Question

How do you solve `x/(2x-6)=2/(x-4)`?

Answer 1

Restrict the domain so that the solutions do not cause division by zero in the original equation.
Multiply both sides by both denominators.
Solve the resulting quadratic.
Check.

Explanation:

Given: `x/(2x-6)=2/(x-4)`

Restrict the values of x so that any solutions that would cause division by zero are discarded:

`x/(2x-6)=2/(x-4);x!=3,x!=4`

Multiply both sides of the equation by `(2x-6)(x-4)`

`(2x-6)(x-4)x/(2x-6)=(2x-6)(x-4)2/(x-4);x!=3,x!=4`

Please observe how the factors cancel:

`cancel(2x-6)(x-4)x/cancel(2x-6)=(2x-6)cancel(x-4)2/cancel(x-4);x!=3,x!=4`

Here is the equation with the cancelled factor removed:

`(x-4)x=(2x-6)2;x!=3,x!=4`

Use the distributive property on both sides:

`x^2-4x=4x-12;x!=3,x!=4`

We can put the quadratic into standard form by adding `12-4x` to both sides:

`x^2-8x+12=0;x!=3,x!=4`

This factors and the restrictions can be dropped:

`(x - 2)(x-6)=0`

`x = 2 and x = 6`

Check:

`2/(2(2)-6)=2/(2-4)`
`6/(2(6)-6)=2/(6-4)`

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

`-1=-1`
`1=1`

This checks.