Question

Determine 2 rational numbers a and b so that?: `a xx b > a -: b > a - b > a + b`

Answer 1

See below.

Explanation:

Equating

`{(a*b = a/b + e_1^2), (a/b = a - b + e_2^2), (a - b = a + b + e_3^2):}`

and solving for `a,b,e_1`

#{(a = -(2 e_2^2 e_3^2 + e_3^4)/(2 (2 + e_3^2))), (b = -e_3^2/2), (e_1 = -1/2 sqrt[e_3^2-2] sqrt[2 e_2^2 + e_3^2]):}#

but `e_1` must be real so `e_3^2 > 2` and we choose `e_3 = sqrt3`

and now solving

`{(a*b = a/b + e_1^2), (a/b = a - b + e_2^2), (a - b = a + b + 3):}`

for `a,b,e_1`

#{(a = -3/10(3 + 2 e_2^2)), (b = -3/2), (e_1 = -1/2 sqrt[3 + 2 e_2^2]):}#

putting now `e_2 = 2` we have

`{(a = -33/10), (b = -3/2), (e_1 = sqrt[11]/2):}`

and finally one of the infinite solutions

`a=-33/10, b = -3/2`