Find the value of x: (2x-1)/(x+1)=2x/(x-1)-(5/x)
cross multiply and simplify
multiply each term by x(x+1)(x-1) , the LCD
x(x-1)(2x-1) = x(x+1)(2x) - 5(x+1)(x-1)
x(2x^2 -3x + 1) = x(2x^2 + 2x) - 5(x^2 - 1)
2x^3 - 3x^2 + x = 2x^3 + 2x^2 - 5x^2 + 5
x = 5
To find the value of x in the given equation: (2x-1)/(x+1) = 2x/(x-1) - (5/x), we can follow these steps:
Step 1: Clear the denominators
Multiply both sides of the equation by the least common denominator (LCD) of (x+1), (x-1), and x to eliminate the fractions.
The LCD of (x+1), (x-1), and x is x(x+1)(x-1).
So, multiply both sides of the equation by x(x+1)(x-1):
x(x+1)(x-1) * (2x-1)/(x+1) = x(x+1)(x-1) * [2x/(x-1) - (5/x)]
Step 2: Simplify the equation
On the left-hand side, x+1 in the denominator cancels out with (x+1) in the numerator, and (x-1) in the denominator cancels out with (x-1) in the numerator:
x(x-1)(2x-1) = x(x+1)(2x) - 5(x+1)(x-1)
Expanding both sides gives:
x(x^2 - x)(2x - 1) = x(x^2 + x)(2x) - 5(x^2 - 1)
Step 3: Simplify the equation further
Expand and collect like terms on both sides:
2x^4 - 5x^2 + 2x^3 - x^2 - 2x + x = 2x^4 + 2x^3 + 2x^3 + 2x^2 - 5x^2 - 5x^2 - 10x - 2x - 5
This simplifies to:
2x^4 + 2x^3 - 6x^2 - 12x - 5 = 2x^4 + 4x^3 - 5x^2 - 12x - 5
Step 4: Combine like terms
Bring all the terms to one side of the equation:
2x^3 - 6x^2 - x^4 = 0
Step 5: Factor the equation
Now, we need to factor the equation to find the potential values of x:
x^2(2x - 1) - x^2(6x - 1) = 0
Factor out the common term x^2:
x^2 (2x - 1 - 6x + 1) = 0
Simplify further:
x^2 (-4x) = 0
Step 6: Solve for x
Set each factor equal to zero and solve for x:
x^2 = 0 => x = 0
-4x = 0 => x = 0
Hence, the value of x that satisfies the equation is x = 0.