# How to solve the following system of inequalities

I came across this system of inequalities in MaplePrimes, where a user complained about Maple taking too long to compute. I would like to see how SageMath performs in this case.

```
x, y, z = var('x y z')
eqns = [
x>=0,
0 <= y,
0 <= z,
x*y*z + x^2 + y^2 + z^2 <= 2*(x*y + x*z + y*z),
2*(x^2 + y^2 + z^2) < x^2*y + x*z^2 + y^2*z - 27
]
solve(eqns, [x, y, z])
```

SageMath provides some solutions, but it seems that the results are not as satisfactory.

```
[[0 < x, 0 < y, 0 < z, x^2*y + y^2*z + x*z^2 - 2*x^2 - 2*y^2 - 2*z^2 - 27 > 0, -x*y*z - x^2 + 2*x*y - y^2 + 2*x*z + 2*y*z - z^2 > 0], [0 < x, 0 < y, 0 < z, x^2*y + y^2*z + x*z^2 - 2*x^2 - 2*y^2 - 2*z^2 - 27 > 0, x*y*z + x^2 - 2*x*y + y^2 - 2*x*z - 2*y*z + z^2 == 0], [x == 0, 0 < y, 0 < z, y^2*z - 2*y^2 - 2*z^2 - 27 > 0, -y^2 + 2*y*z - z^2 > 0], [x == 0, 0 < y, 0 < z, y^2*z - 2*y^2 - 2*z^2 - 27 > 0, y^2 - 2*y*z + z^2 == 0], [x == 0, y == 0, 0 < z, -2*z^2 - 27 > 0, -z^2 > 0], [x == 0, y == 0, 0 < z, -2*z^2 - 27 > 0, z^2 == 0], [x == 0, y == 0, z == 0, -27 > 0, 0 == 0], [x == 0, y == 0, z == 0, -27 > 0, 0 > 0], [x == 0, z == 0, 0 < y, -2*y^2 - 27 > 0, -y^2 > 0], [x == 0, z == 0, 0 < y, -2*y^2 - 27 > 0, y^2 == 0], [y == 0, 0 < x, 0 < z, x*z^2 - 2*x^2 - 2*z^2 - 27 > 0, -x^2 + 2*x*z - z^2 > 0], [y == 0, 0 < x, 0 < z, x*z^2 - 2*x^2 - 2*z^2 - 27 > 0, x^2 - 2*x*z + z^2 == 0], [y == 0, z == 0, 0 < x, -2*x^2 - 27 > 0, -x^2 > 0], [y == 0, z == 0, 0 < x, -2*x^2 - 27 > 0, x^2 == 0], [z == 0, 0 < x, 0 < y, x^2*y - 2*x^2 - 2*y^2 - 27 > 0, -x^2 + 2*x*y - y^2 > 0], [z == 0, 0 < x, 0 < y, x^2*y - 2*x^2 - 2*y^2 - 27 > 0, x^2 - 2*x*y + y^2 == 0]]
```

For example, in the last solution, it appears that x and y are not independent of each other. Many solutions seem to be outputted in a similar form without much simplification.

Here's another example. When we look at the solution obtained by Mathematica, it seems that SageMath is not performing as well. I'm not sure if SageMath has more accurate commands for solving such systems of inequalities.

```
Reduce[{x^2 + y^2 > 1, x^4 - y^4 < 1}, {x, y}]
(*output: (x < -1 && (y < -(-1 + x^4)^(1/4) || y > (-1 + x^4)^(1/4))) || (-1 <=
x <= 1 && (y < -Sqrt[1 - x^2] || y > Sqrt[1 - x^2])) || (x >
1 && (y < -(-1 + x^4)^(1/4) || y > (-1 + x^4)^(1/4)))*)
x, y = var('x y')
inequalities = [
x^2 + y^2 > 1, x^4 - y^4 < 1
]
solution = solve(inequalities, x, y)
print(solution)
# output: [
[-x^4 + y^4 + 1 > 0, x^2 + y^2 - 1 > 0]
]
```

Potentially QEPCAD should be able to solve such systems.