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# Equation with two rational expressions (old example 2)

## 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.