Question 1189926
Here's how to formulate the quadratic inequalities and approach the graphing:

**1. Area Calculations:**

*   **Room 1 Area:** Length * Width = 4m * 3m = 12 m²
*   **Room 2 Area:** Length * Width = 6m * 4m = 24 m²

**2. Tile Area Calculations:**

*   **Glossy 1 Tile Area:** 0.5m * 0.5m = 0.25 m²
*   **Glossy 2 Tile Area:** 1m * 1m = 1 m²
*   **Non-Glossy 1 Tile Area:** 0.5m * 0.5m = 0.25 m²
*   **Non-Glossy 2 Tile Area:** 1m * 1m = 1 m²

**3. Number of Tiles Needed (Variables):**

Let's define variables for the number of each type of tile:

*   g1 = Number of Glossy 1 tiles
*   g2 = Number of Glossy 2 tiles
*   n1 = Number of Non-Glossy 1 tiles
*   n2 = Number of Non-Glossy 2 tiles

**4. Area Inequalities:**

Since the total tile area must be *at least* equal to the room area, we have two inequalities for each room:

*   **Room 1:**
    *   0.25g1 + 1g2 + 0.25n1 + 1n2 ≥ 12
*   **Room 2:**
    *   0.25g1 + 1g2 + 0.25n1 + 1n2 ≥ 24

**5. Cost Inequalities:**

Let's assume a combined budget for both rooms. Let 'B' be the total budget. The cost inequalities are:

*   8g1 + 10g2 + 5n1 + 7n2 ≤ B

**6. Graphing the Solution Sets:**

Graphing these inequalities directly in 4D space (g1, g2, n1, n2) is impossible on a 2D surface.  To visualize, we need to make some simplifications.  Here are a few approaches:

*   **Scenario 1: Fixed Tile Mix:** Assume a fixed ratio of tile types (e.g., equal numbers of glossy and non-glossy tiles). This reduces the variables and makes graphing possible.  For example, if g1 = n1 and g2 = n2, our inequalities would be:

    *   Room 1: 0.5g1 + 2g2 ≥ 12
    *   Room 2: 0.5g1 + 2g2 ≥ 24
    *   Cost: 13g1 + 17g2 ≤ B

    Now you have inequalities in 2D space (g1 and g2) that you can graph.

*   **Scenario 2: Focus on One Room, Two Tile Types:** Consider just Room 1 and two tile types (e.g., Glossy 1 and Glossy 2).  The inequalities become:

    *   0.25g1 + g2 ≥ 12
    *   8g1 + 10g2 ≤ B (budget for Room 1)

    Again, this is a 2D graphing problem (g1 and g2).

*   **Software:** Software like GeoGebra or Wolfram Alpha can handle 3D graphing, which could be used if you can reduce the number of variables to three.

**7. Interpreting the Graphs:**

The solution set (the feasible region) on the graph represents the combinations of tile numbers that satisfy both the area and cost constraints.  Any point within this region represents a possible solution.

**Example (Scenario 2):**

Let's say the budget for Room 1 is $200. Our inequalities are:

*   0.25g1 + g2 ≥ 12
*   8g1 + 10g2 ≤ 200

You would graph these two inequalities on a g1-g2 plane. The overlapping shaded region is your solution set.

**Important Considerations:**

*   **Integer Solutions:**  Since you can't buy fractions of tiles, you're only interested in integer solutions within the feasible region.
*   **Optimization:**  You could add an objective function (e.g., minimize cost) and use linear programming techniques to find the optimal solution.