# Solve for x 1/x+1/(x+4)=1/2.1

1x+1x+4=12.1
Divide 1 by 2.1.
1x+1x+4=0.476190‾
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.
x,x+4,1
Since x,x+4,1 contain both numbers and variables, there are four steps to find the LCM. Find LCM for the numeric, variable, and compound variable parts. Then, multiply them all together.
Steps to find the LCM for x,x+4,1 are:
1. Find the LCM for the numeric part 1,1,1.
2. Find the LCM for the variable part x1.
3. Find the LCM for the compound variable part x+4.
4. Multiply each LCM together.
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 x1 is x itself.
x1=x
x occurs 1 time.
The LCM of x1 is the result of multiplying all prime factors the greatest number of times they occur in either term.
x
The factor for x+4 is x+4 itself.
(x+4)=x+4
(x+4) occurs 1 time.
The LCM of x+4 is the result of multiplying all factors the greatest number of times they occur in either term.
x+4
The Least Common Multiple LCM of some numbers is the smallest number that the numbers are factors of.
x(x+4)
x(x+4)
Multiply each term by x(x+4) and simplify.
Multiply each term in 1x+1x+4=0.476190‾ by x(x+4) in order to remove all the denominators from the equation.
1x⋅(x(x+4))+1x+4⋅(x(x+4))=0.476190‾⋅(x(x+4))
Simplify 1x⋅(x(x+4))+1x+4⋅(x(x+4)).
Simplify each term.
Cancel the common factor of x.
Cancel the common factor.
1x⋅(x(x+4))+1x+4⋅(x(x+4))=0.476190‾⋅(x(x+4))
Rewrite the expression.
x+4+1x+4⋅(x(x+4))=0.476190‾⋅(x(x+4))
x+4+1x+4⋅(x(x+4))=0.476190‾⋅(x(x+4))
Cancel the common factor of x+4.
Factor x+4 out of x(x+4).
x+4+1x+4⋅((x+4)x)=0.476190‾⋅(x(x+4))
Cancel the common factor.
x+4+1x+4⋅((x+4)x)=0.476190‾⋅(x(x+4))
Rewrite the expression.
x+4+x=0.476190‾⋅(x(x+4))
x+4+x=0.476190‾⋅(x(x+4))
x+4+x=0.476190‾⋅(x(x+4))
2x+4=0.476190‾⋅(x(x+4))
2x+4=0.476190‾⋅(x(x+4))
Simplify 0.476190‾⋅(x(x+4)).
Apply the distributive property.
2x+4=0.476190‾⋅(x⋅x+x⋅4)
Simplify the expression.
Multiply x by x.
2x+4=0.476190‾⋅(x2+x⋅4)
Move 4 to the left of x.
2x+4=0.476190‾⋅(x2+4⋅x)
2x+4=0.476190‾⋅(x2+4⋅x)
Apply the distributive property.
2x+4=0.476190‾x2+0.476190‾(4x)
Multiply 4 by 0.476190‾.
2x+4=0.476190‾x2+1.904761‾x
2x+4=0.476190‾x2+1.904761‾x
2x+4=0.476190‾x2+1.904761‾x
Solve the equation.
Since x is on the right side of the equation, switch the sides so it is on the left side of the equation.
0.476190‾x2+1.904761‾x=2x+4
Move all terms containing x to the left side of the equation.
Subtract 2x from both sides of the equation.
0.476190‾x2+1.904761‾x-2x=4
Subtract 2x from 1.904761‾x.
0.476190‾x2-0.095238‾x=4
0.476190‾x2-0.095238‾x=4
Move 4 to the left side of the equation by subtracting it from both sides.
0.476190‾x2-0.095238‾x-4=0
Use the quadratic formula to find the solutions.
-b±b2-4(ac)2a
Substitute the values a=0.476190‾, b=-0.095238‾, and c=-4 into the quadratic formula and solve for x.
0.095238‾±(-0.095238‾)2-4⋅(0.476190‾⋅-4)2⋅0.476190‾
Simplify.
Simplify the numerator.
Raise -0.095238‾ to the power of 2.
x=0.095238‾±0.00907029-4⋅(0.476190‾⋅-4)2⋅0.476190‾
Multiply 0.476190‾ by -4.
x=0.095238‾±0.00907029-4⋅-1.904761‾2⋅0.476190‾
Multiply -4 by -1.904761‾.
x=0.095238‾±0.00907029+7.619047‾2⋅0.476190‾
x=0.095238‾±7.628117912⋅0.476190‾
x=0.095238‾±7.628117912⋅0.476190‾
Multiply 2 by 0.476190‾.
x=0.095238‾±7.628117910.952380‾
Simplify 0.095238‾±7.628117910.952380‾.
x=0.1±2.9
x=0.1±2.9
The final answer is the combination of both solutions.
x=3,-2.8
x=3,-2.8
Solve for x 1/x+1/(x+4)=1/2.1