Simplify each complex fraction. [ (y+3)/y - 4/(y-1) ] / [ y/(y - ...

Question

Simplify each complex fraction. [ (y+3)/y - 4/(y-1) ] / [ y/(y - 1) + 1/y ]

Accepted Answer

Hello. Everyone in this video we're going to be looking at this practice problem where we want to simplify the expression two divided by m minus three plus M. Plus six divided by M. And that is being divided by four divided by m minus M divided by m minus three. So in this case you can see that this expression is a really big fraction but within that fraction the numerator has two fractions and the denominator also has two fractions. So in order to make it easier to simplify, I'm going to simplify the numerator first and then the denominator and then I'll combine them. So if I look at the numerator I have two divided by m minus three plus M plus six divided by. Um So I have two fractions but I can't combine them because they don't have a common denominator but I can make them have a common denominator by multiplying the fraction by the other's denominator. So for this first fraction I need to multiply by em and recall that I need to multiply by M over M. Because that is equivalent to one. So by multiplying my expression by one I'm making sure to not introduce any new numbers into my expression. So my first fraction needs to get multiplied by M over M. And my second fraction needs to get multiplied by minus three Divided by M -3. Because that is the denominator of the first fraction. So once I do this multiplication I have M times two or two M divided by M times minus three Plus M-plus six Times um -3 divided by um Times M -3. So after this multiplication you can see that I now have the same denominator. So I can combine my two fractions. So this becomes two M plus I'll go ahead and multiply this out. I have M times M. Is M squared M times negative three is negative three M. Six times M. Is six M. And finally six times negative three is negative 18. And that's all being divided by The Common Denominator, M. Times M -3. And in my numerator you can see that I can combine some like terms I have to m minus three M. That is negative M plus six M. Is positive five M. So this becomes M squared plus five M minus 18, Divided by M. M -3. And actually I can do the same to my denominator. So if I do M times M that's M squared M times negative three is negative three M. So that is the numerator of my expression. Now we can take a look at the Denominator where I have four divided by M -M divided by M -3. And once again I need to make them have the same denominator in order to combine the two. So multiply the first fraction by a minus three divided by a minus three and the second fraction by M divided by M. And if I do this multiplication, I have four times minus three Divided by M. A -3 minus I'm squared divided by M times M minus three. And once again you can see that the multiplication resulted in the two fractions having the same denominator, so I can combine them. So if I combine them I have four times M. Is four M. Four times negative three is negative 12 divided by still have the minus M squared. And now divided by the common denominator of em Times and -3 and I can go ahead and multiply this out. So this is I'll rearrange the numerator to be negative squared, moving this term to the front Plus four M -12. And if I do M times M in the denominator that becomes M squared and M times negative three is negative three M. So now I've simplified the denominator of my fraction and I can re put them in the bigger fraction. So I have the numerator which was this portion here, so have M squared plus five M minus 18, divided by m squared minus three M. And I have my denominator which was this portion here. So negative M squared plus four M minus 12, divided by m squared minus three M. And an easier way to simplify expressions when we have fractions in both the numerator and denominator is to write it in the linear format. So if I write this in the linear format for division, I'll go ahead and just move the denominator to be the second fraction. And once I do that in this linear format, recall that we can change this division to be a multiplication. If I switch the numerator and denominator of the second fraction. So now the numerator will be M squared minus three M. And my denominator will be negative M squared plus four M minus 12. And from here you can see that the denominator in the first fraction cancels out with the numerator in the second fraction. So that leaves me with M squared plus five M minus 18, divided by negative M squared plus four M minus 12. And from here I can't simplify this any further. So this becomes my final simplification for the expression. And if we look at the answer choices, you can see that answer choice A also has M squared plus five m minus 18, divided by negative M squared plus four m minus 12. So, flee this video home and see you next time

Simplify each complex fraction. [ (y+3)/y - 4/(y-1) ] / [ y/(y - 1) + 1/y ]

Key Concepts

Complex Fractions

Common Denominator

Algebraic Manipulation

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