Proving non-linear inequality doesn't have a solution [Does it have any positive solutions?]

Question

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.

Answer 1

There is no solution, because the inequalities can be rewritten as: $$ 0 < \frac{x_4}{x_5} = \frac{x_4 + x_5}{x_5} - 1 < \frac{x_2 - x_1}{x_0} - 1 < \frac{x_3}{x_3 + x_4} - 1 = -\frac{x_4}{x_3 + x_4} < 0. $$

Answer 2

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 $x_i\in \mathbb{R+}$