+1993 Which statement best reflects the solution(s) of the equation?
x / x−1 − 1 / x−2 = 2x−5 / x^2−3x+2
A - There are two solutions: x = 2 and x = 3.
B- There is only one solution: x = 3.
The solution x = 2 is an extraneous solution.
C -There is only one solution: x = 3.
The solution x = 1 is an extraneous solution.
D - There is only one solution: x = 4.
The solution x = 1 is an extraneous solution.
Solve for x:
x/(x - 1) - 1/(x - 2) = (2 x - 5)/(x^2 - 3 x + 2)
Multiply both sides by x^2 - 3 x + 2:
1 - x + x (x - 2) = 2 x - 5
Expand out terms of the left hand side:
x^2 - 3 x + 1 = 2 x - 5
Subtract 2 x - 5 from both sides:
x^2 - 5 x + 6 = 0
The left hand side factors into a product with two terms:
(x - 3) (x - 2) = 0
Split into two equations:
x - 3 = 0 or x - 2 = 0
Add 3 to both sides:
x = 3 or x - 2 = 0
Add 2 to both sides:
x = 3 or x = 2
x/(x - 1) - 1/(x - 2) ⇒ 2/(2 - 1) - 1/(2 - 2) = ∞^~
(2 x - 5)/(x^2 - 3 x + 2) ⇒ (2 2 - 5)/(2 - 3 2 + 2^2) = ∞^~:
So this solution is incorrect
x/(x - 1) - 1/(x - 2) ⇒ 3/(3 - 1) - 1/(3 - 2) = 1/2
(2 x - 5)/(x^2 - 3 x + 2) ⇒ (2 3 - 5)/(2 - 3 3 + 3^2) = 1/2:
So this solution is correct
The solution is: x = 3 The answer is "B"
Solve for x:
x/(x - 1) - 1/(x - 2) = (2 x - 5)/(x^2 - 3 x + 2)
Multiply both sides by x^2 - 3 x + 2:
1 - x + x (x - 2) = 2 x - 5
Expand out terms of the left hand side:
x^2 - 3 x + 1 = 2 x - 5
Subtract 2 x - 5 from both sides:
x^2 - 5 x + 6 = 0
The left hand side factors into a product with two terms:
(x - 3) (x - 2) = 0
Split into two equations:
x - 3 = 0 or x - 2 = 0
Add 3 to both sides:
x = 3 or x - 2 = 0
Add 2 to both sides:
x = 3 or x = 2
x/(x - 1) - 1/(x - 2) ⇒ 2/(2 - 1) - 1/(2 - 2) = ∞^~
(2 x - 5)/(x^2 - 3 x + 2) ⇒ (2 2 - 5)/(2 - 3 2 + 2^2) = ∞^~:
So this solution is incorrect
x/(x - 1) - 1/(x - 2) ⇒ 3/(3 - 1) - 1/(3 - 2) = 1/2
(2 x - 5)/(x^2 - 3 x + 2) ⇒ (2 3 - 5)/(2 - 3 3 + 3^2) = 1/2:
So this solution is correct
The solution is: x = 3 The answer is "B"
