5/(x+6)+ 2/(x^2+7x+6)= 3/(x+1)
Rewrite x^2 +7x +1 as (x+6)(x+1)
Multiply both sides of the equation by that expression. You get
5(x+1) + 2 = 3(x+6)
Multiply out the parentheses and combine similar terms. You will be left with a simple linear equation for x.
To solve the given equation, we can follow these steps:
1. Start by multiplying both sides of the equation by the common denominator, which is (x + 6)(x + 1). This will eliminate the fractions and make the equation easier to solve.
(x + 6)(x + 1) * (5/(x + 6)) + (x + 6)(x + 1) * (2/(x^2 + 7x + 6)) = (x + 6)(x + 1) * (3/(x + 1))
Simplifying each term separately:
5(x + 1) + 2 = 3(x + 6)
2. Now, distribute the numbers and simplify each side of the equation:
5x + 5 + 2 = 3x + 18
Combine like terms:
5x + 7 = 3x + 18
3. Next, get all the variables on one side and the constants on the other side. Subtract 3x from both sides:
5x - 3x + 7 = 3x - 3x + 18
Simplify:
2x + 7 = 18
4. Subtract 7 from both sides to isolate the variable:
2x + 7 - 7 = 18 - 7
Simplify:
2x = 11
5. Finally, divide both sides by 2 to solve for x:
(2x) / 2 = 11 / 2
Simplify:
x = 11 / 2
Therefore, the solution to the equation 5/(x + 6) + 2/(x^2 + 7x + 6) = 3/(x + 1) is x = 11 / 2.