Kuunika wishes to build a lodge with single and double rooms. He needs to decide the number of each room type he should build to maximize profit. Let $$ x $$ represent the number of single rooms and $$ y $$ the number of double rooms. (a) Write the inequalities which represents each of the following conditions: (i) There must be at least one single room. (ii) There must be at least 10 rooms altogether. (iii) The total number of rooms should not exceed 15 . (iv) The number of double rooms must be at least twice the number of single rooms. (v) The number of double rooms should not be more than 12 . [1] (b) Using a scale 2 cm to 5 units on both axes, draw $$ x $$ and y axes for $$ 0 \leq x \leq 16 $$ and $$ 0 \leq y \leq 16 $$ respectively and shade the un :anted region to indicate ciearly the region where he solution of the inequali ies lies.

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### Problem (a)

**Write the inequalities based on the given conditions:**

Let $$ x $$ represent the number of single rooms and $$ y $$ represent the number of double rooms.

1. **At least one single room:**
   $$
   x \geq 1
   $$

2. **At least 10 rooms altogether:**
   $$
   x + y \geq 10
   $$

3. **Total number of rooms should not exceed 15:**
   $$
   x + y \leq 15
   $$

4. **Number of double rooms must be at least twice the number of single rooms:**
   $$
   y \geq 2x
   $$

5. **Number of double rooms should not be more than 12:**
   $$
   y \leq 12
   $$

### Problem (b)

**Graphical Representation:**

To graph the inequalities and identify the feasible region where the solution lies, follow these steps:

1. **Set Up the Axes:**
   - Draw a coordinate plane with $$ x $$-axis (single rooms) and $$ y $$-axis (double rooms).
   - Use a scale where 2 cm represents 5 units on both axes.
   - Label the axes from 0 to 16 for both $$ x $$ and $$ y $$.

2. **Plot Each Inequality:**

- **$$ x \geq 1 $$:**
     - Draw a vertical line at $$ x = 1 $$.
     - Shade to the right of this line since $$ x $$ must be at least 1.

- **$$ x + y \geq 10 $$:**
     - Rewrite as $$ y \geq 10 - x $$.
     - Plot the line $$ y = 10 - x $$ by finding intercepts:
       - When $$ x = 0 $$, $$ y = 10 $$.
       - When $$ y = 0 $$, $$ x = 10 $$.
     - Shade above this line.

- **$$ x + y \leq 15 $$:**
     - Plot the line $$ y = 15 - x $$ by finding intercepts:
       - When $$ x = 0 $$, $$ y = 15 $$.
       - When $$ y = 0 $$, $$ x = 15 $$.
     - Shade below this line.

- **$$ y \geq 2x $$:**
     - Plot the line $$ y = 2x $$ by finding points:
       - When $$ x = 0 $$, $$ y = 0 $$.
       - When $$ x = 5 $$, $$ y = 10 $$.
     - Shade above this line.

- **$$ y \leq 12 $$:**
     - Draw a horizontal line at $$ y = 12 $$.
     - Shade below this line.

3. **Identify the Feasible Region:**
   - The feasible region is where all shaded areas overlap.
   - This region satisfies all the given inequalities.
   - **Shade the Unwanted Regions:**
     - Clearly shade areas outside the feasible region to indicate they are not solutions.
     - The final feasible region will be a polygon bounded by the intersecting lines of the inequalities.

**Visual Guide:**

While a precise drawing cannot be provided here, the steps above will help you sketch the graph accurately on graph paper or using graphing software. Ensure all lines are clearly drawn with appropriate shading for desired and undesired regions to highlight the feasible solution area effectively.
**Inequalities:** 1. $$ x \geq 1 $$ 2. $$ x + y \geq 10 $$ 3. $$ x + y \leq 15 $$ 4. $$ y \geq 2x $$ 5. $$ y \leq 12 $$ **Graphical Representation:** - Draw a coordinate plane with $$ x $$ (single rooms) and $$ y $$ (double rooms) axes from 0 to 16. - Shade the area where all inequalities overlap to find the feasible region for building the lodge.

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