To solve the equation \( 5\cos(x) = -1.5x \), we can start by rewriting it:

\[
5\cos(x) + 1.5x = 0
\]

This is a transcendental equation and typically requires numerical methods to find solutions, as it cannot be solved algebraically.

### Steps to Solve:

1. **Graphical Method**: Plot the functions \( y = 5\cos(x) \) and \( y = -1.5x \) to find their intersection points.
2. **Numerical Method**: Use numerical solvers such as the Newton-Raphson method or fixed-point iteration to find approximate solutions.

### Finding Solutions:

1. **Using a Graphing Tool**: You can graph both sides to find where they intersect.
2. **Using a Calculator or Software**: You can apply numerical methods to approximate the roots.

Would you like assistance with graphing or using a specific method to find the numerical solution? 

### Related Questions:
1. What are the properties of cosine functions?
2. How can numerical methods be applied to solve transcendental equations?
3. What is the graphical interpretation of a transcendental equation?
4. How does the behavior of \( \cos(x) \) influence the solution of the equation?
5. Are there specific intervals where we should look for solutions?

### Tip:
When dealing with transcendental equations, plotting the functions can provide a good visual insight into potential solutions before applying numerical methods.

Discover how to solve the equation 5cos(x) = -1.5x using graphical and numerical methods. This detailed guide explains the steps to find approximate solutions and covers key concepts in trigonometry and transcendental equations, suitable for high school students.

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