Proving non-linear inequality doesn't have a solution [Does it have any positive solutions?] I am trying to find if following inequalities have a solutions for $x_i\in \mathbb{R+}$
$x_5(x_2-x_1)>x_0(x_4+x_5)$
$x_0x_3>(x_3+x_4)(x_2-x_1)$
How could I prove if this inequality has a solution or not?
I attempted to reach a contradiction by combining the two in-equations but seems like it is not a straight-forward one.
UPDATE:
Some context where this inequality comes from. This was a calculation based on hedged trading.
$x_0$ is the first buy position size, $x_1$ is the second buy size opened with $x_4$ diff from the first position along with a sell position at that price with size of $x_2$. If price moves down with diff of $x_5$ we should be able to make profit, on the other side if price moves up with diff of $x_3$ from the first opened buy we should also have positive profit. These inequalities come from there.

After multiplying left sides and right sides we have
$$
x_5(x_2-x_1)x_0x_3\gt x_0(x_4+x_5)(x_3+x_4)(x_2-x_1)
$$
or
$$
x_3x_5 \gt (x_4+x_5)(x_3+x_4)
$$
which is false because $x_i\in \mathbb{R+}$
