Question

What is the solution to the following rational equation?

Answer 1

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

Answer 2

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

Answer 3

@Hero

Answer 4

Can you factor each of the denominators ?

Answer 5

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

Answer 6

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

Answer 7

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)

Answer 8

okay

Answer 9

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 ?

Answer 10

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

Answer 11

I don't understand...

Answer 12

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
\[(x-3)\cdot \frac{x}{(x-3)(x+3)} - (x-3)\cdot\frac{1}{(x-3)} = (x-3)\cdot\frac{1}{4(x-3)} \]

Answer 13

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

Answer 14

what do you get ?

Answer 15

1?

Answer 16

\[ (x-3)\cdot \frac{x}{(x-3)(x+3)} - (x-3)\cdot\frac{1}{(x-3)} = (x-3)\cdot\frac{1}{4(x-3)} \]

what is the first term simplify to?

Answer 17

x/(x+3)

Answer 18

yes. now the next term ?

Answer 19

do I distribute the negative?

Answer 20

first cancel, then distribute the -1

Answer 21

okay, so it would be -1

Answer 22

yes
now that last term

Answer 23

1/4

Answer 24

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 ?

Answer 25

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

Answer 26

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 ?

Answer 27

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

Answer 28

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

Answer 29

5/4?

Answer 30

yes

you now have
\[ \frac{x}{x+3} = \frac{5}{4} \]

any ideas what to try next ?

Answer 31

no

Answer 32

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 ?

Answer 33

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

Answer 34

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

Answer 35

@phi

Answer 36

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 ?

Answer 37

no...

Answer 38

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

Answer 39

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

Answer 40

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) )

Answer 41

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

Answer 42

yes.

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

Answer 43

do you agree that (x+3)/(x+3) is 1 ?

so
\[ \frac{x(x+3)}{(x+3) } = x \cdot 1 = x \]

Answer 44

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

I would multiply both sides by 4. can you do that ?

Answer 45

you are almost done with this problem.