Answer: two integers are 5 and 14.
Explanation:
Let x be the smaller integer, then the larger integer is x+9. We can set up the equation:
1/x + 2/(x+9) = 6/5
Multiplying both sides by the least common multiple of the denominators:
5(x+9) + 10x = 6x(x+9)
Simplifying and rearranging:
6x^2 - 17x - 45 = 0
We can use the quadratic formula to solve for x:
x = [17 ± √(17^2 - 4(6)(-45))] / (2(6))
x ≈ 4.5 or x ≈ -1.667
Since we are looking for a positive integer, we can discard the negative solution.
Therefore, the smaller integer is x ≈ 4.5, but since it must be a positive integer, the smaller integer is 5. The larger integer is x+9, or 14.
Checking our answer:
1/5 + 2/14 = 6/70 + 20/70 = 26/70 = 13/35
13/35 is equal to 6/5 when multiplied by 6/7:
(13/35) * (6/7) = 78/245 = 6/5
Therefore, our solution is correct, and the two integers are 5 and 14.