suppose the first integer is x, the second one is then x+1
reciprocals are 1/x and 1/(x+1)
1/x +1/(x+1)=-9/20
make the denominators the same:
1(x+1)/x(x+1) + 1x/x(x+1)=-9/20
simplify the demonstrator and add up the numerators: (2x+1)/(x^2+x)=-9/20
20(2x+1)=-9(x^2+x)
40x+20=-9x^2-9x
9x^2 +49x+20=0
factor: (1+5)(9x+4)=0
x=-5 or x=-4/9 (this one doesn't work because it is not an integer)
so the first integer is -5, the second integer is -4