerator
and denominator by 3. 4/9 is the same thing
as 8 over 18. Multiply the numerator
and denominator by 2. 15 minus 8. So this becomes 15 minus
8 over 18, which is equal to 7 over 18. So it works out. 5 over 2x when x is equal to 3,
minus 4 over 3x when x is equal to 3 is indeed
equal to 7/18. So we're done.