https://file.aiquickdraw.com/mathbot/file/1728731265723sktstuvu.jpg

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!

Solve the system of two linear inequalities graphically: y ≤ 7x + 7 and y > 3x + 3.

Learn how to graph the system of two linear inequalities: y ≤ 7x + 7 and y > 3x + 3. This detailed step-by-step guide will help you understand the process of graphing inequalities and shading the solution regions. Suitable for students in Grades 8-10.

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