SOLUTION: (3 over x-5)+1 divided by 1- (4 over x-5) I hope this makes sense. Thanks for any help you can offer. I'm stuck on this one.

Algebra ->  Polynomials-and-rational-expressions -> SOLUTION: (3 over x-5)+1 divided by 1- (4 over x-5) I hope this makes sense. Thanks for any help you can offer. I'm stuck on this one.       Log On


   



Question 86197: (3 over x-5)+1 divided by 1- (4 over x-5)
I hope this makes sense. Thanks for any help you can offer. I'm stuck on this one.

Found 2 solutions by ankor@dixie-net.com, jim_thompson5910:
Answer by ankor@dixie-net.com(22740) About Me  (Show Source):
You can put this solution on YOUR website!
(3 over x-5)+1 divided by 1- (4 over x-5)
:
3%2F%28%28x-5%29%29 + 1
--------------
1 - 4%2F%28%28x-5%29%29
:
simplify the numerator:
%283+%2B+1%28x-5%29%29%2F%28%28x-5%29%29 = %28%28x+%2B+3+-+5%29%29%2F%28%28x-5%29%29 = %28%28x+-+2%29%29%2F%28%28x-5%29%29
:
Simplify the denominator:
%281%28x-5%29+-+4%29%2F%28%28x-5%29%29 = %28%28x+-+5+-+4%29%29%2F%28%28x-5%29%29 = %28%28x+-+9%29%29%2F%28%28x-5%29%29
:
Invert the denominator and multiply:
%28%28x+-+2%29%29%2F%28%28x-5%29%29 * %28%28x-5%29%29%2F%28%28x-9%29%29 = %28%28x+-+2%29%29%2F%28%28x-9%29%29; canceled the (x-5)'s
:
Did this help

Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
There are many ways to solve this problem, here are two routes you could take:

Route 1:

%283%2F%28x-5%29%2B1%29%2F%281-4%2F%28x-5%29%29 Start with the given expression

Let y=x-5

%283%2Fy%2B1%29%2F%281-4%2Fy%29

Multiply the expression by y%2Fy (which is a form of 1)

%28%283%2Fy%2B1%29%2F%281-4%2Fy%29%29%28y%2Fy%29

y%283%2Fy%2B1%29%2Fy%281-4%2Fy%29

%28y%2A%283%2Fy%29%2By%2A1%29%2F%28y%2A1-y%284%2Fy%29%29 Distribute

%283%2By%29%2F%28y-4%29 Multiply and simplify

%283%2B%28x-5%29%29%2F%28%28x-5%29-4%29 Replace y with x-5

%28x-2%29%2F%28x-9%29 Combine like terms


---------------------------------------------------------------------
Route 2:
%283%2F%28x-5%29%2B1%29%2F%281-4%2F%28x-5%29%29 Start with the given expression

%283%2F%28x-5%29%2B%28x-5%29%2F%28x-5%29%29%2F%28%28x-5%29%2F%28x-5%29-4%2F%28x-5%29%29 Convert each "1" into %28x-5%29%2F%28x-5%29

%28%283%2Bx-5%29%2F%28x-5%29%29%2F%28%28x-5-4%29%2F%28x-5%29%29 Combine the fractions

%28%283%2Bx-5%29%2F%28x-5%29%29%2A%28%28x-5%29%2F%28x-5-4%29%29 Multiply the fractions by flipping the 2nd fraction

%28%283%2Bx-5%29%2Fcross%28%28x-5%29%29%29%2A%28cross%28%28x-5%29%29%2F%28x-5-4%29%29 Notice these terms cancel

%283%2Bx-5%29%2F%28x-5-4%29

Now combine like terms

%28x-2%29%2F%28x-9%29 which looks just like the original answer.


So no matter what route you take, the expression

%283%2F%28x-5%29%2B1%29%2F%281-4%2F%28x-5%29%29

simplifies to

%28x-2%29%2F%28x-9%29


Check:
One way we could verify this is to graph both expressions as equations like this

Graph the first equation y=%283%2F%28x-5%29%2B1%29%2F%281-4%2F%28x-5%29%29 (just set the expression equal to y)
graph of y=%283%2F%28x-5%29%2B1%29%2F%281-4%2F%28x-5%29%29

Now graph the 2nd equation y=%28x-2%29%2F%28x-9%29
+graph%28+500%2C+500%2C+-10%2C+10%2C+-10%2C+10%2C+%28x-2%29%2F%28x-9%29%29+ graph of y=%28x-2%29%2F%28x-9%29

Notice how they trace out the same path. If they were plotted on the same graph, one would overlap another perfectly. This means they are equivalent (minus the discontinuous hole) and this verifies our answer.