## Step 1: Define the symbolic representation of the variables
The variables are ['hours worked by Peggy', 'hours worked by Laura', 'hours worked by George', 'hours worked by Mary']. Let's denote them as $x_1, x_2, x_3, x_4$ respectively.

## Step 2: Convert the objective function into a symbolic representation
The objective function to minimize is $7.35x_1 + 3.76x_2 + 4.62x_3 + 9.64x_4$.

## Step 3: Define the constraints in symbolic notation
The constraints given are:
1. $22x_1 + 4x_3 \geq 49$
2. $22x_1 + 16x_4 \geq 66$
3. $10x_2 + 16x_4 \geq 72$
4. $10x_2 + 4x_3 + 16x_4 \geq 71$
5. $22x_1 + 10x_2 + 16x_4 \geq 71$
6. $10x_2 + 4x_3 + 16x_4 \geq 69$
7. $22x_1 + 10x_2 + 16x_4 \geq 69$
8. $22x_1 + 10x_2 + 4x_3 + 16x_4 \geq 69$
9. $10x_1 - 3x_4 \geq 0$
10. $5x_3 - 5x_4 \geq 0$

## 4: Consider the variable bounds and types
- $x_1, x_2, x_3$ can be non-integer (or continuous), but $x_4$ must be an integer.

## 5: Formulate the problem in Gurobi
We will use Gurobi to solve this linear programming problem.

```python
import gurobi as gp

# Define the model
m = gp.Model("optimization_problem")

# Define the variables
x1 = m.addVar(name="hours_worked_by_Peggy", lb=0)  # No upper bound given
x2 = m.addVar(name="hours_worked_by_Laura", lb=0)  # No upper bound given
x3 = m.addVar(name="hours_worked_by_George", lb=0)  # No upper bound given
x4 = m.addVar(name="hours_worked_by_Mary", lb=0, vtype=gp.GRB.INTEGER)  # Must be integer

# Define the objective function
m.setObjective(7.35*x1 + 3.76*x2 + 4.62*x3 + 9.64*x4, gp.GRB.MINIMIZE)

# Add constraints
m.addConstr(22*x1 + 4*x3 >= 49)
m.addConstr(22*x1 + 16*x4 >= 66)
m.addConstr(10*x2 + 16*x4 >= 72)
m.addConstr(10*x2 + 4*x3 + 16*x4 >= 71)
m.addConstr(22*x1 + 10*x2 + 16*x4 >= 71)
m.addConstr(10*x2 + 4*x3 + 16*x4 >= 69)
m.addConstr(22*x1 + 10*x2 + 16*x4 >= 69)
m.addConstr(22*x1 + 10*x2 + 4*x3 + 16*x4 >= 69)
m.addConstr(10*x1 - 3*x4 >= 0)
m.addConstr(5*x3 - 5*x4 >= 0)

# Optimize the model
m.optimize()

# Print the solution
if m.status == gp.GRB.OPTIMAL:
    print("Objective: ", m.objVal)
    print("Hours worked by Peggy: ", x1.varValue)
    print("Hours worked by Laura: ", x2.varValue)
    print("Hours worked by George: ", x3.varValue)
    print("Hours worked by Mary: ", x4.varValue)
else:
    print("The model is infeasible")
```

## 6: Symbolic Representation
The symbolic representation is as follows:
```json
{
    'sym_variables': [
        ('x1', 'hours worked by Peggy'),
        ('x2', 'hours worked by Laura'),
        ('x3', 'hours worked by George'),
        ('x4', 'hours worked by Mary')
    ],
    'objective_function': '7.35*x1 + 3.76*x2 + 4.62*x3 + 9.64*x4',
    'constraints': [
        '22*x1 + 4*x3 >= 49',
        '22*x1 + 16*x4 >= 66',
        '10*x2 + 16*x4 >= 72',
        '10*x2 + 4*x3 + 16*x4 >= 71',
        '22*x1 + 10*x2 + 16*x4 >= 71',
        '10*x2 + 4*x3 + 16*x4 >= 69',
        '22*x1 + 10*x2 + 16*x4 >= 69',
        '22*x1 + 10*x2 + 4*x3 + 16*x4 >= 69',
        '10*x1 - 3*x4 >= 0',
        '5*x3 - 5*x4 >= 0'
    ]
}
```