The system of two linear inequalities shown in the image is:

$y \leq 7x + 7 \quad \text{and} \quad y > 3x + 3$

Let's break this down into the steps as suggested in the question:

Step 1: Graph the first inequality $y \leq 7x + 7$.

1. Find the boundary line for $y = 7x + 7$. This line is straight, and its slope is 7 with a y-intercept of 7. To graph the line:

   • Start by plotting the point (0, 7) since the y-intercept is 7.
   • Use the slope $7$ to find another point. The slope means rise over run (rise = 7, run = 1), so from (0, 7), move 1 unit to the right and 7 units up to get the point (1, 14).

2. Determine the boundary type:

   • Since the inequality is $\leq$ (less than or equal), the boundary line should be solid because it includes the points on the line.

3. Shade the region:

   • You will shade the region below the line $y = 7x + 7$ because the inequality is $y \leq 7x + 7$ (less than or equal means below the line).

Now, you can enter two points on the graph corresponding to the line (e.g., (0, 7) and (1, 14)) and choose a solid line. Finally, shade the appropriate region (below the line).

Once you finish this step, you'll proceed to graph the second inequality $y > 3x + 3$ in a similar way.

Would you like further details on the second inequality and overall solution? Let me know if you have any questions.

Here are some follow-up questions to help deepen your understanding:

1. What happens if the inequality symbol is switched to $\geq$ or $\leq$?
2. How do you graph inequalities with negative slopes?
3. What if the system involves $x$-related inequalities like $x \geq 5$?
4. Can two inequalities have overlapping shaded regions, and what does that represent?
5. How do you handle inequalities involving absolute values?

Tip: Always check whether the inequality involves a strict inequality ( $>$ or $<$) or an inclusive one ( $\geq$ or $\leq$), as this affects the type of boundary line!