`3x + 2y le 12, x ge 1, y ge 2`

Explanation

To solve the inequalities \(3x + 2y \leq 12\), \(x \geq 1\), and \(y \geq 2\) using the graphical method, follow these steps:

### Step 1: Rewrite the inequalities
We start with the given inequalities:
1. \(3x + 2y \leq 12\)
2. \(x \geq 1\)
3. \(y \geq 2\)

### Step 2: Convert the inequalities into equations
To graph the inequalities, we first convert them into equations:
1. \(3x + 2y = 12\)
2. \(x = 1\)
3. \(y = 2\)

### Step 3: Find the intercepts of the equation \(3x + 2y = 12\)
To find the intercepts, set \(x = 0\) to find the y-intercept and set \(y = 0\) to find the x-intercept:
- For the x-intercept:
\[ 3x + 2(0) = 12 \implies 3x = 12 \implies x = 4 \]
So, the x-intercept is \((4, 0)\).

- For the y-intercept:
\[ 3(0) + 2y = 12 \implies 2y = 12 \implies y = 6 \]
So, the y-intercept is \((0, 6)\).

### Step 4: Plot the lines on a graph
- Draw the line \(3x + 2y = 12\) using the intercepts \((4, 0)\) and \((0, 6)\).
- Draw the vertical line \(x = 1\).
- Draw the horizontal line \(y = 2\).

### Step 5: Determine the shaded regions
- For \(x \geq 1\): Shade the region to the right of the line \(x = 1\).
- For \(y \geq 2\): Shade the region above the line \(y = 2\).
- For \(3x + 2y \leq 12\): Shade the region below the line \(3x + 2y = 12\).

### Step 6: Identify the feasible region
The feasible region is where all shaded areas overlap. This region is bounded by:
- The line \(x = 1\)
- The line \(y = 2\)
- The line \(3x + 2y = 12\)

### Step 7: Identify the vertices of the feasible region
The vertices of the feasible region can be found by solving the equations of the lines:
1. Intersection of \(x = 1\) and \(y = 2\) gives the point \((1, 2)\).
2. Intersection of \(x = 1\) and \(3x + 2y = 12\):
\[ 3(1) + 2y = 12 \implies 3 + 2y = 12 \implies 2y = 9 \implies y = 4.5 \]
So, the point is \((1, 4.5)\).
3. Intersection of \(y = 2\) and \(3x + 2y = 12\):
\[ 3x + 2(2) = 12 \implies 3x + 4 = 12 \implies 3x = 8 \implies x = \frac{8}{3} \]
So, the point is \(\left(\frac{8}{3}, 2\right)\).

### Step 8: Conclusion
The solution to the system of inequalities is the shaded region bounded by the points \((1, 2)\), \((1, 4.5)\), and \(\left(\frac{8}{3}, 2\right)\).

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