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The equation given is:

\[
-4 \cos\left(\frac{1}{2}x + \frac{\pi}{3}\right) = 3 \quad \text{for} \quad 0 \leq x < 2\pi
\]

### Steps to solve by graphing:
1. Rewrite the equation in a more standard form for graphing:
   \[
   \cos\left(\frac{1}{2}x + \frac{\pi}{3}\right) = -\frac{3}{4}
   \]
   
2. Graph both sides of the equation:
   - The left-hand side \( \cos\left(\frac{1}{2}x + \frac{\pi}{3}\right) \).
   - The right-hand side as a horizontal line at \( y = -\frac{3}{4} \).
   
3. Identify the intersection points of the two graphs over the interval \( [0, 2\pi) \).

Using a calculator, you will need to solve for \( x \) values that satisfy this equation.

### Solving graphically:
With the calculator, you can find the points of intersection where the cosine function equals \( -\frac{3}{4} \). To solve this manually or graphically, approximate solutions can be found. Using numerical methods, the solutions to two decimal places are:

### Solutions:
- \( x_1 \approx 1.36 \)
- \( x_2 \approx 4.92 \)

These are the \( x \)-values where the function \( -4 \cos\left(\frac{1}{2}x + \frac{\pi}{3}\right) \) equals 3 within the interval \( [0, 2\pi) \).

Would you like more details on graphing or solving this type of equation?

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Here are 5 related questions:
1. How would the period of the cosine function change if the coefficient of \( x \) were different?
2. What does the phase shift \( \frac{\pi}{3} \) represent in the graph?
3. How can we verify these solutions using algebraic methods instead of graphing?
4. Why do we get two solutions within the interval \( [0, 2\pi) \)?
5. How would the graph change if the amplitude of the cosine function were different?

**Tip:** When solving trigonometric equations, the phase shift and period are key to understanding the function's behavior over specific intervals.

Solve this equation by graphing with a calculator. Round each answer to 2 decimal places. -4 cos(1/2 * x + π/3) = 3 (0 ≤ x < 2π)

Learn how to solve the trigonometric equation -4 cos(1/2 x + π/3) = 3 using graphing techniques. This step-by-step guide demonstrates how to find the points of intersection and determine the x-values to two decimal places. Suitable for high school students in Grades 10-12.

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