Question

Solve 5 over x minus 5 equals the quantity of x over x minus 5, minus five fourths for x and determine if the solution is extraneous or not

Answer 1

5/(x-5)=x/(x-5)-5/4.
Multiply through by 4(x-5):
20=4x-5(x-5).
20=4x-5x+25=25-x, x=5.
This is an extraneous solution because when x=5, the denominators become zero so division by zero occurs, and the result is meaningless.

Answer 2

ANSWER

The solution

`x = 5`
is extraneous.

Step-by-step explanation

The given equation is


`(5)/(x - 5) = (x)/(x - 5) - (5)/(4)`

We multiply the above equation with the least common multiple of denominators which is

`4(x - 5)`


This implies that,


`4(x - 5) * (5)/(x - 5) = 4(x - 5) * (x)/(x - 5) - (5)/(4) * 4(x - 5)`



This simplifies to


`20 = 4x- 5x + 25`


`20 - 25 = 4x - 5x`


This implies that,



`- 5 = - x`



Therefore

`x = 5`


But this solution does not satisfy the original equation.




`(5)/(5 - 5) \\e (5)/(5 - 5) - (5)/(4)`



`(5)/(0) \\e (5)/(0) - (5)/(4)`


Hence

`x = 5`
is an extraneous solution.