Question

How do you solve `4 /(x+1 )+ 3 /(x - 4) = 2 /( x +1)`?

Answer 1

`x=1`

Explanation:

Assuming you want to solve for x, find the common denominator for the left side, `(x+1)(x-4)` and combine them into a single rational equation. Since `4/(x+1)` and `3/(x-4)` combine, it turns into `(7x-13)/((x+1)(x-4))` by multiplying the numerator `4*(x-4)` and `3*(x+1)`, adding them and combining like terms . You want to get rid of any denominators in the problem to make it simpler, so multiply that common denominator to both sides of the equation,
`(7x-13)/((x+1)(x-4))*(x+1)(x+4)` and same with the right side. On the left the common denominator gets cancelled so you're left with `7x-13`. On the right the `(x+1)` gets cancelled and you have `2(x-4)`. Multiply that and overall you have `7x-13=2x-8` left. then bring the `x's` on one side and the rest on the other side. `7x-2x=-8+13` , add and finally you have, `5x=5`, `x=1`. To check you can plug `x` in the original equation.

Answer 2

`x=1`

Explanation:

`4/(x+1)+3/(x-4)=2/(x+1)`

Our strategy will be to eliminate quotients.

This equation will look a lot less scary if we multiply both sides by `x+1`.

`4+(3(x+1))/(x-4)=2`

Now subtract 4 from both sides of the equation.

`(3(x+1))/(x-4)=-2`

Now multiply both side s of the equation by `x-4`.

`3(x+1)=-2(x-4)`

Hey! No more quotients! Now apply the distributive property.

`3x+3=-2x+8`

Add `2x` to both sides of this equation.

`5x+3=8`

Subtract 3 from both sides of this equation.

`5x=5`

Divide both sides of this equation by 5.

`x=1`

We can check our answer.

`4/(1+1)+3/(1-4)` =?= `2/(1+1)`

`4/2+3/-3` =?= `2/2`

`2-1` =?= `1`

`1=1`

Wadya know? Algebra WORKS!