## Step 1: Define the symbolic representation of the problem
The problem involves two variables: 'hours worked by Jean' and 'hours worked by Mary'. Let's denote 'hours worked by Jean' as $x_1$ and 'hours worked by Mary' as $x_2$. The objective function to minimize is $5.11x_1 + 6.35x_2$.

## Step 2: List the constraints in symbolic notation
The constraints given are:
- $5x_1 + 1x_2 \geq 19$
- $12x_1 + 4x_2 \geq 18$
- $10x_1 + 7x_2 \geq 21$
- $12x_1 + 14x_2 \geq 10$
- $8x_1 - x_2 \geq 0$
- $5x_1 + 1x_2 \leq 34$
- $12x_1 + 4x_2 \leq 31$
- $10x_1 + 7x_2 \leq 52$
- $12x_1 + 14x_2 \leq 42$

## 3: Provide the symbolic representation of the problem
The symbolic representation is:
```json
{
'sym_variables': [('x1', 'hours worked by Jean'), ('x2', 'hours worked by Mary')],
'objective_function': '5.11*x1 + 6.35*x2',
'constraints': [
    '5*x1 + 1*x2 >= 19',
    '12*x1 + 4*x2 >= 18',
    '10*x1 + 7*x2 >= 21',
    '12*x1 + 14*x2 >= 10',
    '8*x1 - 1*x2 >= 0',
    '5*x1 + 1*x2 <= 34',
    '12*x1 + 4*x2 <= 31',
    '10*x1 + 7*x2 <= 52',
    '12*x1 + 14*x2 <= 42'
]
}
```

## 4: Convert the problem into Gurobi code
Now, let's write the Gurobi code for this 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_Jean', lb=0)  # Assuming hours cannot be negative
    x2 = model.addVar(name='hours_worked_by_Mary', lb=0)  # Assuming hours cannot be negative

    # Define the objective function
    model.setObjective(5.11 * x1 + 6.35 * x2, gurobi.GRB.MINIMIZE)

    # Add constraints
    model.addConstr(5 * x1 + 1 * x2 >= 19)
    model.addConstr(12 * x1 + 4 * x2 >= 18)
    model.addConstr(10 * x1 + 7 * x2 >= 21)
    model.addConstr(12 * x1 + 14 * x2 >= 10)
    model.addConstr(8 * x1 - 1 * x2 >= 0)
    model.addConstr(5 * x1 + 1 * x2 <= 34)
    model.addConstr(12 * x1 + 4 * x2 <= 31)
    model.addConstr(10 * x1 + 7 * x2 <= 52)
    model.addConstr(12 * x1 + 14 * x2 <= 42)

    # Optimize the model
    model.optimize()

    # Print the status
    if model.status == gurobi.GRB.Status.OPTIMAL:
        print("Optimal solution found.")
        print(f"Hours worked by Jean: {x1.varValue}")
        print(f"Hours worked by Mary: {x2.varValue}")
        print(f"Objective function value: {model.objVal}")
    elif model.status == gurobi.GRB.Status.INFEASIBLE:
        print("No feasible solution exists.")
    else:
        print("No optimal solution found.")

solve_optimization_problem()
```