Question

How do you solve the rational equation `1/(x-1)+3/(x+1)=2`?

Answer 1

`x = 0 , x= 2`

Explanation:

Step 1 : Identify the restricted value.

This is done by set the denominator equal to zero like this

`x-1= 0 <=> x= 1`
`x+1 = 0 <=> x = -2`

The idea of restricted value, is to narrow down what value our variable can't be (aka domain)

Step 2: Multiply the equation by `color(red)(LCD)`

`1/(x-1) + 3/(x+1) = 2`

`color(red)((x-1)(x+1))(1/(x-1)) +color(red)( (x-1)(x+1))(3/(x+1)) = 2color(red)((x-1)(x+1)`

`color(red)(cancel(x-1)(x+1))(1/cancel(x-1)) +color(red)( (x-1)cancel(x+1))(3/cancel(x+1)) = 2color(red)((x-1)(x+1)`

`(x+1) + 3(x-1) = 2(x-1)(x+1)`

Step 3: Multiply and combine like terms

`x+1+3x -3 = 2(x^2-x+x-1)`

`4x -2 = 2(x^2 -1)`

`4x -2 = 2x^2 -2`

`0 =2x^2-4x`

Step 4: Solve the quadratic equation

`2x^2 -4x = 0`
`2x(x-2) = 0`

`2x = 0 => color(blue)(x = 0)`

`x-2 = 0 => color(blue)(x = 2)`

Step 5 Check your solution ..

Check to see if the answer from Step 4 is the same a restricted value.

If it's not, the the solution is `x = 0, x= 2`