What is the solution to the following rational equation?

- anonymous

What is the solution to the following rational equation?

- Stacey Warren - Expert brainly.com

Hey! We 've verified this expert answer for you, click below to unlock the details :)

- katieb

I got my questions answered at brainly.com in under 10 minutes. Go to brainly.com now for free help!

- anonymous

\[ \frac{ x }{ x ^{2} -9}-\frac{ 1 }{ x-3 }=\frac{ 1 }{ 4x-12 }\]

- anonymous

@ash2326 @hba @Hero @nincompoop @phi @satellite73

- anonymous

@Hero

Looking for something else?

Not the answer you are looking for? Search for more explanations.

## More answers

- phi

Can you factor each of the denominators ?

- anonymous

I know how to factor the first one... (x+3)(x-3)

- anonymous

the second one can't be and the last one I think is x-3

- phi

the last one you can factor 4 from each term. you get 4(x-3)
(notice if you "distribute" the 4 you get 4x -12 so you know that is correct)

- anonymous

okay

- phi

now to solve, we could find a common denominator. Or, which is almost the same thing,
is "clear the denominator"
so far you have
\[ \frac{x}{(x-3)(x+3)} - \frac{1}{(x-3)} = \frac{1}{4(x-3)} \]
multiply both sides and all terms by (x-3)
can you do that ?

- phi

you put (x-3) next to each term . It multiplies the top of each fraction
notice you can cancel when the same thing is in the top and bottom

- anonymous

I don't understand...

- phi

multiply both sides of the equation by (x-3). That means multiply *every term* by (x-3)
to multiply, write (x-3) next to each term, like this
\[(x-3)\cdot \frac{x}{(x-3)(x+3)} - (x-3)\cdot\frac{1}{(x-3)} = (x-3)\cdot\frac{1}{4(x-3)} \]

- phi

you can think of (x-3) as
\[ \frac{(x-3)}{1} \]
and the (x-3) multiplies the top
but notice you have
\[ \frac{(x-3)}{(x-3)} \]
in each term. This is the same as 1 (anything divided by itself is 1)
so you can "cancel" x-3 from the top and bottom

- phi

what do you get ?

- anonymous

1?

- phi

\[ (x-3)\cdot \frac{x}{(x-3)(x+3)} - (x-3)\cdot\frac{1}{(x-3)} = (x-3)\cdot\frac{1}{4(x-3)} \]
what is the first term simplify to?

- anonymous

x/(x+3)

- phi

yes. now the next term ?

- anonymous

do I distribute the negative?

- phi

first cancel, then distribute the -1

- anonymous

okay, so it would be -1

- phi

yes
now that last term

- anonymous

1/4

- phi

so what do we now have ?
\[ \frac{x}{x+3} - 1 = \frac{1}{4} \]
we could do a few different things, but "combining like terms" is always a good step
Can you add +1 to both sides and then simplify ?

- anonymous

I don't know how... -1 doesn't have a like term

- phi

the like terms are the "pure numbers" you know you can add two numbers together.
so add +1 to both sides, like this
\[ \frac{x}{x+3} - 1 +1 = \frac{1}{4} +1\]
can you simplify ?

- anonymous

\[\frac{ x }{ x+3 }=1\frac{ 1 }{ 4 }\]

- phi

can you make 1 and 1/4 an "improper fraction" ?

- anonymous

5/4?

- phi

yes
you now have
\[ \frac{x}{x+3} = \frac{5}{4} \]
any ideas what to try next ?

- anonymous

no

- phi

to solve these things you should have some ideas on what to do next.
one thing you could say to yourself is " (x+3) in the denominator is ugly"
If I multiply both sides of the equation by (x+3), it will cancel in the first term.
try multiplying both terms by (x+3)
what do you get ?

- phi

to multiply, write (x+3) next to each term

- anonymous

\[x=\frac{ 5 }{ 4 }\]?

- anonymous

@phi

- phi

Did you write (x+3) time each term in
\[ \frac{x}{x+3} = \frac{5}{4} \]
there are two terms (one on the left side of =, and the other on the right side.
write (x+3) next to each term.
can you do that ?

- anonymous

no...

- phi

just write (x+3) next to each term.

- anonymous

\[\frac{ x(x+3) }{( x+3)(x+3)}=\frac{ 5(x+3) }{ 4(x+3) }\]

- phi

that is correct (you multiplied each term by 1). but not what we want to do.
just multiply each term by (x+3) (NOT (x+3)/(x+3) )

- anonymous

\[\frac{x(x+3) }{ x+3 }=\frac{ 5(x+3) }{ 4 }\]

- phi

yes.
now notice on the left side of =, you have (x+3)/(x+3)
you can "cancel" those.
what do you get ?

- phi

do you agree that (x+3)/(x+3) is 1 ?
so
\[ \frac{x(x+3)}{(x+3) } = x \cdot 1 = x \]

- phi

you now have
\[ x = \frac{ 5(x+3) }{ 4 } \]
I would multiply both sides by 4. can you do that ?

- phi

you are almost done with this problem.

Looking for something else?

Not the answer you are looking for? Search for more explanations.