 This video talked about fractions and we're just going to solve equations that involve fractions. So we look at this and the first thing we need to do is find the least common denominator. So we look at 4, 12 and 8 and we're looking for something that they have in common. If we look at 12 we can think that that's 12 and then times 2 is 24 and times 3 is 36 and times 4 is 48. We know that that's 4 and 12 have that in common but we can also know that 8 times 6 is 48 so 48 is going to be at least common denominator. Seems like a big number but remember we're going to reduce. So I'm going to take this 48 but I'm really going to say 48 times x over 4 minus 48 times that 1 over 12 is equal to 48 times x over 8. You've got to distribute the 48 to all three parts of the equation. So remember now we can reduce this and we can say that 4 goes into 48 12 times So now we just have 12 times x and then 12 goes into 48 4 times so we have a minus 4 times 1 or just minus 4 equal to 8 goes into 48 6 times and so we have 6 times x. Now we're ready to solve. When you multiply by the least common denominator when you get done you should end up with an equation that has no fractions which is exactly what we have here. So now we are ready to combine like terms and add and subtract across the equal sign to do that. Now I have x's on both sides but a constant is only on this side so even though it's going to give me a negative x value I'm going to subtract 12x from both sides. So this will give me 0 and then minus 4 is equal to 6x minus 12x would be a negative 6x. It's okay to have a negative. We're just going to divide by a negative and just watch our signs. So we're going to divide both sides by that negative 6 so we can get x by itself positive and 4 over 6 reduces to 2 over 3. So we think that it's 2 thirds. So let's see if we can check this here. 2 thirds. It's kind of tricky to check when you have a fraction but just remember that that's really dividing the same thing as multiplying by the reciprocal. So we say 2 thirds divided by 8 here. This is really 2 thirds times 1 over 4. The reciprocal of 4. Minus 1 over 12 is equal to 2 thirds times the reciprocal of 8 or 1 over 8. This gives me 2 over 12 minus 1 over 12 which is equal to 2 times 1 is 2 and 3 times 8 is 24. 2 minus 1 would be 1 over 12 on this side. 2 over 24 reduces to 1 over 12 so we know that our answer is positive 2 thirds. Look at this fun example then. I have lots of things going on here but the first thing I always have to do is find all these common denominator and between 12 and 6 I can make 6 become 12 so 12 is going to be my least common denominator. So I want to multiply this whole equation by 12. Now look real closely. There's one term on this side and one term on this side because this parenthesis, my fraction times this quantity and this addition is not making me have a different term. This is all multiplication so it's one term. So I'm going to have 12 and then times 7 over 12 and then times my 3x plus 5 and equal to 12 times my 5 over 6 times my x minus 2. Okay, here's my 712 times 3x plus 5 but the reason why I put this in parenthesis is because remember we can multiply two things at a time. So I'm going to take these first two because my 12's become a factor of 1 and now I have 7 times 3x plus 5 and again I can reduce this 6. In fact, let's get a different color here so you see what's going on. The 6 goes into 12 two times and so now I have 2 times 5 times that x minus 2 so it's really going to be 10 times x minus 2 and when I distribute on this side I'm going to have 21x plus 35 and on this side I will have 10x minus 20 and I'm going to move my x first and 21 is positive and it's larger than my 10 so I'm going to subtract 10x from both sides so I can try to keep my x positive whenever possible. So that gives me 11x plus 35 equal to this is 0 and then negative 20 so subtract our constant do that addition property before we start dividing things so I still have my 11x, this is 0 and this is going to be a negative 55 and now we're ready to divide off the coefficient and that tells me that x should be equal to negative 5 so let's check. 7 over 12 times 3 times my negative 5 which I found out x to be plus 5 is equal to 5 over 6 times my negative 5 which I found out x to be minus 2 so this is 7 over 12 this is negative 15 plus 5 and I'll just simplify this side 7 over 12 is going to give me negative 10 negative 15 plus 5 so I have negative 70 over 12 and I think it's going to be okay to leave it that way so I have 5 over 6 and negative 5 minus 2 is going to be negative 7 so when I multiply I'm going to have negative 35 over 6 well it looks like I might be able to reduce this 70 and 12 and they're both even so they're divisible by 2 70 divided by 2 would be negative 35 and 12 divided by 2 would be 6 and so we see that we really did find that right answer of x equal negative 5 one more this one's really important to watch and be very careful when you see a more than one term in a numerator like this you want to put parentheses around it before you do anything else so you don't forget to multiply correctly now we're ready to look for the least common denominator and again I have a 3 and a 6 and I can make 3 I can multiply by 2 and get to 6 so 6 will be the least common denominator and I'm going to then multiply this whole thing by 6 there's one term there's two terms and there's a third term on the other side so 6 times a plus 7 over 3 minus 6 times my 4 is equal to 6 times my a over 6 well 3 goes into 6 twice so I'm going to have 2 times a plus 7 and negative 6 times 4 is going to be a negative 24 and the 6 and the 6 remember this is a numerator and denominator so the common factor in f1 and I just have a on that side so I cleared my fraction so I'm headed in the right direction and now I just have to distribute the 2 and then go on from there 2a plus 14 minus 24 will equal a I've got like terms right here so my 2a is going to be minus 10 equal to a and since I only have an a on this side I'm going to subtract my 2a from both sides so I have negative 10 equal to negative a and we divide by negative 1 and a is equal to 10 and just in case we'll double check so 10 plus 7 over 3 minus 4 is supposed to be equal to 10 over 6 well this is 17 over 3 and if I'm going to be able to subtract I need to make this 4 have a denominator of 3 so I have to multiply by 3 over 3 so that's going to give me 12 over 3 and then that's going to be equal to my 10 over 6 which I may have to reduce but we'll look and see 17 minus 12 then on this side is 5 and over 3 equal 10 over 6 and if I reduce 10 over 6 they're both divisible by 2 so it is 5 over 3 equal to 5 over 3 and now I know that a really is 10