Solve 𝑥 minus three divided by four plus one-third equals two 𝑥 plus three divided by two for 𝑥.
So here we have this rational equation, an equation full of fractions. On the left-hand side of our equation, we can see that we are adding these two fractions. And any time you add or subtract fractions, you need a common denominator.
Now since we have a fraction on the right-hand side of the equation, we need a common denominator between all three fractions, so the common denominator will be the smallest number that four, three, and two can all go into, and that would be 12: four goes into 12, three goes into 12, and two goes into 12.
So here’s our original equation, and if we want our new denominator to be 12, we have to multiply four by three to get to 12. And whatever you do to the bottom, you’d also have to multiply to the top. For our second fraction, we’d have to multiply three by four to get to 12, so we need to multiply the numerator by 12.
And then lastly, we have to multiply two by six to get to 12, so we also need to multiply it by six at the top. So why is it that when you multiply the bottom by a number, you have to multiply the top by a number? Well, you can’t just take an equation and multiply anywhere you want; we have to keep it balanced.
So if you look, three over three is just one, and multiplying by one doesn’t actually change anything, same thing with four over four and the same thing with six divided by six. So when we say six divided by six and six over six, it means the same thing. So multiplying by one to a piece of an equation isn’t going to make it unbalanced, so it’s okay.
So let’s go ahead and start distributing and multiplying. So first we take three times 𝑥 and three times negative three, so we have three 𝑥 minus nine and on the bottom we have 12, and then four times one is four and four times three is 12 which it should make sense; four twelfths reduces to one-third. And then lastly, we get 12𝑥 plus 18 over 12, so now we have a common denominator between all three fractions, so let’s go ahead and add the fractions on the left of the equation.
So when we add fractions, we keep the common denominator — that’s the whole point — and then we combine like terms for the numerators, so we have a three 𝑥 and then there’s negative nine and a four. Add those together; we get negative five.
Now in order to solve, we cross-multiply. So we find the cross-product by taking 12 times three 𝑥 minus five and 12 times 12𝑥 plus 18. Now notice we’re multiplying by 12 to both sides of the equation, so we can actually divide both sides by 12, and they disappear. This makes it a little bit faster. Instead of distributing, this is the exact same thing because we don’t have to distribute 12 to both sides because that’s doing the exact same thing to both sides of the equation, just making each side larger by 12.
And also remember it would be okay to go ahead and distribute both sides by 12; we would still get the exact same answer, and we can look at that at the end. Subtracting three 𝑥 from both sides, and now we need to subtract 18 from both sides, so we have negative 23 equals nine 𝑥, and now we divide both sides by nine, so 𝑥 equals negative 23 ninths.
So like we said before, we could also do this by not dividing by 12 to both sides. We could go ahead and distribute the 12. So let’s just double-check and make sure that it works this way, and it should. So first we distributed 12 to both sides, and now let’s combine the 𝑥s, and we get 108𝑥.
Now let’s subtract 216, and we get negative 276. And now we divide by 108 and we get negative 276 divided by 108, and this actually reduces to be negative 23 over nine because if we take out a factor of 12 to negative 276 and 108, we get negative 23 ninths.