We are given a vector space $(V, \oplus, \otimes)$ with non-standard operations defined as:

1. Vector addition: $u \oplus v := uv - 5(u + v) + 30$
2. Scalar multiplication: $a \otimes u := (u - 5)a + 5$

We are tasked with finding the additive inverse of an element $x$, denoted $\ominus x$, which satisfies the equation:

$x \oplus (\ominus x) = 0$

Here, 0 is the additive identity of the vector space. The additive identity in a vector space is the element such that:

$x \oplus 0 = x \quad \text{for all} \quad x \in V$

Let’s first determine the additive identity by solving for $0$. We set $u \oplus 0 = u$ and use the definition of $\oplus$:

$u \oplus 0 = u \cdot 0 - 5(u + 0) + 30 = u$

Simplifying:

$0 - 5u - 5 \cdot 0 + 30 = u$

$-5u + 30 = u$

$30 = 6u$

$u = 5$

Thus, the additive identity is $5$.

Step 1: Find the Additive Inverse

Now we find $\ominus x$ such that $x \oplus (\ominus x) = 5$. Using the definition of $\oplus$, we get:

$x \oplus (\ominus x) = x \cdot (\ominus x) - 5(x + \ominus x) + 30 = 5$

Simplify:

$x \cdot (\ominus x) - 5(x + \ominus x) + 30 = 5$

$x \cdot (\ominus x) - 5(x + \ominus x) = -25$

Let $y = \ominus x$. We now have the equation:

$xy - 5(x + y) = -25$

Expanding:

$xy - 5x - 5y = -25$

Rearranging:

$xy - 5x - 5y = -25$

We need to solve this equation for $y$ in terms of $x$. Factoring it can help:

$(x - 5)(y - 5) = 25$

Thus, we find:

$y - 5 = \frac{25}{x - 5}$

$y = 5 + \frac{25}{x - 5}$

Hence, the additive inverse of $x$ is:

$\ominus x = 5 + \frac{25}{x - 5}$

---

Would you like a further breakdown of the steps? Here are 5 related questions:

1. How does scalar multiplication behave in this vector space?
2. What are the properties of the non-standard vector addition in this space?
3. How would you find the scalar multiple of a vector using this operation?
4. Can you derive the identity for scalar multiplication in this vector space?
5. How would this operation affect the solution of a system of linear equations?

Tip: In unusual vector spaces like this, always start by understanding how the identity elements behave under the given operations.