## Step 1: Define the symbolic representation of the variables
The variables are 'hours worked by Peggy' and 'hours worked by Bill', which we can denote as $x_1$ and $x_2$ respectively.

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

## Step 3: List all the constraints
The constraints given are:
- $6x_1 + 6x_2 \geq 63$
- $14x_1 + 6x_2 \geq 34$
- $9x_1 + 15x_2 \geq 50$
- $5x_1 - 9x_2 \geq 0$
- $6x_1 + 6x_2 \leq 132$
- $14x_1 + 6x_2 \leq 159$
- $9x_1 + 15x_2 \leq 151$

## 4: Provide the symbolic representation of the problem
```json
{
    'sym_variables': [('x1', 'hours worked by Peggy'), ('x2', 'hours worked by Bill')],
    'objective_function': '2*x1 + 3*x2',
    'constraints': [
        '6*x1 + 6*x2 >= 63',
        '14*x1 + 6*x2 >= 34',
        '9*x1 + 15*x2 >= 50',
        '5*x1 - 9*x2 >= 0',
        '6*x1 + 6*x2 <= 132',
        '14*x1 + 6*x2 <= 159',
        '9*x1 + 15*x2 <= 151'
    ]
}
```

## 5: Write the Gurobi code to solve the optimization problem
```python
import gurobi

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

    # Define the variables
    x1 = model.addVar(name='hours_worked_by_Peggy', lb=0)  # Assuming hours cannot be negative
    x2 = model.addVar(name='hours_worked_by_Bill', lb=0)  # Assuming hours cannot be negative

    # Define the objective function
    model.setObjective(2 * x1 + 3 * x2, gurobi.GRB.MINIMIZE)

    # Define the constraints
    model.addConstr(6 * x1 + 6 * x2 >= 63)
    model.addConstr(14 * x1 + 6 * x2 >= 34)
    model.addConstr(9 * x1 + 15 * x2 >= 50)
    model.addConstr(5 * x1 - 9 * x2 >= 0)
    model.addConstr(6 * x1 + 6 * x2 <= 132)
    model.addConstr(14 * x1 + 6 * x2 <= 159)
    model.addConstr(9 * x1 + 15 * x2 <= 151)

    # Optimize the model
    model.optimize()

    # Print the solution
    if model.status == gurobi.GRB.OPTIMAL:
        print('Optimal solution found.')
        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()
```