## Step 1: Define the symbolic representation of the variables
The variables are defined as follows: 
- 'x1' corresponds to 'hours worked by Jean'
- 'x2' corresponds to 'hours worked by John'
- 'x3' corresponds to 'hours worked by Peggy'
- 'x4' corresponds to 'hours worked by George'
- 'x5' corresponds to 'hours worked by Mary'

## Step 2: Define the objective function in symbolic notation
The objective function to minimize is: $5x_1 + 3x_2 + 3x_3 + 2x_4 + 6x_5$

## Step 3: Define the constraints in symbolic notation
The constraints are:
- $16x_2 + 17x_3 \geq 56$
- $22x_1 + 17x_3 \geq 33$
- $17x_3 + 23x_5 \geq 55$
- $22x_1 + 16x_2 + 17x_3 \geq 51$
- $22x_1 + 16x_2 + 17x_3 + 2x_4 + 23x_5 \geq 51$
- $-2x_1 + 8x_5 \geq 0$
- $16x_2 + 2x_4 \leq 254$
- $16x_2 + 17x_3 \leq 96$
- $22x_1 + 16x_2 \leq 170$
- $22x_1 + 2x_4 \leq 94$
- $17x_3 + 23x_5 \leq 87$
- $22x_1 + 23x_5 \leq 202$
- $2x_4 + 23x_5 \leq 62$
- $17x_3 + 2x_4 \leq 244$

## 4: Specify variable bounds and types
- $x_1$ is an integer
- $x_2$ is an integer
- $x_3$ is an integer
- $x_4$ is continuous
- $x_5$ is continuous

## 5: Write down the symbolic representation of the problem
```json
{
    'sym_variables': [
        ('x1', 'hours worked by Jean'), 
        ('x2', 'hours worked by John'), 
        ('x3', 'hours worked by Peggy'), 
        ('x4', 'hours worked by George'), 
        ('x5', 'hours worked by Mary')
    ], 
    'objective_function': '5*x1 + 3*x2 + 3*x3 + 2*x4 + 6*x5', 
    'constraints': [
        '16*x2 + 17*x3 >= 56',
        '22*x1 + 17*x3 >= 33',
        '17*x3 + 23*x5 >= 55',
        '22*x1 + 16*x2 + 17*x3 >= 51',
        '22*x1 + 16*x2 + 17*x3 + 2*x4 + 23*x5 >= 51',
        '-2*x1 + 8*x5 >= 0',
        '16*x2 + 2*x4 <= 254',
        '16*x2 + 17*x3 <= 96',
        '22*x1 + 16*x2 <= 170',
        '22*x1 + 2*x4 <= 94',
        '17*x3 + 23*x5 <= 87',
        '22*x1 + 23*x5 <= 202',
        '2*x4 + 23*x5 <= 62',
        '17*x3 + 2*x4 <= 244'
    ]
}
```

## 6: Implement the optimization problem using Gurobi
```python
import gurobi

def optimize_problem():
    model = gurobi.Model()

    # Define variables
    x1 = model.addVar(name="x1", vtype=gurobi.GRB.INTEGER)  # hours worked by Jean
    x2 = model.addVar(name="x2", vtype=gurobi.GRB.INTEGER)  # hours worked by John
    x3 = model.addVar(name="x3", vtype=gurobi.GRB.INTEGER)  # hours worked by Peggy
    x4 = model.addVar(name="x4")  # hours worked by George
    x5 = model.addVar(name="x5")  # hours worked by Mary

    # Objective function
    model.setObjective(5 * x1 + 3 * x2 + 3 * x3 + 2 * x4 + 6 * x5, gurobi.GRB.MINIMIZE)

    # Constraints
    model.addConstr(16 * x2 + 17 * x3 >= 56)
    model.addConstr(22 * x1 + 17 * x3 >= 33)
    model.addConstr(17 * x3 + 23 * x5 >= 55)
    model.addConstr(22 * x1 + 16 * x2 + 17 * x3 >= 51)
    model.addConstr(22 * x1 + 16 * x2 + 17 * x3 + 2 * x4 + 23 * x5 >= 51)
    model.addConstr(-2 * x1 + 8 * x5 >= 0)
    model.addConstr(16 * x2 + 2 * x4 <= 254)
    model.addConstr(16 * x2 + 17 * x3 <= 96)
    model.addConstr(22 * x1 + 16 * x2 <= 170)
    model.addConstr(22 * x1 + 2 * x4 <= 94)
    model.addConstr(17 * x3 + 23 * x5 <= 87)
    model.addConstr(22 * x1 + 23 * x5 <= 202)
    model.addConstr(2 * x4 + 23 * x5 <= 62)
    model.addConstr(17 * x3 + 2 * x4 <= 244)

    # Solve the model
    model.optimize()

    # Print the solution
    if model.status == gurobi.GRB.OPTIMAL:
        print("Optimal solution found.")
        print(f"Hours worked by Jean: {x1.varValue}")
        print(f"Hours worked by John: {x2.varValue}")
        print(f"Hours worked by Peggy: {x3.varValue}")
        print(f"Hours worked by George: {x4.varValue}")
        print(f"Hours worked by Mary: {x5.varValue}")
        print(f"Objective function value: {model.objVal}")
    else:
        print("No optimal solution found.")

optimize_problem()
```