Question

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

Answer 1

Ultimately, you're solving for `x`. Firstly, add all the terms and equate the equation to `0`. Then, using quadratic formula, solve for `x`.

Explanation:

Solving implies we determine the value of the variable. To do this, we must isolate and solve for `x`.

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

Before we do that, let's simplify the left side. To do this, let's make the `6` have the same denominator as the other term.

`[6xx (x-1)/(x-1) ] + 1/(x-1) = 6/(x-2)`

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

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

Now we can add the two terms.

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

`(6x-5)/(x-1) = 6/(x-2)`

Now we can cross-multiply. Cross-multiplying is done by:

`(6x-5)(x-2)=(6)(x-1)`

Now we simplify both sides. On the left side, we have to expand the brackets.

`6x^2-12x-5x+10=6x-6`

`6x^2-17x+10=6x-6`

Now, we are going to bring the terms to one side and simplify again. We are also going to equate the expression to `0`.

`6x^2-17x+10-6x+6=0`

`6x^2-23x+16=0`

Now we are going to factor the equation. To avoid any complications, we will just use the quadratic formula.

`x = (-b+-sqrt(b^2-4ac))/(2a)`

`x = (-[-23]+-sqrt([-23]^2-4[6][16]))/(2[6])`

Simplify.

`x = (23+-sqrt(145))/(12)`

Now we solve for `x`. Remember there will be two answers because of the `+-`.

`x (+) ~~ 2.92`

`x (-) ~~ 0.91`

We can double check our work by graphing the function and finding the zeros.

Hope this helps :)