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.

EXPLANATION

The given equation is

$\frac{5}{x - 5} = \frac{x}{x - 5} - \frac{5}{4}$

We multiply the above equation with the least common multiple of denominators which is
$4(x - 5)$


This implies that,

$4(x - 5) \times \frac{5}{x - 5} = 4(x - 5) \times \frac{x}{x - 5} - \frac{5}{4} \times 4(x - 5)$



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.



$\frac{5}{5 - 5} \ne \frac{5}{5 - 5} - \frac{5}{4}$


$\frac{5}{0} \ne \frac{5}{0} - \frac{5}{4}$


Hence
$x = 5$
is an extraneous solution.