To solve the equation $\cos(x) + 0.3x = 0$, we need to use a numerical method since there is no simple algebraic solution to this kind of transcendental equation.

Steps:

1. Rearrange the equation: $\cos(x) = -0.3x$
2. We will solve this equation using an approximation method such as the Newton-Raphson method or using graphing/numerical solvers.

Let me compute an approximate solution using a numerical method.The solution to the equation $\cos(x) + 0.3x = 0$ is approximately:

$x \approx -1.2019$

Would you like more details on the method used or have any other questions?

Here are 5 related questions to deepen your understanding:

1. How does the Newton-Raphson method work for solving equations numerically?
2. Can transcendental equations always be solved algebraically?
3. What are other numerical methods to solve non-linear equations?
4. How does the graph of $\cos(x) + 0.3x$ look around $x = -1.2$?
5. Why is the cosine function periodic, and how does that affect the number of solutions?

Tip: For transcendental equations like this one, numerical solvers are often the most efficient approach since exact algebraic solutions are rare.