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

Jul 3, 2017

Make the denominators common.

$v = 5.2$

#### Explanation:

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

−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 $4 v - 27 = - 6$

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

$v = 5.2$

Jul 3, 2017

$v = \frac{21}{4} = 5 \frac{1}{4} = 5.25$

$\frac{21}{4}$ is probably the best answer to give.

#### Explanation:

$- \frac{3}{2 v - 12} + 2 = - \frac{3}{v - 6}$

$- \frac{3}{2 \left(v - 6\right)} + 2 = - \frac{3}{v - 6}$

Re-arrange the equation:

$\frac{3}{\left(v - 6\right)} - \frac{3}{2 \left(v - 6\right)} = - 2$

Make a common denominator and make equivalent fractions.

$\frac{3}{\left(v - 6\right)} \times \frac{2}{2} - \frac{3}{2 \left(v - 6\right)} = - 2$

$\frac{6 - 3}{2 \left(v - 6\right)} = - 2$

Simplify and cross-multiply.

$6 - 3 = - 2 \left(2 \left(v - 6\right)\right)$

$3 = - 4 \left(v - 6\right)$

$3 = - 4 v + 24$

$4 v = 21$

$v = \frac{21}{4}$

or, $v = 5.25$

Jul 3, 2017

$v = 5 \frac{1}{4}$

#### 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.

Add $\frac{3}{2 v - 12}$ to both sides: moves it to the right of = and changes its sign.

Add $\frac{3}{v - 6}$ to both sides: moves it to the left of = and changes its sign.

$\frac{3}{v - 6} + 2 = \frac{3}{2 v - 12} \leftarrow \text{ All positive}$

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

Note that $2 v - 12 \text{ "->" } 2 \left(v - 6\right)$
This is similar to the denominator of $\frac{3}{v - 6}$ so let the common denominator be $2 v - 12$

Also note that 2 is the same as $\frac{2}{1}$. 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)

$\frac{6}{2 v - 12} + \frac{4 v - 24}{2 v - 12} = \frac{3}{2 v - 12}$

If this is true then it is also true that:

$6 + 4 v - 24 = 3 \leftarrow \text{ Only the numerators}$
..........................................................................................
For the 'purists'
$\frac{6}{2 v - 12} + \frac{4 v - 12}{2 v - 12} = \frac{3}{2 v - 12}$

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

$6 + 4 v - 24 = 3 \leftarrow \text{ Only the numerators}$
................................................................................................

$4 v - 18 = 3$

$4 v = 21$
$v = \frac{21}{4} = 5 \frac{1}{4}$