Please solve and check all proposed solutions. Show work for solving and checking:
(x-2)/(x-1) + (2)/(x^2-1) = 0
(x-2)/(x-1) + (2)/(x^2-1) = 0
(x-2)/(x-1) + (2)/((x-1)(x+1)) = 0
Take out the common factor 1/(x-1)
(1/(x-1)((x-2)+2/(x+1)) = 0
Solve for (x-2)+2/(x+1) = 0
to give x=0;
The common factor of 1/(x-1) tells us that x=1 is NOT a solution, because it would make the expression on the left side indefinite.
Another way to do the problem is to transpose one of the terms to the right-hand-side and cross multiply. Solve the resulting equation to get 0 and 1 as roots. Reject 1 as a root when back-substituted into the original equation gives infinite values.
To solve this equation, we'll follow these steps:
Step 1: Factor the denominator (x^2 - 1).
Step 2: Find the least common denominator (LCD).
Step 3: Multiply every term on both sides of the equation by the LCD.
Step 4: Simplify the equation.
Step 5: Solve for x.
Step 6: Check the solutions.
Let's begin:
Step 1: Factor the denominator(x^2 - 1).
The denominator can be factored as (x - 1)(x + 1).
Step 2: Find the least common denominator (LCD).
In this equation, the LCD is (x - 1)(x + 1).
Step 3: Multiply every term on both sides of the equation by the LCD.
(x - 2)/(x - 1) * (x + 1)(x - 1) + (2)/(x^2 - 1) * (x + 1)(x - 1) = 0
After multiplication, we get:
(x - 2)(x + 1) + 2 = 0
Step 4: Simplify the equation.
To simplify, let's multiply (x -2) by (x + 1) using the distributive property.
x^2 - 2x + x - 2 + 2 = 0
x^2 - x = 0
Step 5: Solve for x.
To solve this quadratic equation, we'll set it equal to zero and factor it.
x(x - 1) = 0
This equation can be solved by setting each factor equal to zero:
x = 0 or x - 1 = 0
So, we find two possible solutions: x = 0 and x = 1.
Step 6: Check the solutions.
To check the solutions, substitute them back into the original equation and see if both sides are equal.
Checking x = 0:
Left side = [(0 - 2)/(0 - 1)] + [(2)/(0^2 - 1)] = (-2)/(-1) + (2)/(-1) = 2 - 2 = 0
Right side = 0 (since 0 + 0 = 0)
Checking x = 1:
Left side = [(1 - 2)/(1 - 1)] + [(2)/(1^2 - 1)] = (-1)/0 + 2/0
Since division by zero is undefined, x = 1 is not a valid solution.
Therefore, the only valid solution is x = 0.