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The system of inequalities you are given is:

1. \( y < \frac{1}{3}x - 2 \)
2. \( y \leq -\frac{1}{6}x - 5 \)

We need to graph these inequalities on the provided coordinate plane to find their solution region, which is the area that satisfies both inequalities. Let’s break this down:

### 1. First Inequality: \( y < \frac{1}{3}x - 2 \)
- The line equation is \( y = \frac{1}{3}x - 2 \), which is a line with slope \( \frac{1}{3} \) and y-intercept at \( -2 \).
- The inequality is strict (less than), so we will draw a **dashed line**.
- We shade the region below the line because the inequality is \( y < \).

### 2. Second Inequality: \( y \leq -\frac{1}{6}x - 5 \)
- The line equation is \( y = -\frac{1}{6}x - 5 \), which has a slope of \( -\frac{1}{6} \) and a y-intercept at \( -5 \).
- The inequality is less than or equal to, so we draw a **solid line**.
- We shade the region below the line because the inequality is \( y \leq \).

### Graphing the Lines:
- Plot the points and draw the lines based on the given equations.
- For \( y = \frac{1}{3}x - 2 \), a couple of points to plot are:
  - At \( x = 0 \), \( y = -2 \) (point \( (0, -2) \)).
  - At \( x = 3 \), \( y = 0 \) (point \( (3, 0) \)).
- For \( y = -\frac{1}{6}x - 5 \), some points are:
  - At \( x = 0 \), \( y = -5 \) (point \( (0, -5) \)).
  - At \( x = 6 \), \( y = -6 \) (point \( (6, -6) \)).

### Finding a Point in the Solution Region:
After graphing, the solution region is where the shading overlaps. A possible point in this region can be found by looking at the graph. For instance, the point \( (0, -6) \) should satisfy both inequalities:

1. \( y = -6 \) for \( y < \frac{1}{3}(0) - 2 = -2 \), so \( -6 < -2 \) is true.
2. \( y = -6 \) for \( y \leq -\frac{1}{6}(0) - 5 = -5 \), so \( -6 \leq -5 \) is true.

Thus, \( (0, -6) \) is a point in the solution set.

Would you like more detailed steps or another point from the solution? 

Here are 5 related questions to expand your understanding:

1. How do you solve systems of inequalities algebraically?
2. What is the geometric interpretation of the slope in these inequalities?
3. How would the solution region change if one of the inequalities were a strict "greater than"?
4. How does shading help in determining the solution to a system of inequalities?
5. How do you find the intersection points of two linear equations graphically?

**Tip**: Always check if the point chosen lies within the correct region by substituting it back into the inequalities.

Solve the system of inequalities y < (1/3)x - 2 and y ≤ -(1/6)x - 5 graphically. State the coordinates of a point in the solution set.

Solve the system of inequalities y < (1/3)x - 2 and y ≤ -(1/6)x - 5. Graph each inequality and determine the solution region. Learn how to graph lines and interpret inequalities in Grades 9-11 math.

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