6. Solve the equation. Check for extraneous solutions.
2/x^2-1=4/x+1
2/(x^2 - 1) = 4/(x + 1)
Rewrite as:
2/(x^2 - 1) = 4/(x + 1)*(x - 1)/(x - 1)
this step makes no sense
It should be
2/((x + 1)*(x - 1)) = 4/(x+1)
1/(x-1) = 2 (for x≠1)
x-1 = 1/2
x = 3/2
Apologies for the mistake in the previous solution. Let's correct it:
Given equation:
2/(x^2 - 1) = 4/(x + 1)
Rewrite the equation:
2/((x + 1)(x - 1)) = 4/(x + 1)
Simplify the expression:
1/(x - 1) = 2
Now solve for x:
x - 1 = 1/2
x = 1/2 + 1
x = 3/2
Now, we need to check if x = 3/2 is an extraneous solution by substituting it back into the original equation:
2/((3/2)^2 - 1) = 4/(3/2 + 1)
2/(9/4 - 1) = 4/(3/2 + 1)
2/(9/4 - 4/4) = 4/(3/2 + 1)
2/(5/4) = 4/(5/2)
2 * 4/5 = 4 * 2/5
8/5 = 8/5
Since the equation holds true when x = 3/2, there are no extraneous solutions, and the solution to the equation is x = 3/2.
To solve the equation, we first need to get a common denominator:
2/(x^2 - 1) = 4/(x + 1)
Rewrite as:
2/(x^2 - 1) = 4/(x + 1)*(x - 1)/(x - 1)
Now our equation becomes:
2/(x^2 - 1) = 4(x - 1)/((x + 1)(x - 1))
Simplify:
2/(x^2 - 1) = 4(x - 1)/(x^2 -1)
Now, cross multiply:
2(x^2 - 1) = 4(x - 1)
2x^2 - 2 = 4x - 4
2x^2 - 4x + 2 = 0
This is a quadratic equation, we can solve using the quadratic formula:
x = [-(-4) +/- sqrt((-4)^2 - 4*2*2)] / 2*2
x = [4 +/- sqrt(16 - 16)] / 4
x = [4 +/- sqrt(0)] / 4
x = [4 +/- 0] / 4
There seems to be only one solution:
x = 1
We need to check for extraneous solutions by substituting x = 1 back into the original equation:
2/1^2 - 1 = 4/1 + 1
2/0 = 4/2
No solution can be divided by zero, so x = 1 is an extraneous solution.
Therefore, the original equation has no real solutions.