Question

How do you solve `\frac { 1} { x - 2} + \frac { 6} { x } = \frac { - 12} { x ^ { 2} - 2x }`?

Answer 1

There is no solution.
`x !=0`

Explanation:

With algebraic fractions, the first step is to factorise the denominators.

`1/(x-2) +6/x = (-12)/(x^2-2x)`

`1/(x-2) +6/x = (-12)/(x(x-2))`

Lets' identify invalid values for `x` first.

A denominator may NOT be equal to `0`
`x!= 0 and x !=2`

It is an equation, so you can get rid of the denominators by multiplying each term by the LCM of the denominators.
In this case it is `color(red)(x(x-2))`

`(color(red)(x(x-2))xx1)/((x-2)) +(color(red)(x(x-2))xx6)/x = (-12xxcolor(red)(x(x-2)))/(x(x-2))`

Cancel the like factors:

`(color(red)(xcancel((x-2)))xx1)/cancel((x-2)) +(color(red)(cancelx(x-2))xx6)/cancelx = (-12xxcancel(color(red)(x(x-2))))/cancel((x(x-2)))`

This leaves us without any denominators.

`x +6(x-2) = -12`

`x +6x-12 = -12`

`7x = -12+12`

`7x =0`

`x =0`

This is an invalid solution, which means that there is no solution to this equation.