The sum of the reciprocals of two consecutive even integers is 9/40. This can be represented by the equation shown. 1/x+1/x+2=9/40
use the rational equation to determine the integers. show all work
First, we need to clear the denominators by multiplying both sides of the equation by the common denominator of x(x+2):
1/x + 1/(x+2) = 9/40
40x(x+2)(1/x) + 40x(x+2)(1/(x+2)) = 40x(x+2)(9/40)
40(x+2) + 40x = 9x(x+2)
Distribute the 40 on the left:
40x + 80 + 40x = 9x(x+2)
Combine like terms:
80x + 80 = 9x^2 + 18x
Move all terms to one side:
9x^2 - 62x - 80 = 0
Now we can factor this quadratic equation:
(9x + 8)(x - 10) = 0
Setting each factor equal to zero gives us two possible solutions:
9x + 8 = 0 or x - 10 = 0
Solving for x gives us:
x = -8/9 or x = 10
We can disregard the negative solution since we are looking for even integers. Thus, the two consecutive even integers are:
x = 10
x+2 = 12
Therefore, the two consecutive even integers are 10 and 12.
your wrong
I apologize for the error in my previous answer. Here is the correct solution:
The equation is:
1/x + 1/(x+2) = 9/40
To solve for x, we can cross multiply and simplify:
40(x+2) + 40x = 9x(x+2)
80x + 80 = 9x^2 + 18x
9x^2 - 62x - 80 = 0
We can factor the quadratic equation:
(9x + 4)(x - 10) = 0
Setting each factor equal to zero gives us two possible solutions:
9x + 4 = 0 or x - 10 = 0
Solving for x gives us:
x = -4/9 or x = 10
Since we are looking for consecutive even integers, x must be even. Thus, we can disregard the solution x = -4/9. The only valid solution is x = 10.
Therefore, the two consecutive even integers are:
x = 10
x+2 = 12
Hence, the two consecutive even integers that satisfy the given condition are 10 and 12.
To determine the integers, we will first solve the rational equation step by step.
Given equation: 1/x + 1/(x + 2) = 9/40
Step 1: Multiply through by the common multiple of the denominators, which is 40x(x + 2), to eliminate the fractions.
40(x + 2) + 40x = 9x(x + 2)
Step 2: Distribute and simplify both sides of the equation.
40x + 80 + 40x = 9x^2 + 18x
80x + 80 = 9x^2 + 18x
Step 3: Move all terms to one side of the equation.
9x^2 + 18x - 80x - 80 = 0
9x^2 - 62x - 80 = 0
Step 4: Factor the quadratic equation.
(9x + 8)(x - 10) = 0
Step 5: Set each factor equal to zero and solve for x.
9x + 8 = 0 or x - 10 = 0
For 9x + 8 = 0,
9x = -8
x = -8/9
For x - 10 = 0,
x = 10
So, the two consecutive even integers are -8/9 and 10.