Question

How do you solve `1 /(x-4) = 6/( x+6)`?

Answer 1

`x = 6`

Explanation:

Right from the start, you know that you need

`x - 4 !=0 implies x != 4`

and

`x + 6 != 0 implies x != -6`

Your goal here is to find the common denominator of the two fractions, multiply each of them accordingly to get this denominator, then focus solely on the numerators.

The common denominator of `(x-4)` and `(x+6)` is

`(x-4)(x+6)`

This means that you need to multiply the left-hand side of the equation by `1= ((x+6))/((x+6))` and the right-hand side of the equation by `1 = ((x-4))/((x-4))`.

This will get you

`1/(x-4) * ((x+6))/((x+6)) = 6/(x+6) * ((x-4))/((x-4))`

`(x+6)/((x-4)(x+6)) = (6 * (x-4))/((x-4)(x+6))`

Now you can focus solely on the numerators. You have

`x+6 = 6(x-4)`

`x + 6 = 6x - 24`

`-5x = -30 implies x = ((-30))/((-5)) = color(green)(6)`

Since `x=6 !in {4, -6}`, this will be a valid solution to the equation.

Do a quick check to make sure that the calculations are correct

`1/((6)-4) = 6/((6) + 6)`

`1/2 = 6/12color(white)(x)color(green)(sqrt())`