Simplify each complex rational expression. (1/x-1/2)/(1/3-x/6)

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Hello. Today we're gonna be simplifying a complex rational equation. So the complex rational equation is two over x minus three over to all of that over one half minus x minus over three. So let's go ahead and start by just simplifying our numerator and denominator separately. So in the numerator we have two over X -3/2. So in order to combine these quantities together we need to go ahead and get a similar denominator in the denominator we have X and two. So that means the lowest common denominator between these two fractions is going to be two X. So in order to get to X in both of these fractions we're gonna multiply the left fraction by 2/2 on both the numerator and denominator. And we're gonna multiply the right fraction by X over X. In the numerator. And denominator. Doing so is going to give me 4/ X minus three X over two X. And since we have a common denominator of two X we can combine this fraction into a single fraction. Doing so is going to give me four minus three X. All over two X. So that's our numerator simplified. Let's go ahead and simplify our denominator. Now in the denominator we have a half minus X over three. So we need to get a lowest common denominator between two and three. While the lowest common denominator is going to be six. So in order to get six in the denominator we're gonna multiply the left fraction by 3/3 and multiply the right fraction by 2/2. Doing so is gonna give me 3/ minus two X over six. And since we have a common denominator we can go ahead and combine this into a single fraction. Doing so is going to give me 3 -2 x. All over six. Now let's go ahead and put this back into the complex fraction. Doing so is going to give me four minus three X. All of that over two X. All of that over 3 -2 x over six. Now, in order to reduce the complex fraction let's go ahead and multiply both the numerator and denominator by the reciprocal of the denominator. The reciprocal of the denominator is gonna be 6/ minus two X. And we're gonna multiply that to the numerator as well. So the reason why we do this is because this allows us to completely cancel this denominator to just one, essentially eliminating the complex fraction. And we are thus left with 4 - x times six over two x times 3 -2 x. Now we have six in the numerator and two X. In the denominator six and two are multiples of each other. So we can go ahead and reduce this to three and cancel out the two in the denominator we are then left with three times four minus three X over X times three minus two X. Now let's go ahead and rewrite both of these quantities so that the X term is in front of the common term. So reordering this is going to give us three times negative three X plus four. All of that over X times negative two X plus three. Now, ideally we don't want this first uh the first number to be negative. So let's go ahead and factor out a negative one from both the numerator and denominator. Doing so is gonna give me three times negative one times three X minus four. All that over X times negative one times two X minus three. And since we have a negative one in both the numerator and denominator, we can just go ahead and cancel that out. That thus leaves us with three times three X minus four over X times two X minus three. And that is going to be our solution because that's the simplest form of this complex fraction. That also means that the answer to this problem is going to be a so I so I hope this video helps you simplify complex fractions and I'll go ahead and see you all in the next video

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