## Step 1: Define the symbolic representation of the problem
The problem involves two variables: 'hours worked by Laura' and 'hours worked by Peggy', which we can denote as $x_1$ and $x_2$ respectively. The objective function to minimize is $9x_1 + 6x_2$. The constraints are:
- $14.34x_1 + 1.9x_2 \geq 47$
- $6.17x_1 + 4.28x_2 \geq 34$
- $5.43x_1 + 6.99x_2 \geq 44$
- $2x_1 - x_2 \geq 0$
- $14.34x_1 + 1.9x_2 \leq 73$
- $6.17x_1 + 4.28x_2 \leq 100$
- $5.43x_1 + 6.99x_2 \leq 82$
- $x_1$ is an integer
- $x_2$ can be a real number

## Step 2: Convert the problem into a Gurobi-compatible format
We need to define the variables, the objective function, and the constraints in a way that Gurobi can understand.

## 3: Write down the symbolic representation
The symbolic representation of the problem is:
```json
{
'sym_variables': [('x1', 'hours worked by Laura'), ('x2', 'hours worked by Peggy')],
'objective_function': '9*x1 + 6*x2',
'constraints': [
    '14.34*x1 + 1.9*x2 >= 47',
    '6.17*x1 + 4.28*x2 >= 34',
    '5.43*x1 + 6.99*x2 >= 44',
    '2*x1 - x2 >= 0',
    '14.34*x1 + 1.9*x2 <= 73',
    '6.17*x1 + 4.28*x2 <= 100',
    '5.43*x1 + 6.99*x2 <= 82',
    'x1 >= 0', 'x1 is an integer',
    'x2 >= 0'
]
}
```

## 4: Implement the problem in Gurobi
Now, let's implement this problem using Gurobi in Python.

```python
import gurobi

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

    # Define the variables
    x1 = model.addVar(name="hours_worked_by_Laura", vtype=gurobi.GRB.INTEGER)
    x2 = model.addVar(name="hours_worked_by_Peggy")

    # Set the objective function
    model.setObjective(9 * x1 + 6 * x2, gurobi.GRB.MINIMIZE)

    # Add constraints
    model.addConstr(14.34 * x1 + 1.9 * x2 >= 47)
    model.addConstr(6.17 * x1 + 4.28 * x2 >= 34)
    model.addConstr(5.43 * x1 + 6.99 * x2 >= 44)
    model.addConstr(2 * x1 - x2 >= 0)
    model.addConstr(14.34 * x1 + 1.9 * x2 <= 73)
    model.addConstr(6.17 * x1 + 4.28 * x2 <= 100)
    model.addConstr(5.43 * x1 + 6.99 * x2 <= 82)
    model.addConstr(x1 >= 0)
    model.addConstr(x2 >= 0)

    # Optimize the model
    model.optimize()

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

solve_optimization_problem()
```