Final answer:
To find the consecutive even integers, we will set up an equation using the given sum of their reciprocals. By finding a common denominator, simplifying, and solving the resulting quadratic equation, we can determine the values of the integers.
Step-by-step explanation:
To find the consecutive even integers, let's assume that the first even integer is x. The next consecutive even integer would be x + 2.
The sum of their reciprocals is given as 3/2, so we can write the equation:
1/x + 1/(x + 2) = 3/2
To solve this equation, we need to find a common denominator. In this case, the common denominator is 2x(x + 2). Multiplying every term by this common denominator, we get:
2(x + 2) + 2x = 3x(x + 2)
Expanding and simplifying, we have:
2x + 4 + 2x = 3x^2 + 6x
Combining like terms, we get:
4x + 4 = 3x^2 + 6x
Subtracting 4x + 4 from both sides, we get:
3x^2 + 6x - 4x - 4 = 0
3x^2 + 2x - 4 = 0
Using a factoring, quadratic formula, or completing the square, we can solve for x:
x ≈ 0.796, x ≈ -2.129
Since we are looking for even integers, we can discard the negative solution. Therefore, the first even integer is approximately 0.796. The next consecutive even integer is x + 2 ≈ 0.796 + 2 = 2.796.