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. (1 point)

Answer 1

At x=5 it is extraneous

and at
`x=(4)/(5)` is a verified solution.

Explanation:

Given : Expression
`(5)/(x-5)=(x)/(x-5)-(5)/(4)x`

To determine : The solution is extraneous or not?

Solution :

We solve the given expression

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

Taking LCM

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

(x-5) cancel from both sides,

`5=(4x-5x^2+25x)/(4)`

`20=29x-5x^2`

`5x^2-29x+20=0`

Apply middle term split,

`5x^2-25x-4x+20=0`

`5x(x-5)-4(x-5)=0`

`(x-5)(5x-4)=0`

`\text{Either }(x-5)=0\text{ or }(5x-4)=0`

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

Extraneous is when we get the solution mathematically correct.

Substituting x = 5 gives denominators of 0, which is extraneous.

Substituting
`x=(4)/(5)` gives a valid equation, so this is the verified solution.

Answer 2

Assuming the equation is:
5/(x-5) = x/(x-5) - 5x/4
We first multiply by the LCD: 4(x-5)
20 = 4x - 5x(x-5)
20 = 4x - 5x^2 + 25x
5x^2 - 29x + 20 = 0
(5x - 4)(x - 5) = 0
x = 4/5, 5
Substituting x = 5 gives denominators of 0, which is extraneous.
Substituting x = 4/5 gives a valid equation, so this is the only correct solution.