3b-2/b+1=4-b+2/b-1
To solve the equation 3b - 2 / b + 1 = 4 - b + 2 / b - 1, we need to simplify and rearrange the equation to isolate the variable b. Here are the steps to solve this equation:
Step 1: Simplify the equation by using the distributive property. Apply the distributive property to the terms containing b in the numerators.
3b - 2 / b + 1 = 4 - b + 2 / b - 1 becomes
(3b(b - 1) - 2) / (b + 1) = (4(b - 1) - b + 2) / (b - 1)
Step 2: Clear the fractions by multiplying both sides of the equation by the least common multiple (LCM) of the denominators, which is (b + 1)(b - 1).
[b + 1)(b - 1)][(3b(b - 1) - 2) / (b + 1)] = [(b + 1)(b - 1)][(4(b - 1) - b + 2) / (b - 1)]
This eliminates the fractions, giving us:
(3b(b - 1) - 2)(b - 1) = (4(b - 1) - b + 2)(b + 1)
Step 3: Expand and simplify both sides of the equation.
Expanding both sides gives us:
3b(b^2 - b - b + 1) - 2(b - 1) = 4(b^2 - b - 1) - b(b + 1) + 2(b + 1)
Simplifying further, we have:
3b(b^2 - 2b + 1) - 2b + 2 = 4b^2 - 4b - 4 - b^2 - b + 2b + 2
Step 4: Simplify the equation by combining like terms.
Expanding and simplifying the equation further, we get:
3b^3 - 6b^2 + 3b - 2b + 2 = 4b^2 - 4b - 4 - b^2 - b + 2b + 2
Combining like terms on both sides of the equation gives us:
3b^3 - 6b^2 + b^2 + 3b - 2b - 2b + 4b - 4b^2 - b + b^2 + 2 = 0
This simplifies to:
3b^3 - 5b^2 + 2b - 2 = 0
Step 5: Solve the simplified equation.
At this point, the equation is a cubic equation in terms of b. To solve it, you can use various methods, such as factoring, synthetic division, or numerical methods. However, this specific cubic equation doesn't appear to have an easily factorizable solution.
One way to find an approximate solution is by using a numerical method, such as graphing the equation or using a calculator or computer program to find the value(s) of b that satisfy the equation.
Alternatively, you can use software or an online calculator capable of solving cubic equations to find the solutions directly.
Please note that the exact solutions may involve irrational numbers or complex numbers, depending on the nature of the equation.