Question

How do you simplify `2/( x - 4) + 3/ ( x - 2) = (6x)/ ( x ^ { 2} - 6x + 8)`?

Answer 1

`x=-16`

Explanation:

First, you need to find the least common denominator, just like when you add a fraction with no variables.
The common denominator of `(2)/(x-4)` and `(3)/(x-2)` is `(x-4)` `(x-2)`.
If you factor the bottom of the right side of the equation, `x^2-6x+8`, you will also get the answer `(x-4)` `(x-2)`.

Now, multiply the numerator of each fraction by the part of the LCD not already in the denominator, like so:
`(2(x-2))/((x-2)(x-4))`+`(3(x-4))/((x-2)(x-4))`
You can now combine the fractions because they have the same denominator.
`((2(x-2))+(3(x-4)))/(x^2-6x+8)`
Multiply/expand with distribution.
`(5x-16)/(x^2-6x+8)`
Multiply both sides by the denominator to remove it.
`((5x-16)/(x^2-6x+8))(x^2-6x+8)=((6x)/(x^2-6x+8))(x^2-6x+8)`

Simplify
`5x-16=6x`

Add 16 to both sides.
`5x=6x+16`

Subtract 6x from both sides.
`5x-6x=6x+16-6x`

Simplify
`-x=16`

Divide by `-1` to isolate `x`.
`(-x)/(-1)=(16)/(-1)`

Simplify

`x=-16`

Answer 2

`color(magenta)(x=-16`

Explanation:

`2/(x-4)+3/(x-2)=(6x)/(x^2-6x+8)`

`:.2/(x-4)+3/(x-2)=(6x)/((x-2)(x-4))`

`:.(2(x-2)+3(x-4)=6x)/((x-2)(x-4))`

multiply both sides by`(x-2)(x-4)`

`:.2x-4+3x-12=6x`

`:.2x+3x-6x=4+12`

`:.-x=16`

`:.color(magenta)(x=-16`

~~~~~~~~~~~~~~~~~~

check:-

substitute`x=-16`

`:.2/((-16)-4)+3/((-16)-2)=(6(-16))/((-16)^2-6(-16)+8)`

`:.2/-20+3/-18=-96/(256+96+8)`

`:.-0.1+(-0.166666666)=-96/360`

`:.color(magenta)(-0.266666666=-0.266666666`