Question

Simplify this expression and state any restrictions on the variables. (x+1/x^2+2x-8) - (x/4x-8) please show me step by step!

Answer 1

\[\frac{x+1}{x^2+2x-8} - \frac{x}{4x-8}\]
OK, where are you getting stuck?

Answer 2

\[\frac{ x+1 }{ x^{2}+2x-8 }-\frac{ x }{ 4x-8}\]

Answer 3

I have no idea how to do these!!

Answer 4

Start by factoring the denominators. you need to find a comon denominator and the only way to do so is with factors.

Answer 5

\[\frac{ x + 1 }{ (x -2)(x-4) } - \frac{ x }{ 4(x-2) }\]

Answer 6

Ta da... factors. Hehe. You see what superdav815 has done there?

Answer 7

yeah but the fraction part confuses me, i stink at fractions :(

Answer 8

\[\frac{ 4(x+1) }{ 4(x-2)(x+4) } - \frac{ x(x+4) }{ 4(x-2)(x+4) }\]

Answer 9

you can do the rest

Answer 10

Well, it is the same old common deniminator thing. Just like when you want to add 1/2 and 1/3.

Answer 11

What you look for in the denominator of each fraction is overlap, what is there and what is missing. That is how superdav815 got what he did. Do you understamd that, or would you like me to show it to you a different way?

Answer 12

like i know how to solve 4(x+1) and problems like that by itself but i don't get hoe to do it when you start putting them on top of each other lol

Answer 13

No problem at all! Then you know the foundation of it,, you jsut need some specifics. Are you OK with the factoring part?

Answer 14

factoring is when you solve 4(x+1) thats called factoring right?

Answer 15

No, that is called solving. Facroing is when you turn \(x^2+2x-8\) into \((x−2)(x−4)\). Another name for it could be un-multiplying.

Answer 16

oh lol sorry yeah i understand how to do that, i just forget how to start sometimes

Answer 17

NP. The next part is one of the ones that might confuse you, lowest comon denominator. I am doing something to deal with that, but have you heard of it and are you at least somewhat familliar with the idea?

Answer 18

yeah

Answer 19

Good, good. That process is the key to these. Almost done with my drawing. Then we can talk about any parts you do not get.

Answer 20

okay

Answer 21

OK. Here is is!

Answer 22

That is the foundation behind what happend that let superdav815 just post those equations. He found the lowest common denominator, or LCD, then looked for what each side was missing and multiplied by one to add it in. Now the one he used is things like \(\frac{4}{4}\), but that is still one.

Answer 23

yeah i understand that part now, so what do I do next?

Answer 24

Subtract.

Answer 25

What does the top become?

Answer 26

4x+5-x^2+4x?

Answer 27

I think you distributed things wrong.

Answer 28

darn -_- what did i do wrong?

Answer 29

Lets take it apart and try again.
\(4(x+1)-[x(x+4)]\)

So lets grab: \(4(x+1)\) What does that distribute to?

Answer 30

4x+4

Answer 31

Yes, good.
Now: \(x(x+4)\)

Answer 32

1x^2+4x

Answer 33

Perfect.
OK, if I put those back in I get:
\(4x+4-[x^2+4x]\)
But I still need to distribute the \(-1\) that is hiding outside the [.

Answer 34

how do you do that again?

Answer 35

Multiply the negative through. \(-[x^2+4x]\) So the negativd distributes.

Answer 36

-1x^2+3x

Answer 37

When you distribute a nagative, it changes the sign of each term inside. Not the value, just the sign.

Answer 38

okay so it would be -1x^2-4x

Answer 39

YES! OK, so, let me put that all back in the top, we are just looking at the top of the fraction and ignoring the bottom for now.
\(4x+4-x^2-4x\)
Now, see any like terms to combined?

Answer 40

1x^2+4-x^2

Answer 41

\(1x^2+4-x^2\) ? Not sure how that happned. I was taling about the \(4x\) and \(-4x\). The rest, well, is unique.

Answer 42

It looks like you are having a little trouble with polynomials. I have done tons of them and still make mistakes at times, so I am not going to say it is surprising. I think we all have troubles with one part of math or another. However, I have a sneaking suspicion at this point it is just getting frustrating.

My aim with this top of the fraction is this:
\(4x+4-x^2-4x\implies 4x-4x+4-x^2 \implies 4-x^2\)

Do you see what I did there?

Answer 43

i think so

Answer 44

It is just the 4x and -4x that cancel. OK. Let me put what is left back into the fraction. There is still a little work to do, but it is close to done.

Answer 45

That gets us to:
\[\frac{4-x^2}{4(x-2)(x+4)}\]

Answer 46

Now, all that is left is the simplify part. All the subtracting is done.

Answer 47

okay 4x-8+4x+16 -> 8x^2+22?

Answer 48

@e.mccormick

Answer 49

@superdav815 Hehe. What, not going to draw that in the machine like mine?

Answer 50

srry

Answer 51

@sophiasmommy94 Oh, you multiplied it out. Actually, the top factors a bit. Look at the pic SD uploaded. It has the end in there.

Answer 52

@superdav815 Just teasing. Nice handwriting.

Answer 53

lol thank you

Answer 54

forgot to add in the picture: another restriction is \[x \neq -4\]

Answer 55

Yah, from the original. Both the start and the end must be true.

Answer 56

You two have been such a BIG help!! Thank you Thank you Thank you!!! Do any of you have time for one more question?

Answer 57

whats the question

Answer 58

I need to head to bed. But I did one to share one last thing on this one. What the restrictions mean. See, people hear it is invalid, or does not exist, but that does not always mean much. Here is a graph of the two equations that were subtracted:
https://www.desmos.com/calculator/qldnssqthu
Notice the big gaps where the graph suddenly shoots up and down at the same time? Those are the restriction points for the original graph.
I could also add the answer, or you can, and see where it suddenly goes funky.

Answer 59

simplify: \[\frac{ 1 }{ 1+\frac{ x }{ y } }\]

Answer 60

this is what i got so far 1/(1+y/x)= 1/(x/x+y/x)= 1/(x+y/x)

Answer 61

Thank you @e.mccormick

Answer 62

np. Have fun!

Answer 63

Watch your common denominator in that first step you did. It is a y they need to share, not an x.

Answer 64

\[\frac{ 1 }{ 1 +\frac{ x }{ y } } = \frac{ 1 (y)}{( 1 +\frac{ x }{ y } )(y) }\]

Answer 65

do you get that last step? I just multiplied numerator and denominator by y

Answer 66

yeah, so it's simplified?

Answer 67

im guessing the answer they want is this \[\frac{ y }{ y+x }\]

Answer 68

okay, thank you!!

Answer 69

can you write out the steps so i can put them in my notebook for future refrence?

Answer 70

@superdav815

Answer 71

thank you!!