## Step 1: Define the symbolic representation of the variables
The variables are 'hours worked by John', 'hours worked by Peggy', and 'hours worked by Bill'. Let's denote them as $x_0$, $x_1$, and $x_2$ respectively.

## Step 2: Define the symbolic representation of the problem
The objective function to minimize is $5x_0 + 3x_1 + 4x_2$.

## Step 3: List the constraints
The constraints given are:
- $9x_0 - 3x_1 \geq 0$
- $7x_0 + 9x_1 + 2x_2 \geq 12$
- $7x_0 + 9x_1 + 2x_2 \geq 12$ is redundant with $7x_0 + 9x_1 + 2x_2 \geq 12$ and $x_0 + x_1 + x_2$ does not have a direct lower bound but $7x_0 + 2x_2 \geq 12$ and $9x_1 + 2x_2 \geq 20$ imply it.
- $9x_1 + 2x_2 \geq 20$
- $7x_0 + 2x_2 \geq 12$
- $11x_0 + 4x_1 + 9x_2 \geq 29$
- $11x_0 + 4x_1 \geq 29 - 9x_2$ is covered by $11x_0 + 4x_1 + 9x_2 \geq 29$
- $4x_1 + 9x_2 \geq 18$
- $7x_0 + 9x_1 + 2x_2 \leq 73$
- $4x_1 + 9x_2 \leq 69$

## 4: Write down the symbolic representation
```json
{
    'sym_variables': [
        ('x0', 'hours worked by John'), 
        ('x1', 'hours worked by Peggy'), 
        ('x2', 'hours worked by Bill')
    ], 
    'objective_function': '5*x0 + 3*x1 + 4*x2', 
    'constraints': [
        '9*x0 - 3*x1 >= 0',
        '7*x0 + 2*x2 >= 12',
        '9*x1 + 2*x2 >= 20',
        '11*x0 + 4*x1 + 9*x2 >= 29',
        '4*x1 + 9*x2 >= 18',
        '7*x0 + 9*x1 + 2*x2 <= 73',
        '4*x1 + 9*x2 <= 69'
    ]
}
```

## 5: Implement the problem in Gurobi
```python
import gurobi

def solve_optimization_problem():
    # Create a new model
    model = gurobi.Model()

    # Define the variables
    x0 = model.addVar(name='hours_worked_by_John', lb=0)
    x1 = model.addVar(name='hours_worked_by_Peggy', lb=0)
    x2 = model.addVar(name='hours_worked_by_Bill', lb=0)

    # Define the objective function
    model.setObjective(5 * x0 + 3 * x1 + 4 * x2, gurobi.GRB.MINIMIZE)

    # Add constraints
    model.addConstr(9 * x0 - 3 * x1 >= 0)
    model.addConstr(7 * x0 + 2 * x2 >= 12)
    model.addConstr(9 * x1 + 2 * x2 >= 20)
    model.addConstr(11 * x0 + 4 * x1 + 9 * x2 >= 29)
    model.addConstr(4 * x1 + 9 * x2 >= 18)
    model.addConstr(7 * x0 + 9 * x1 + 2 * x2 <= 73)
    model.addConstr(4 * x1 + 9 * x2 <= 69)

    # Optimize the model
    model.optimize()

    # Print the solution
    if model.status == gurobi.GRB.OPTIMAL:
        print('Optimal solution found.')
        print(f'Hours worked by John: {x0.varValue}')
        print(f'Hours worked by Peggy: {x1.varValue}')
        print(f'Hours worked by Bill: {x2.varValue}')
        print(f'Objective function value: {model.objVal}')
    else:
        print('No optimal solution found.')

solve_optimization_problem()
```