Solving Rational Equations...help!!

**mOoGaLy** · January 5th 2007, 11:41 AM

Okay, I don't know how to do this what so ever and I'm getting frustrated because my book isn't showing in great detail what I'm suppose to be doing!

I need help with 5 equations... this is one of them.

8/x+2 + 8/2 = 5

Please help me!!!

**galactus** · January 5th 2007, 11:56 AM

Originally Posted by mOoGaLy

Okay, I don't know how to do this what so ever and I'm getting frustrated because my book isn't showing in great detail what I'm suppose to be doing!

I need help with 5 equations... this is one of them.

8/x+2 + 8/2 = 5

Please help me!!!

You haven't used parentheses, so your post is rather ambiguous.

Is this it:

$\frac{8}{x+2}+\frac{8}{2}=5$

If so, why not write 8/2 as 4?.

Anyway, multiply by $x+2$ :

$(x+2)\frac{8}{x+2}+4(x+2)=5(x+2)$

$8+4(x+2)=5(x+2)$

Now, can you finish?.

**mOoGaLy** · January 5th 2007, 12:11 PM

Originally Posted by galactus

You haven't used parentheses, so your post is rather ambiguous.

Is this it:

$\frac{8}{x+2}+\frac{8}{2}=5$

If so, why not write 8/2 as 4?.

Anyway, multiply by $x+2$ :

$(x+2)\frac{8}{x+2}+4(x+2)=5(x+2)$

$8+4(x+2)=5(x+2)$

Now, can you finish?.

Yeah, that's the equation. and what I did was...

$8+4(x+2)=5(x+2)$

$12(x+2)=5(x+2)$

Now, do I multiply 12 to x & 2 to get
12x + 24 = 5x + 10

The answer I got is
$x=-2$

?

**galactus** · January 5th 2007, 12:35 PM

No, check yourself. If you put your answer back into the equation, do you get 5?. I don't think so.

You got a little discombobulated on your distribution.

Let's try your answer first:

$\underbrace{\frac{8}{\underbrace{-2+2}_{\text{oops, division by 0, a big no-no}}}+4=5}_{\text{bad, the universe will\\collapse in on itself}}$

Now, let's do it this way:

$8+4(x+2)=5(x+2)$

$8+4x+8=5x+10$

$16+4x=5x+10$

Subtract 4x from both sides:

$16=x+10$

Subtract 10 from both sides:

$x=6$

See?.

Actually, you're not seeing the forest for the trees. Don't let the algebra and x's psyche you out.

Look at it for a second. Follow:

$\frac{8}{x+2}+4=5$

What plus 4 equals 5. Well, you don't have to be a brilliant mathematician to see that. 4+1=5

Therefore, $\frac{8}{x+2}$ must be 1. What makes that 1?. When x=6. See?.

Relax. It'll be OK.

**mOoGaLy** · January 5th 2007, 12:46 PM

Ohh. Parenthasis first..
That seems so easy! Thank you!!

$\frac{2}{3x}+\frac{2}{3}=\frac{8}{x+6}$

Do I multiply $x+6$ by all the equations?

**galactus** · January 5th 2007, 01:23 PM

You're partially correct. You use the GCD, which is $3x(x+6)$

See?. You have more in the denominators than just x+6. There's a 3x there, also.

$(x+6)(3x)\frac{2}{3x}+(x+6)(3x)\frac{2}{3}=(x+6)(3 x)\frac{8}{x+6}$

Cancel what needs cancelled and you get:

$2(x+6)+2(x+6)(x)=8(3x)$

Distribute:

$2x+12+2x^{2}+12x=24x$

$14x+12+2x^{2}=24x$

$2x^{2}-10x+12=0$

Divide by 2:

$x^{2}-5x+6=0$

Now, can you solve the quadratic?.

**mOoGaLy** · January 5th 2007, 01:42 PM

Woah woah, why do you divide by 2?

Okay, well using the equation $x^{2}-5x+6=0$
I used the quadratic formula and I am stuck at
(doesn't know the codes to make this look correct)
x = -5 plus/minus symbol 1 all over 2

**galactus** · January 5th 2007, 02:04 PM

I divided by 2 because I could and wanted to. All the terms were divisible by 2

The quadratic $x^{2}-5x+6=0$ can be factored.

Can you factor?. You can use the quadratic formula, too, but this one is easily factorable.

$(x-3)(x-2)=0$

**mOoGaLy** · January 5th 2007, 02:07 PM

Originally Posted by galactus

I divided by 2 because I could and wanted to. All the terms were divisible by 2

The quadratic $x^{2}-5x+6=0$ can be factored.

Can you factor?. You can use the quadratic formula, too, but this one is easily factorable.

$(x-3)(x-2)=0$

I forgot all about those.
So $x=3 or x=2$ ?

**mOoGaLy** · January 5th 2007, 02:47 PM

$\frac{x-3}{x-4}+4=\frac{3x}{x}$

$(x-4)(x)\frac{x-3}{x-4}+(x-4)(x)4=(x-4)(x)\frac{3x}{x}$

I multipied them

$x(x-3)+4(x-4)(x)=3x(x-4)$ but I had a feeling that I was suppose to cancel out the 4's but I chose not to..

Next, $x^{2}-3x+4x^{2}-16x=3x^{2}-12x$

And I moved the $3x^{2}-12x$
to the left side of the equation...

$x(x-3)+4(x-4)(x)=3x(x-4)=3x^{2}-12x$

$5x^{2}-3x{2}-19x-12x$

I'm stuck at $2x^{2}-21x$

I did something wrong, didn't I? =( I am horrible at this!

**galactus** · January 5th 2007, 03:01 PM

$(x-4)(x)\frac{x-3}{x-4}+(x-4)(x)4=(x-4)(x)\frac{3x}{x}$

I multipied them

$x(x-3)+4(x-4)(x)=3x(x-4)$ but I had a feeling that I was suppose to cancel out the 4's but I chose not to..

Next, $x^{2}-3x+4x^{2}-16x=3x^{2}-12x$

And I moved the $3x^{2}-12x$
to the left side of the equation...

$x(x-3)+4(x-4)(x)=3x(x-4)=3x^{2}-12x$ $\leftarrow\text{good to here}$

$5x^{2}-19x=3x^{2}-12x$

$2x^{2}-7x=0$

Factor:

$x(2x-7)=0$

See the solution now?.

$5x^{2}-3x{2}-19x12x$

I'm stuck at $2x^{2}-21x$

I did something wrong, didn't I? =( I am horrible at this!

**mOoGaLy** · January 5th 2007, 03:18 PM

You lost me. How does $x(2x-7)=0$ end up into x = a number?

**mOoGaLy** · January 5th 2007, 03:19 PM

$2x^{2}=7$ ??

**topsquark** · January 5th 2007, 06:08 PM

Originally Posted by mOoGaLy

You lost me. How does $x(2x-7)=0$ end up into x = a number?

$x(2x-7) = 0$

Since the RHS is 0, at least one of x or 2x - 7 must also be 0. Thus:
$x = 0$ or $2x - 7 = 0$

Thus $x = 0$ or $x = \frac{7}{2}$

Now be careful. Your original equation canNOT work with x = 0. (The RHS of your original equation has us dividing 0/0, which is not allowed.) So your only solution is $x = \frac{7}{2}$ .

-Dan

**mOoGaLy** · January 7th 2007, 02:37 PM

Oh! Okay, thank you so much galactus & topsquark!! You've been loads of help.

If I need anymore, I'll be back lol

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