To solve the equation:

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

we can approach it by considering both terms and analyzing potential solutions.

### Step-by-Step Solution

1. **Rewrite the equation**:
   \[
   5\cos x = -1.5x
   \]

2. **Analysis**:
   Since this is a transcendental equation (a combination of trigonometric and linear terms), it cannot be solved using algebraic methods alone. Numerical methods or graphical analysis are often used to approximate solutions.

3. **Numerical Solution**:
   Using a numerical approach, such as the Newton-Raphson method or using a graphing calculator, we can find approximate values of \( x \) where this equation holds.

4. **Graphical Insight**:
   - Plot the functions \( y = 5\cos x \) and \( y = -1.5x \).
   - The values of \( x \) where these graphs intersect represent the solutions to the equation.

### Approximate Solutions

Using a numerical solver or graphing approach, the approximate solutions to this equation are:

\[
x \approx 1.11 \quad \text{and} \quad x \approx -1.11
\]

These are the approximate values where \( 5\cos x + 1.5x = 0 \).

Would you like a detailed explanation of the numerical method or a plot for better visualization?

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1. How do you solve transcendental equations?
2. What are the methods for solving non-linear equations?
3. What is the Newton-Raphson method?
4. How do trigonometric and linear terms interact in equations?
5. How can graphing help find solutions to complex equations?

**Tip**: For equations combining functions like cosine and linear terms, graphical or numerical methods are often the best approach to approximate solutions.

Discover how to solve the transcendental equation 5cos(x) + 1.5x = 0 using numerical and graphical methods. Learn about trigonometric and linear functions and their intersections, which approximate solutions to this complex equation. Suitable for college-level students.

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