How do you solve #(x+1)/(x-1)=2/(2x-1)+2/(x-1)#?

Question

Nathan Axel · Accepted Answer

#x=3/2#

We have an equation, #{x+1}/{x-1}=2/{2x-1}+2/{x-1}#

For simplicity, which you might find to your liking, I'll be taking #x-1# as #y#, so #y=x-1#, turning the equation to

#{x+1}/y=2/{y+x}+2/y# (Note, 2x-1=x+x-1=x+y)

Now, we have two terms, one on either side of the equation to have the same denominator, being #{x+1}/y# and #2/y#. so what I'll be doing is that I'll subtract on both sides by #2/y#, hence removing the #2/y# term from the right side of the equation and bringing it to the left. That gives us

#{x+1-2}/y=2/{x+y}# Since the terms on the left-hand side already had the same denominator, I already grouped them together.

By solving the sum on the numerator of the left-hand side, we get #{x-1}/y=2/{x+y}#

Now, what is the numerator of the left-hand side similar to? If you go back up to the top of the answer, you'll see that I equated #y=x-1=>{x-1}/y=1#

So that makes the left-hand side of the equation as #1=2/{x+y}#

Multiplying by #x+y#, gives us #x+y=2#

Now, we know that #y=x-1#, so substituting back into the equation, give us #x+x-1=2=>2x-1=2#

Now, I'm sure, figuring out the solution will be simple.

Eva Abramson · Answer

undefined To solve the equation (x+1)/(x-1) = 2/(2x-1) + 2/(x-1), we can follow these steps:

1. Start by finding a common denominator for the fractions on the right side of the equation. The common denominator will be (x-1)(2x-1).

2. Multiply both sides of the equation by the common denominator to eliminate the fractions.

3. Simplify the equation by distributing and combining like terms.

4. Solve for x by isolating the variable on one side of the equation.

5. Check the solution by substituting the value of x back into the original equation to ensure it satisfies the equation.

Here are the steps in detail:

1. The common denominator is (x-1)(2x-1).

2. Multiply both sides of the equation by (x-1)(2x-1):

(x+1)(x-1)(2x-1) = 2(x-1)(x-1) + 2(2x-1)(x-1)

3. Simplify the equation:

(x+1)(2x-1) = 2(x-1)^2 + 2(2x-1)(x-1)

Expand and simplify both sides:

2x^2 - x + 2x - 1 = 2(x^2 - 2x + 1) + 2(2x^2 - 3x + 1)

4. Continue simplifying:

2x^2 + x - 1 = 2x^2 - 4x + 2 + 4x^2 - 6x + 2

Combine like terms:

2x^2 + x - 1 = 6x^2 - 6x + 4

5. Rearrange the equation to have all terms on one side:

0 = 6x^2 - 2x^2 + 6x - x + 4 + 1

Combine like terms:

0 = 4x^2 + 5x + 5

6. Set the equation equal to zero and attempt to factor or use the quadratic formula to solve for x. However, in this case, the equation cannot be factored easily, so we will use the quadratic formula:

x = (-b ± √(b^2 - 4ac)) / (2a)

For our equation, a = 4, b = 5, and c = 5.

Substitute these values into the quadratic formula:

x = (-5 ± √(5^2 - 4(4)(5))) / (2(4))

Simplify:

x = (-5 ± √(25 - 80)) / 8

x = (-5 ± √(-55)) / 8

Since we have a negative value under the square root, the equation has no real solutions. Therefore, there are no values of x that satisfy the original equation.

Thus, the solution to the equation (x+1)/(x-1) = 2/(2x-1) + 2/(x-1) is no solution.