Answer:
To solve the equation:
3/(k+1) - 1/(2k+2) = 5
We need to first simplify the left-hand side of the equation by finding a common denominator. The common denominator is (k+1)(2k+2) = 2(k+1)(k+1).
So we have:
[3(2k+2) - (k+1)] / [2(k+1)(k+1)] = 5
Simplifying the numerator, we get:
[6k + 6 - k - 1] / [2(k+1)(k+1)] = 5
Combining like terms, we get:
[5k + 5] / [2(k+1)(k+1)] = 5
Multiplying both sides by the denominator, we get:
5k + 5 = 10(k+1)(k+1)
Expanding the right-hand side, we get:
5k + 5 = 10k^2 + 20k + 10
Subtracting 5k + 5 from both sides, we get:
0 = 10k^2 + 15k + 5
Dividing both sides by 5, we get:
0 = 2k^2 + 3k + 1
Factorizing the equation, we get:
0 = (2k + 1) (k + 1)
Solving for k, we get:
k = -1 or k = -1/2
Therefore, the solutions to the equation are k = -1 and k = -1/2.