# Inequality Solving

E1 = 1/(2 k (c - m) m) (2 c dt m + c k m + c dt k m - k m^2 -
dt k m^2 - \[Sqrt](4 c^4 dt^2 + 8 c^4 dt^2 k + 4 c^4 dt^2 k^2 +
4 c^3 dt k m - 12 c^3 dt^2 k m + 4 c^4 dt^2 k m +
4 c^3 dt k^2 m - 12 c^3 dt^2 k^2 m + 4 c^4 dt^2 k^2 m -
4 c^2 dt k m^2 + 4 c^2 dt^2 k m^2 - 8 c^3 dt^2 k m^2 +
c^2 k^2 m^2 - 10 c^2 dt k^2 m^2 + 13 c^2 dt^2 k^2 m^2 -
12 c^3 dt^2 k^2 m^2 + 4 c^2 dt^2 k m^3 - 2 c k^2 m^3 +
8 c dt k^2 m^3 - 6 c dt^2 k^2 m^3 + 12 c^2 dt^2 k^2 m^3 +
k^2 m^4 - 2 dt k^2 m^4 + dt^2 k^2 m^4 -
4 c dt^2 k^2 m^4)) /. {dt -> 1};

E2 = 1/(2 k (c - m) m) (2 c dt m + c k m + c dt k m - k m^2 -
dt k m^2 + \[Sqrt](4 c^4 dt^2 + 8 c^4 dt^2 k + 4 c^4 dt^2 k^2 +
4 c^3 dt k m - 12 c^3 dt^2 k m + 4 c^4 dt^2 k m +
4 c^3 dt k^2 m - 12 c^3 dt^2 k^2 m + 4 c^4 dt^2 k^2 m -
4 c^2 dt k m^2 + 4 c^2 dt^2 k m^2 - 8 c^3 dt^2 k m^2 +
c^2 k^2 m^2 - 10 c^2 dt k^2 m^2 + 13 c^2 dt^2 k^2 m^2 -
12 c^3 dt^2 k^2 m^2 + 4 c^2 dt^2 k m^3 - 2 c k^2 m^3 +
8 c dt k^2 m^3 - 6 c dt^2 k^2 m^3 + 12 c^2 dt^2 k^2 m^3 +
k^2 m^4 - 2 dt k^2 m^4 + dt^2 k^2 m^4 -
4 c dt^2 k^2 m^4)) /. {dt -> 1};

Solve[Abs[E1] < 1 && Abs[E2] < 1, {c, m, k}]


I am not able to get reduced condition in simplest form of c, k, m.

I did try with 'Solve' but it is taking indefinite time. Help is solicited.

• It looks extremely complicated. Just try Reduce[E1 < 1, {c, m, k}] to see what I mean. – Lotus Feb 24 '17 at 7:24
• Even Reducing one argument is taking too long time. Any smart idea is required. – Sk Sarif Hassan Feb 24 '17 at 7:46
• What would help is if you have additional constraints for the variables. – Lotus Feb 24 '17 at 7:49
• dt can be taken as 1 or 0.005. c, k, m are all real numbers. – Sk Sarif Hassan Feb 24 '17 at 8:02
• It gets better if we have c > 0 etc..Reduce[{E1 < 1, c > 0, m > 0, k > 0}, {c, m, k}] – Lotus Feb 24 '17 at 8:04