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The problem involves graphing two inequalities and determining the coordinates of a point in the solution set. The two inequalities are:

1. \( y \leq -2x - 2 \)
2. \( y \geq \frac{1}{3}x + 5 \)

### Steps for solving:
1. **Graph the boundary lines**:
   - For \( y = -2x - 2 \), this is a straight line with slope \( -2 \) and y-intercept \( -2 \). The inequality \( y \leq -2x - 2 \) means the region below or on the line.
   - For \( y = \frac{1}{3}x + 5 \), this is a straight line with slope \( \frac{1}{3} \) and y-intercept \( 5 \). The inequality \( y \geq \frac{1}{3}x + 5 \) means the region above or on the line.

2. **Shade the regions**:
   - Shade the region **below** the line \( y = -2x - 2 \) (including the line because it's \( \leq \)).
   - Shade the region **above** the line \( y = \frac{1}{3}x + 5 \) (including the line because it's \( \geq \)).

3. **Find the solution region**:
   - The solution region is where the shaded areas overlap. Any point in this region will satisfy both inequalities.

4. **Testing points**:
   - You can test a point like (0, 0) to see if it lies in the solution set by substituting it into both inequalities.

Would you like me to graph the solution for better visualization or walk you through testing specific points?

Here are 5 related questions you can explore:
1. How do you graph inequalities with a non-integer slope?
2. What changes if the inequalities were strict (e.g., \( y < -2x - 2 \))?
3. How do you find the intersection point of the two lines?
4. How do inequalities affect systems of equations?
5. Can you describe the effect of changing the slope or y-intercept of these lines?

**Tip:** Always shade in the direction indicated by the inequality symbol when graphing.

Graph the inequalities and determine the coordinates of a point in the solution set: y ≤ -2x - 2 and y ≥ (1/3)x + 5.

Learn how to graph two linear inequalities, y ≤ -2x - 2 and y ≥ (1/3)x + 5, and find the solution set. This problem covers inequalities and linear equations suitable for Grades 8-10. Step-by-step solution includes graphing the boundary lines and shading the appropriate regions.

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