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To solve this problem, we need to graph the system of linear inequalities given:

1. \( 6x - y \leq -18 \)
2. \( y \geq 5 \)

### Step 1: Graphing the first inequality \( 6x - y \leq -18 \)

1. **Convert the inequality to an equation**: \( 6x - y = -18 \). This is the boundary line. 
2. **Find two points to plot the line**:
   - When \( x = 0 \), \( 6(0) - y = -18 \) gives \( y = 18 \), so one point is \( (0, 18) \).
   - When \( y = 0 \), \( 6x - 0 = -18 \) gives \( x = -3 \), so another point is \( (-3, 0) \).

   Plot these points on the graph and draw a line through them. Since the original inequality is \( \leq \), the line should be **solid** to indicate that points on the line are included in the solution.

3. **Shade the appropriate region**: Test a point not on the line, such as \( (0, 0) \). 
   - Substitute \( (0, 0) \) into the inequality: \( 6(0) - 0 \leq -18 \), which simplifies to \( 0 \leq -18 \). This is false, so shade the region on the opposite side of the line from \( (0, 0) \).

### Step 2: Graphing the second inequality \( y \geq 5 \)

1. Draw the line \( y = 5 \). Since the inequality is \( \geq \), the line should be **solid**.
2. Shade the region above the line because \( y \) is greater than or equal to 5.

### Step 3: Finding the solution region

The solution to the system is the overlapping shaded region from both inequalities.

Would you like a detailed step-by-step graphing explanation? Here are some related questions to expand on this topic:

1. How do you determine whether to use a solid or dashed line for graphing inequalities?
2. What is the significance of testing a point when shading regions for inequalities?
3. How can systems of linear inequalities be applied to real-world problems?
4. How do you solve systems of inequalities graphically if more than two inequalities are given?
5. What happens if there is no overlapping region when graphing multiple inequalities?

**Tip:** Always double-check the region you shade by testing a point in the shaded area to ensure it satisfies the original inequality.

Solve the system of linear inequalities graphically: 6x - y ≤ -18 and y ≥ 5.

Learn how to graph a system of linear inequalities, including 6x - y ≤ -18 and y ≥ 5, step by step. This problem focuses on graphing techniques, identifying solution regions, and understanding boundary lines. Suitable for students in Grades 9-12.

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