# Solve for n 1/(n+1)=1/(n+6)+(n-4)/(n^2+7n+6)

1n+1=1n+6+n-4n2+7n+6
Factor n2+7n+6 using the AC method.
Consider the form x2+bx+c. Find a pair of integers whose product is c and whose sum is b. In this case, whose product is 6 and whose sum is 7.
1,6
Write the factored form using these integers.
1n+1=1n+6+n-4(n+1)(n+6)
1n+1=1n+6+n-4(n+1)(n+6)
Find the LCD of the terms in the equation.
Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.
n+1,n+6,(n+1)(n+6)
The LCM is the smallest positive number that all of the numbers divide into evenly.
1. List the prime factors of each number.
2. Multiply each factor the greatest number of times it occurs in either number.
The number 1 is not a prime number because it only has one positive factor, which is itself.
Not prime
The LCM of 1,1,1 is the result of multiplying all prime factors the greatest number of times they occur in either number.
1
The factor for n+1 is n+1 itself.
(n+1)=n+1
(n+1) occurs 1 time.
The factor for n+6 is n+6 itself.
(n+6)=n+6
(n+6) occurs 1 time.
The factor for n+1 is n+1 itself.
(n+1)=n+1
(n+1) occurs 1 time.
The factor for n+6 is n+6 itself.
(n+6)=n+6
(n+6) occurs 1 time.
The LCM of n+1,n+6,n+1,n+6 is the result of multiplying all factors the greatest number of times they occur in either term.
(n+1)(n+6)
(n+1)(n+6)
Multiply each term by (n+1)(n+6) and simplify.
Multiply each term in 1n+1=1n+6+n-4(n+1)(n+6) by (n+1)(n+6) in order to remove all the denominators from the equation.
1n+1⋅((n+1)(n+6))=1n+6⋅((n+1)(n+6))+n-4(n+1)(n+6)⋅((n+1)(n+6))
Cancel the common factor of n+1.
Cancel the common factor.
1n+1⋅((n+1)(n+6))=1n+6⋅((n+1)(n+6))+n-4(n+1)(n+6)⋅((n+1)(n+6))
Rewrite the expression.
n+6=1n+6⋅((n+1)(n+6))+n-4(n+1)(n+6)⋅((n+1)(n+6))
n+6=1n+6⋅((n+1)(n+6))+n-4(n+1)(n+6)⋅((n+1)(n+6))
Simplify 1n+6⋅((n+1)(n+6))+n-4(n+1)(n+6)⋅((n+1)(n+6)).
Simplify each term.
Cancel the common factor of n+6.
Factor n+6 out of (n+1)(n+6).
n+6=1n+6⋅((n+6)(n+1))+n-4(n+1)(n+6)⋅((n+1)(n+6))
Cancel the common factor.
n+6=1n+6⋅((n+6)(n+1))+n-4(n+1)(n+6)⋅((n+1)(n+6))
Rewrite the expression.
n+6=n+1+n-4(n+1)(n+6)⋅((n+1)(n+6))
n+6=n+1+n-4(n+1)(n+6)⋅((n+1)(n+6))
Cancel the common factor of (n+1)(n+6).
Cancel the common factor.
n+6=n+1+n-4(n+1)(n+6)⋅((n+1)(n+6))
Rewrite the expression.
n+6=n+1+n-4
n+6=n+1+n-4
n+6=n+1+n-4
n+6=2n+1-4
Subtract 4 from 1.
n+6=2n-3
n+6=2n-3
n+6=2n-3
n+6=2n-3
Solve the equation.
Move all terms containing n to the left side of the equation.
Subtract 2n from both sides of the equation.
n+6-2n=-3
Subtract 2n from n.
-n+6=-3
-n+6=-3
Move all terms not containing n to the right side of the equation.
Subtract 6 from both sides of the equation.
-n=-3-6
Subtract 6 from -3.
-n=-9
-n=-9
Multiply each term in -n=-9 by -1
Multiply each term in -n=-9 by -1.
(-n)⋅-1=(-9)⋅-1
Multiply (-n)⋅-1.
Multiply -1 by -1.
1n=(-9)⋅-1
Multiply n by 1.
n=(-9)⋅-1
n=(-9)⋅-1
Multiply -9 by -1.
n=9
n=9
n=9
Solve for n 1/(n+1)=1/(n+6)+(n-4)/(n^2+7n+6)

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