Final answer:
To find the two consecutive integers, you can assume that the smaller integer is x, and the larger integer is x + 1. The sum of their reciprocals can be expressed as an equation, which can be solved using algebraic methods. The two integers in this case are -1 and -3.
Step-by-step explanation:
To find the two consecutive integers, we can assume that the smaller integer is x, and the larger integer is x + 1. The reciprocal of a number is 1 divided by the number. So, the sum of their reciprocals is:
1/x + 1/(x + 1) = -7/12
To solve this equation, we can first find a common denominator:
(x + 1)/x(x + 1) + x/x(x + 1) = -7/12
Next, we can combine the fractions:
(2x + 1)/(x(x + 1)) = -7/12
To get rid of the denominators, we can cross-multiply:
12(2x + 1) = -7(x(x + 1))
Expanding and simplifying:
24x + 12 = -7x^2 - 7x
Arranging the equation in quadratic form:
7x^2 + 31x + 12 = 0
Using the quadratic formula:
x = (-31 ± sqrt(31^2 - 4 * 7 * 12)) / (2 * 7)
Simplifying further:
x = (-31 ± sqrt(961 - 336)) / 14
x = (-31 ± sqrt(625)) / 14
x = (-31 ± 25) / 14
This gives us two possible values for x:
x = (-31 + 25) / 14 = -1/2
x = (-31 - 25) / 14 = -3
So, the two consecutive integers are -1 and -3.