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- 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.

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