A factory employs unskilled workers at R1 000 per week and skilled workers at R2 000 per week. The payroll should not exeeed R90 000 per week. To keep going, at least 50 operators are necessary, Furthermore, regulations determine that the number of skilled workers should be at least half that of the unskilled workers. Let $$ x $$ be the number of unskilled workers and $$ y $$ the number of skilled workers. The following inequalities are obtained: $$ 1000 x+2000 y \leq 90000 $$ $$ x+y \geq 50 $$ $$ y \geq \frac{x}{2} $$ $$ x: y \geq 0 $$ $$ x: y \in N_{0} $$ 7.1 $$ 7.2 \quad $$ Sketeh these inequalities on ADDENDUM B (attached). $$ 7.3 \quad $$ Shade and indicate the feasible region on the sketeh made in QUESTION 7.1 . that should be employed.

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To solve the problem, we need to analyze the inequalities provided and determine the feasible region for the number of unskilled workers $$ x $$ and skilled workers $$ y $$. Let's break down the inequalities step by step.

### Known Conditions
1. **Payroll Constraint**: $$ 1000x + 2000y \leq 90000 $$
2. **Minimum Workers**: $$ x + y \geq 50 $$
3. **Skilled Workers Regulation**: $$ y \geq \frac{x}{2} $$
4. **Non-negativity**: $$ x \geq 0 $$ and $$ y \geq 0 $$
5. **Non-negative integers**: $$ x, y \in \mathbb{N}_0 $$

### Step 1: Rewrite the Inequalities
1. **Payroll Constraint**: 
   $$
   1000x + 2000y \leq 90000 \implies x + 2y \leq 90
   $$
2. **Minimum Workers**: 
   $$
   x + y \geq 50
   $$
3. **Skilled Workers Regulation**: 
   $$
   y \geq \frac{x}{2}
   $$

### Step 2: Graph the Inequalities
To graph these inequalities, we will find the boundary lines and then determine the feasible region.

1. **For $$ x + 2y = 90 $$**:
   - When $$ x = 0 $$: $$ 2y = 90 \implies y = 45 $$ (Point: $$ (0, 45) $$)
   - When $$ y = 0 $$: $$ x = 90 $$ (Point: $$ (90, 0) $$)

2. **For $$ x + y = 50 $$**:
   - When $$ x = 0 $$: $$ y = 50 $$ (Point: $$ (0, 50) $$)
   - When $$ y = 0 $$: $$ x = 50 $$ (Point: $$ (50, 0) $$)

3. **For $$ y = \frac{x}{2} $$**:
   - When $$ x = 0 $$: $$ y = 0 $$ (Point: $$ (0, 0) $$)
   - When $$ x = 100 $$: $$ y = 50 $$ (Point: $$ (100, 50) $$)

### Step 3: Determine the Feasible Region
Now we will find the intersection points of the lines to identify the vertices of the feasible region.

1. **Intersection of $$ x + 2y = 90 $$ and $$ x + y = 50 $$**:
   $$
   \begin{align*}
   x + 2y &= 90 \
   x + y &= 50 \implies y = 50 - x \
   \end{align*}
   $$
   Substituting $$ y $$ in the first equation:
   $$
   x + 2(50 - x) = 90 \implies x + 100 - 2x = 90 \implies -x = -10 \implies x = 10
   $$
   Then, $$ y = 50 - 10 = 40 $$ (Point: $$ (10, 40) $$)

2. **Intersection of $$ x + 2y = 90 $$ and $$ y = \frac{x}{2} $$**:
   $$
   x + 2\left(\frac{x}{2}\right) = 90 \implies x + x = 90 \implies 2x = 90 \implies x = 45
   $$
   Then, $$ y = \frac{45}{2} = 22.5 $$ (Point: $$ (45, 22.5) $$)

3. **Intersection of $$ x + y = 50 $$ and $$ y = \frac{x}{2} $$**:
   $$
   x + \frac{x}{2} = 50 \implies \frac{3x}{2} = 50 \implies 3x = 100 \implies x = \frac{100}{3} \approx 33.33
   $$
   Then, $$ y = 50 - \frac{100}{3} \approx 16.67 $$ (Point: $$ \left(\frac{100}{3}, \frac{50}{3}\right) $$)

### Step 4: Identify the Feasible Region
The feasible region is bounded by the lines and the axes, and it must satisfy all inequalities. The vertices of the feasible region are:
- $$ (10, 40) $$
- $$ (45, 22.5) $$
- $$ (50, 0) $$
- $$ (0, 50) $$

### Step 5: Sketch the Graph
You would sketch the lines based on the points calculated and shade the area that satisfies all inequalities. The feasible region will be the area where all shaded regions overlap.

### Step 6: Indicate the Feasible Region
On your sketch, shade the area that satisfies all the inequalities, ensuring to label the vertices and the axes clearly.

### Conclusion
The feasible region represents the combinations of unskilled and skilled workers that the factory can employ while adhering to the constraints provided.
The feasible region for the number of unskilled and skilled workers is the area where all the given inequalities overlap. It includes the points where the constraints intersect, such as (10, 40), (45, 22.5), (50, 0), and (0, 50).

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