To determine which system of linear inequalities matches the graph, let's analyze the inequalities provided in each option. Here’s a breakdown of what you need to look for:

Key elements to check:

1. Inequality Direction:

   • $y \leq$ or $y \geq$ indicates if the solution region lies below or above the line.
   • $y \leq$ means shading below the line (or on the line if equality holds).
   • $y \geq$ means shading above the line (or on the line if equality holds).

2. Slope and Y-Intercept:

   • The inequalities are in the form $y \leq mx + b$ or $y \geq mx + b$, where $m$ is the slope and $b$ is the y-intercept.

   • For the line $y = 3x - 9$:

     • Slope $m = 3$
     • Y-intercept $b = -9$

   • For the line $y = -45x + 7$:

     • Slope $m = -45$
     • Y-intercept $b = 7$

Now, let's interpret each option:

• Option A:

  • $y \leq 3x - 9$ (shading below this line)
  • $y \geq -45x + 7$ (shading above this line)

• Option B:

  • $y \geq 3x - 9$ (shading above this line)
  • $y \geq -45x + 7$ (shading above this line)

• Option C:

  • $y \geq 3x - 9$ (shading above this line)
  • $y \leq -45x + 7$ (shading below this line)

• Option D:

  • $y \leq 3x - 9$ (shading below this line)
  • $y \leq -45x + 7$ (shading below this line)

Choosing the correct system:

The correct system will have:

• One inequality with $y \leq$ and one with $y \geq$ if the solution region is bounded between the two lines.

Thus, Option A is the most likely answer:

• $y \leq 3x - 9$ (shading below)
• $y \geq -45x + 7$ (shading above)

This option suggests the solution region is between the two lines, matching a typical intersection of two linear inequalities.

Would you like a detailed graph explanation or further clarification?
Here are five related questions to expand your understanding:

1. How do we determine if an inequality includes the boundary line?
2. What happens when the slopes of two lines are equal in a system of inequalities?
3. How do you graph a system of linear inequalities step by step?
4. How can you tell if a point lies in the solution region of a system of inequalities?
5. What does it mean if two inequalities have no overlapping solution region?

Tip: When graphing inequalities, use test points (like the origin) to confirm which side of the line should be shaded.