Final answer:
To solve the equation 5/(x-5) = x/(x-5) - 5/4, we simplify and find that x = 25/9. The solution is not extraneous.
Step-by-step explanation:
To solve the equation, let's first simplify it:
5/(x-5) = x/(x-5) - 5/4
We can multiply both sides of the equation by (x-5) to eliminate the denominator:
5 = x - 5 - (5/4)(x-5)
Now, let's simplify and solve for x:
5 = x - 5 - (5/4)x + 25/4
Combining like terms, we have:
(9/4)x = 25/4
Multiplying both sides by 4/9, we get:
x = 25/9
To determine if the solution is extraneous, we need to check if it satisfies the original equation:
5/(25/9-5) = (25/9)/(25/9-5) - 5/4
Simplifying, we find that both sides of the equation are equal, so the solution x = 25/9 is not extraneous.