anonymous
  • anonymous
Solve: (x)/(x-2) + (1)/(x-4) = (2)/(x^2 - 6x + 8)
Mathematics
  • Stacey Warren - Expert brainly.com
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SOLVED
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katieb
  • katieb
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lgbasallote
  • lgbasallote
multiply with LCM this time it's (x-2)(x-4) that is equal to x^2 -6x +8 x(x-4) + x-2 = 2 x^2 - 4x + x -2 -2 =0 x^2 - 3x -4 =0 (x-4)(x+1) = 0 x = 4 x=-1
anonymous
  • anonymous
What is the LCD? Let's start there.
anonymous
  • anonymous
Least Common Denominator?

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anonymous
  • anonymous
That's right - first we want to identify the Least Common Denominator.
anonymous
  • anonymous
Can you tell what it is?
anonymous
  • anonymous
Well I don't really know how to tell D:
anonymous
  • anonymous
Alright - I'll explain how to find it. :)
anonymous
  • anonymous
When you are asked to solve rational equations like this, the first thing you want to do is factor everything.
anonymous
  • anonymous
Factor them by what? LCD?
anonymous
  • anonymous
Factor each polynomial, all the numerators and denominators. I'll show you:
anonymous
  • anonymous
This is what you were given: \[\frac{x}{x-2} + \frac{1}{x-4} = \frac{2}{x^2 - 6x + 8} \]Now, first identify 3 fractions. Each fraction has a numerator and denominator, but only one of these pieces can be factored. The denominator on the right-most fraction.
anonymous
  • anonymous
To factor it, we're looking specifically at this: \[x^2-6x+8\]Here, you want to ask yourself, what two numbers multiply to +8 and add to -6.
anonymous
  • anonymous
is it because it is a trinomial?
anonymous
  • anonymous
That's right! :) Thanks for all the medals, by the way.
anonymous
  • anonymous
Can you see how it would factor?
anonymous
  • anonymous
It would factor like this: We're looking for numbers that multiply to 8 and add to -6. So the numbers we are looking for are -4 and -2. So this trinomial would factor like this: (x-4)(x-2) And you can always FOIL those back out to check to see that you get what you started with.
anonymous
  • anonymous
I was eating late dinner sorry lol and your welcome. thanks for the tutor! (:
anonymous
  • anonymous
Happy to help! You ready for the next step?
anonymous
  • anonymous
Yes!
anonymous
  • anonymous
Alright. :)
anonymous
  • anonymous
you make it easier than my online class really is :P
anonymous
  • anonymous
So we can rewrite the original question in factored form, like this:\[\frac{x}{x-2} + \frac{1}{x-4} = \frac{2}{(x-4)(x-2)} \]
anonymous
  • anonymous
The reason we do this is because we are eventually going to multiply both sides of this by the LCD in order to completely clear the fractions. But, we have to know what factors all the pieces are made from in order to do that first.
anonymous
  • anonymous
So at this point we can see that the LCD is: (x-4)(x-2)
anonymous
  • anonymous
alright so what do i do with the LCD
anonymous
  • anonymous
From here, you're going to multiply each fraction by the LCD over 1 - like this:
anonymous
  • anonymous
\[\frac{(x-4)(x-2)}{1}\frac{x}{x-2} + \frac{(x-4)(x-2)}{1}\frac{1}{x-4} = \frac{(x-4)(x-2)}{1}\frac{2}{(x-4)(x-2)} \]
anonymous
  • anonymous
It's a long process, but now comes the good part - you get to cancel.
anonymous
  • anonymous
So, looking at the first piece specifically, what is going to cancel? \[\frac{(x-4)(x-2)}{1}\frac{x}{x-2}\]
anonymous
  • anonymous
I cancelled in my head xD thats my favorite part of this once i ACTUALLY get to this part without making a mistake :P
anonymous
  • anonymous
Hey - even better! :)
anonymous
  • anonymous
so you would cance the (x-2)
anonymous
  • anonymous
Exactly!
anonymous
  • anonymous
then your left with (x-4)x/1 right?
anonymous
  • anonymous
Yes, although because the denominator is 1, we don't need to write it anymore.
anonymous
  • anonymous
So then because you were able to cancel in your head, you see how we're left with: \[x(x-4)+1(x-2)=2\]
anonymous
  • anonymous
The canceling is the payoff for picking the right LCD. If you've done your job right, at this point, there should be no fractions left.
anonymous
  • anonymous
oh yah forgot that second piece :P
anonymous
  • anonymous
Oh, be sure to remember all of them. We have 3 pieces to this equation. :)
anonymous
  • anonymous
So from here, what do we do?
anonymous
  • anonymous
Bring the equation together?
anonymous
  • anonymous
That's right - distribute, combine like terms... What will that leave you with?
anonymous
  • anonymous
(x-4)((x-2)(x-2)
anonymous
  • anonymous
I'm not sure what you did there.
anonymous
  • anonymous
Here's where you want to go with that:
anonymous
  • anonymous
First, use the distributive properto to get rid of those parenthesis:\[x(x-4)+1(x-2)=2\] Then combine the -4x with the +x\[x^2-4x+x-2=2\] Now, recognize that this thing is going to end up quadratic, so we want to set it equal to zero. \[x^2-3x-2=2\]You can subtract 2 from both sides to do that. \[x^2-3x-4=0\]
anonymous
  • anonymous
WOW!
anonymous
  • anonymous
Do you need me to explain any of those steps a little better?
anonymous
  • anonymous
I think that was the best you could explain mate xD
anonymous
  • anonymous
Excellent!
anonymous
  • anonymous
Alright, so we're left looking at this: \[x^2-3x-4=0\] Now you want to factor that. What does it leave you with?
anonymous
  • anonymous
(x-4)(x+3) x=4 x= -3 ?
anonymous
  • anonymous
Well, it can't be (x-4)(x-3) because if you tried to FOIL that out, you'd get: \[x^2-7x+12\]
anonymous
  • anonymous
We're looking for numbers that multiply to -4 and add to -3
anonymous
  • anonymous
Oops I meant (x-4)(x+1)
anonymous
  • anonymous
That's right! That's perfect! :)
anonymous
  • anonymous
So we're left with:\[(x-4)(x+1)=0\]
anonymous
  • anonymous
Now we can use the Zero-Product Property: \[x-4=0\]&\[x+1=0\]
anonymous
  • anonymous
This is GREAT xD
anonymous
  • anonymous
So, what do you get for the answers?
anonymous
  • anonymous
x = 4 and x = -1 ?
anonymous
  • anonymous
There's a little trick to this one. Because these are rational equations, we have to check to see if those are good answers. According to the equation, those are our answers, but they may turn out to be bad x's. We have to plug them in to the original equation and make sure they don't cause division by zero.
anonymous
  • anonymous
Only one of those answers is correct. The other is a bad x.
anonymous
  • anonymous
So, my question for you, which is the right one, and which is the bad one?
anonymous
  • anonymous
how would i know D: lol
anonymous
  • anonymous
Well, only one of them will cause division by zero with our original equation.
anonymous
  • anonymous
am i suppsed to plug it in and find out?
anonymous
  • anonymous
You could plug them in and find out that way, or you can just eyeball it. Have a look: When we factored the original equation we were left with this:\[\frac{x}{x-2}+\frac{1}{x-4}=\frac{2}{(x-4)(x-2)}\]
anonymous
  • anonymous
Our two solutions are: x=4 and x=-1
anonymous
  • anonymous
Now, the -1 won't cause any problems. But that 4 will because look at the second fraction. If we plug a 4 in for x, what does the bottom become?
anonymous
  • anonymous
0
anonymous
  • anonymous
Exactly - and we can't ever divide by 0. So that makes x=4 a "bad x". Because x=4 causes division by 0 in our original equation, we have to throw it out. So this question only has 1 answer. x=-1 :)
anonymous
  • anonymous
So you always want to check your answers for these to make sure they're not bad x's.
anonymous
  • anonymous
Sometimes there are no bad x's, sometimes (like this time) there is 1 bad x, and other times, both solutions are bad x's.
anonymous
  • anonymous
And that's how they're done! :) I hope that helped!!
anonymous
  • anonymous
THANKS!! xD
anonymous
  • anonymous
You're welcome!!

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