1.     The equations for inequalities are  2x+y-6≥0

                            x+y-3≤0

                            y≤2

(1)   Draw them on the graph and label the required region as S

(2)   Calculate the area of the required region S

### Step 1: Graph Interpretation

The graph above shows the lines corresponding to the inequalities:

- \( y = -2x + 6 \) (blue line)
- \( y = -x + 3 \) (green line)
- \( y = 2 \) (red line)

The shaded gray region represents the intersection of these inequalities, which we label as region \( S \).

### Step 2: Calculate the Area of Region S

To find the area of region \( S \), we first need to identify the vertices of the polygon that forms this region. Let's determine these vertices by solving the system of equations at their intersection points.

#### Finding Intersection Points:

1. **Intersection of \( y = -2x + 6 \) and \( y = -x + 3 \):**
   \[
   -2x + 6 = -x + 3 \implies x = 3, \quad y = 0
   \]
   Vertex: \( (3, 0) \)

2. **Intersection of \( y = -x + 3 \) and \( y = 2 \):**
   \[
   2 = -x + 3 \implies x = 1, \quad y = 2
   \]
   Vertex: \( (1, 2) \)

3. **Intersection of \( y = -2x + 6 \) and \( y = 2 \):**
   \[
   2 = -2x + 6 \implies x = 2, \quad y = 2
   \]
   Vertex: \( (2, 2) \)

The vertices of the region \( S \) are \( (1, 2) \), \( (2, 2) \), and \( (3, 0) \).

#### Calculating the Area:
We can calculate the area of this triangle using the formula:

\[
\text{Area} = \frac{1}{2} \left| x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) \right|
\]
Where the vertices are \( (x_1, y_1) = (1, 2) \), \( (x_2, y_2) = (2, 2) \), \( (x_3, y_3) = (3, 0) \).

Let's calculate this next.### Step 2: Area of Region S

The area of the required region \( S \) is \( 1 \) square unit.

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Do you want any further details or have any questions? Here are some related problems you might find interesting:

1. How would the area change if the inequality \( y \leq 2 \) was replaced by \( y \leq 3 \)?
2. Can you determine the coordinates of a point that lies within the region \( S \)?
3. What would be the area if one of the inequalities was reversed?
4. How would the graph change if the inequality \( 2x + y - 6 \geq 0 \) was replaced by \( 2x + y - 6 \leq 0 \)?
5. What is the significance of the intersection points in the context of linear programming?

**Tip:** When dealing with systems of inequalities, graphing can help you visually identify the feasible region where all conditions are met.

The equations for inequalities are 2x + y - 6 ≥ 0, x + y - 3 ≤ 0, y ≤ 2. (1) Draw them on the graph and label the required region as S (2) Calculate the area of the required region S

Learn how to graph linear inequalities 2x + y - 6 ≥ 0, x + y - 3 ≤ 0, y ≤ 2 and calculate the area of the intersected region S. This problem covers key concepts in coordinate geometry and is suitable for students in Grades 9-10.

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