To solve and graph the system of inequalities given:

1. $x - 3y \geq -3$
2. $2x + y \leq 6$

we'll follow these steps:

Step 1: Convert each inequality into its boundary line

Inequality 1: $x - 3y = -3$

Solve for $y$: $x - 3y = -3 \Rightarrow -3y = -x - 3 \Rightarrow y = \frac{1}{3}x + 1$ This line has a slope of $\frac{1}{3}$ and a y-intercept of $(0, 1)$.

Inequality 2: $2x + y = 6$

Solve for $y$: $2x + y = 6 \Rightarrow y = -2x + 6$ This line has a slope of $-2$ and a y-intercept of $(0, 6)$.

Step 2: Determine the shading direction for each inequality

1. For $x - 3y \geq -3$, the inequality is satisfied by shading above the line $y = \frac{1}{3}x + 1$.
2. For $2x + y \leq 6$, the inequality is satisfied by shading below the line $y = -2x + 6$.

Step 3: Graph the inequalities

On the graph:

1. Draw the line $y = \frac{1}{3}x + 1$ as a solid line because it includes points on the line ($\geq$ indicates "greater than or equal to").
2. Draw the line $y = -2x + 6$ as a solid line as well ($\leq$ includes points on the line).

Step 4: Identify the solution region

The solution region is the area where the shaded regions of both inequalities overlap. This overlapping region represents all the points $(x, y)$ that satisfy both inequalities.

Would you like a detailed explanation of each step?

Here are some questions to expand on this topic:

1. What are some real-world examples of systems of inequalities?
2. How can we find the points of intersection for these inequalities?
3. What happens if one of the inequalities has a ">" or "<" instead?
4. How do changes in slope and y-intercept affect the solution region?
5. How can systems of inequalities be solved graphically and algebraically?

Tip: Always start by turning inequalities into boundary lines to help visualize the solution area.