## Step 1: Define the symbolic representation of the variables
The variables are 'hours worked by Mary' and 'hours worked by Peggy', 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 $6.86x_1 + 6.82x_2$.

## Step 3: List all the constraints in symbolic notation
The constraints are:
- $14x_1 + 3x_2 \geq 27$
- $5x_1 + 12x_2 \geq 25$
- $4x_1 + 9x_2 \geq 23$
- $7x_1 + 9x_2 \geq 18$
- $12x_1 + 13x_2 \geq 36$
- $x_1 - 10x_2 \geq 0$
- $14x_1 + 3x_2 \leq 68$
- $5x_1 + 12x_2 \leq 103$
- $4x_1 + 9x_2 \leq 54$
- $7x_1 + 9x_2 \leq 49$
- $12x_1 + 13x_2 \leq 53$

## 4: Provide the symbolic representation in JSON format
```json
{
    'sym_variables': [('x1', 'hours worked by Mary'), ('x2', 'hours worked by Peggy')],
    'objective_function': '6.86*x1 + 6.82*x2',
    'constraints': [
        '14*x1 + 3*x2 >= 27',
        '5*x1 + 12*x2 >= 25',
        '4*x1 + 9*x2 >= 23',
        '7*x1 + 9*x2 >= 18',
        '12*x1 + 13*x2 >= 36',
        'x1 - 10*x2 >= 0',
        '14*x1 + 3*x2 <= 68',
        '5*x1 + 12*x2 <= 103',
        '4*x1 + 9*x2 <= 54',
        '7*x1 + 9*x2 <= 49',
        '12*x1 + 13*x2 <= 53'
    ]
}
```

## 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_Mary', lb=0)  # No upper bound for hours worked
    x2 = model.addVar(name='hours_worked_by_Peggy', lb=0)  # No upper bound for hours worked

    # Define the objective function
    model.setObjective(6.86 * x1 + 6.82 * x2, gurobi.GRB.MINIMIZE)

    # Add constraints
    model.addConstr(14 * x1 + 3 * x2 >= 27)
    model.addConstr(5 * x1 + 12 * x2 >= 25)
    model.addConstr(4 * x1 + 9 * x2 >= 23)
    model.addConstr(7 * x1 + 9 * x2 >= 18)
    model.addConstr(12 * x1 + 13 * x2 >= 36)
    model.addConstr(x1 - 10 * x2 >= 0)
    model.addConstr(14 * x1 + 3 * x2 <= 68)
    model.addConstr(5 * x1 + 12 * x2 <= 103)
    model.addConstr(4 * x1 + 9 * x2 <= 54)
    model.addConstr(7 * x1 + 9 * x2 <= 49)
    model.addConstr(12 * x1 + 13 * x2 <= 53)

    # Optimize the model
    model.optimize()

    # Print the solution
    if model.status == gurobi.GRB.OPTIMAL:
        print('Optimal solution found.')
        print(f'Hours worked by Mary: {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()
```