To solve this optimization problem, we will first create a symbolic representation of the problem. Let's define the variables as follows:

- $x_1$: hours worked by Laura
- $x_2$: hours worked by Peggy

The objective function is to minimize $9x_1 + 6x_2$.

The constraints are:
- $14.34x_1 + 1.9x_2 \geq 47$
- $14.34x_1 + 1.9x_2 \geq 47$ (same as above, so we'll consider it only once)
- $6.17x_1 + 4.28x_2 \geq 34$
- $6.17x_1 + 4.28x_2 \geq 34$ (same as above, so we'll consider it only once)
- $5.43x_1 + 6.99x_2 \geq 44$
- $5.43x_1 + 6.99x_2 \geq 44$ (same as above, so we'll consider it only once)
- $2x_1 - x_2 \geq 0$
- $14.34x_1 + 1.9x_2 \leq 73$
- $6.17x_1 + 4.28x_2 \leq 100$
- $5.43x_1 + 6.99x_2 \leq 82$
- $x_1$ must be an integer (non-fractional)
- $x_2$ can be a fraction

Here is the symbolic representation in JSON format:
```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'
    ]
}
```

Now, let's implement this problem in Gurobi using Python:
```python
from gurobipy import *

# Create a model
m = Model("Optimization Problem")

# Define variables
x1 = m.addVar(vtype=GRB.INTEGER, name="hours_worked_by_Laura")
x2 = m.addVar(vtype=GRB.CONTINUOUS, name="hours_worked_by_Peggy")

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

# Add constraints
m.addConstr(14.34*x1 + 1.9*x2 >= 47, "productivity_rating")
m.addConstr(6.17*x1 + 4.28*x2 >= 34, "work_quality_rating")
m.addConstr(5.43*x1 + 6.99*x2 >= 44, "dollar_cost_per_hour")
m.addConstr(2*x1 - x2 >= 0, "hours_worked_constraint")
m.addConstr(14.34*x1 + 1.9*x2 <= 73, "productivity_rating_upper_bound")
m.addConstr(6.17*x1 + 4.28*x2 <= 100, "work_quality_rating_upper_bound")
m.addConstr(5.43*x1 + 6.99*x2 <= 82, "dollar_cost_per_hour_upper_bound")

# Optimize model
m.optimize()

# Print solution
if m.status == GRB.OPTIMAL:
    print("Optimal solution found:")
    print("Hours worked by Laura:", x1.x)
    print("Hours worked by Peggy:", x2.x)
    print("Objective function value:", m.objVal)
else:
    print("No optimal solution found")
```