Question

How do you solve `-\frac { 3} { 2v - 12} + 2= - \frac { 3} { v - 6}`?

Answer 1

Make the denominators common.

`v=5.2`

Explanation:

First make the denominators common. The common denominator is `2v-12` so each part must be multiplied by a fraction (equal to 1) that will make the denominator `2v-12`.

`−3/(2v−12)+2*(2v−12)/(2v−12)=−3/(v−6)*2/2`

this gives us `−3/(2v−12)+(4v−24)/(2v−12)=−6/(2v−12)`

combine to get `(-3+4v-24)/(2v−12)=-6/(2v−12)`

add to get `(4v-27)/(2v−12)=-6/(2v−12)`

now cross multiply `(4v-27)(2v−12)=-6(2v−12)`

divide both sides by `(2v−12)` to get `4v-27=-6`

add `27` to both sides; `4v=21`

`v=5.2`

Answer 2

`v = 21/4 = 5 1/4 = 5.25`

`21/4` is probably the best answer to give.

Explanation:

`-3/(2v-12)+2 = -3/(v-6)`

`-3/(2(v-6))+2 = -3/(v-6)`

Re-arrange the equation:

`3/((v-6)) -3/(2(v-6)) = -2`

Make a common denominator and make equivalent fractions.

`3/((v-6))xx 2/2 -3/(2(v-6)) = -2`

`(6 -3)/(2(v-6)) = -2`

Simplify and cross-multiply.

`6-3 = -2(2(v-6))`

`3 = -4(v-6)`

`3 = -4v+24`

`4v = 21`

`v = 21/4`

or, `v = 5.25`

Answer 3

`v=5 1/4`

Explanation:

If you make all the denominators (bottom value) the same then you only need to consider the top values (numerators) to get the same answer.

Lets get rid of the negatives to start with.

Add `3/(2v-12)` to both sides: moves it to the right of = and changes its sign.

Add `3/(v-6)` to both sides: moves it to the left of = and changes its sign.

`3/(v-6)+2=3/(2v-12) larr" All positive"`

We need to look for a common structure of the denominator that they can all share.

Note that `2v-12" "->" "2(v-6)`
This is similar to the denominator of `3/(v-6)` so let the common denominator be `2v-12`

Also note that 2 is the same as `2/1`. This is not normally shown.

`color(green)([3/(v-6)color(red)(xx1)]+[2color(red)(xx1)]=3/(2v-12)`

`color(green)([3/(v-6)color(red)(xx2/2)]+[2/1color(red)(xx(2v-12)/(2v-12))]=3/(2v-12)`

`6/(2v-12)+(4v-24)/(2v-12)=3/(2v-12)`

If this is true then it is also true that:

`6+4v-24=3 larr" Only the numerators"`
..........................................................................................
For the 'purists'
`6/(2v-12)+(4v-12)/(2v-12)=3/(2v-12)`

Multiply all of both sides by `2v-12` giving:

`6+4v-24=3 larr" Only the numerators"`
................................................................................................

`4v-18=3`

Add 18 to both sides

`4v=21`

divide both sides by 4

`v=21/4=5 1/4`