Draw the graphs of the following system of inequations and indicate the solution set.
`2x+3y ge 6, 2x +y ge 4, x ge 4 and y le 3. `

Answer

To solve the given system of inequalities and draw the graphs, we will follow these steps:

### Step 1: Convert inequalities to equations
We will start by converting each inequality into an equation to find the boundary lines.

1. **First Inequality**: \(2x + 3y \geq 6\)
- Convert to equation: \(2x + 3y = 6\)

2. **Second Inequality**: \(2x + y \geq 4\)
- Convert to equation: \(2x + y = 4\)

3. **Third Inequality**: \(x \geq 4\)
- Convert to equation: \(x = 4\)

4. **Fourth Inequality**: \(y \leq 3\)
- Convert to equation: \(y = 3\)

### Step 2: Find intercepts for each line
Next, we will find the intercepts for each line to plot them.

1. **For \(2x + 3y = 6\)**:
- **x-intercept**: Set \(y = 0\)
\[ 2x + 3(0) = 6 \implies x = 3 \quad \text{(Point: (3, 0))} \]
- **y-intercept**: Set \(x = 0\)
\[ 2(0) + 3y = 6 \implies y = 2 \quad \text{(Point: (0, 2))} \]

2. **For \(2x + y = 4\)**:
- **x-intercept**: Set \(y = 0\)
\[ 2x + 0 = 4 \implies x = 2 \quad \text{(Point: (2, 0))} \]
- **y-intercept**: Set \(x = 0\)
\[ 2(0) + y = 4 \implies y = 4 \quad \text{(Point: (0, 4))} \]

3. **For \(x = 4\)**: This is a vertical line at \(x = 4\).

4. **For \(y = 3\)**: This is a horizontal line at \(y = 3\).

### Step 3: Graph the lines
Now, we will plot the lines on a coordinate system.

- **Plot the line for \(2x + 3y = 6\)** using points (3, 0) and (0, 2).
- **Plot the line for \(2x + y = 4\)** using points (2, 0) and (0, 4).
- **Draw the vertical line** for \(x = 4\).
- **Draw the horizontal line** for \(y = 3\).

### Step 4: Determine the feasible regions
We will now determine the feasible regions for each inequality:

1. **For \(2x + 3y \geq 6\)**: The region above the line (away from the origin).
2. **For \(2x + y \geq 4\)**: The region above the line (away from the origin).
3. **For \(x \geq 4\)**: The region to the right of the line \(x = 4\).
4. **For \(y \leq 3\)**: The region below the line \(y = 3\).

### Step 5: Find the intersection of the regions
The solution set is where all the shaded regions overlap.

1. The area that satisfies \(2x + 3y \geq 6\) and \(2x + y \geq 4\) is above both lines.
2. The area that satisfies \(x \geq 4\) is to the right of the vertical line.
3. The area that satisfies \(y \leq 3\) is below the horizontal line.

### Step 6: Identify the solution set
The intersection of all these regions gives us the solution set. This will be a polygonal region bounded by the lines we have drawn.

### Final Graph
The final graph will show the lines and the shaded area representing the solution set.