## Step 1: Define the symbolic representation of the variables
The variables are 'hours worked by Hank' and 'hours worked by George', which we can denote as $x_1$ and $x_2$ respectively.

## Step 2: Define the objective function in symbolic notation
The objective function to maximize is $1 \times x_1 + 9 \times x_2$.

## Step 3: Define the constraints in symbolic notation
The constraints are:
- $4x_1 + 2x_2 \geq 22$ (total combined likelihood to quit index)
- $7x_1 + 2x_2 \geq 30$ (total combined dollar cost per hour)
- $3x_1 - 4x_2 \geq 0$
- $4x_1 + 2x_2 \leq 55$ (total combined likelihood to quit index upper bound)
- $7x_1 + 2x_2 \leq 51$ (total combined dollar cost per hour upper bound)
- $x_1 \in \mathbb{Z}$ (Hank's hours are a whole number)
- $x_2 \in \mathbb{Z}$ (George's hours are a whole number)

## 4: Convert the problem into a Gurobi-compatible format
We will use Gurobi's Python API to model and solve this problem.

## 5: Write the Gurobi code
```python
import gurobi as gp

# Create a new model
m = gp.Model("optimization_problem")

# Define the variables
x1 = m.addVar(name="hours_worked_by_Hank", vtype=gp.GRB.INTEGER)
x2 = m.addVar(name="hours_worked_by_George", vtype=gp.GRB.INTEGER)

# Define the objective function
m.setObjective(x1 + 9 * x2, gp.GRB.MAXIMIZE)

# Define the constraints
m.addConstr(4 * x1 + 2 * x2 >= 22, name="likelihood_to_quit_index")
m.addConstr(7 * x1 + 2 * x2 >= 30, name="dollar_cost_per_hour")
m.addConstr(3 * x1 - 4 * x2 >= 0, name="hours_worked_constraint")
m.addConstr(4 * x1 + 2 * x2 <= 55, name="likelihood_to_quit_index_upper_bound")
m.addConstr(7 * x1 + 2 * x2 <= 51, name="dollar_cost_per_hour_upper_bound")

# Optimize the model
m.optimize()

# Print the solution
if m.status == gp.GRB.OPTIMAL:
    print("Optimal solution found.")
    print(f"Hours worked by Hank: {x1.varValue}")
    print(f"Hours worked by George: {x2.varValue}")
    print(f"Objective function value: {m.objVal}")
else:
    print("No optimal solution found.")
```

## 6: Provide the symbolic representation of the problem
```json
{
    'sym_variables': [('x1', 'hours worked by Hank'), ('x2', 'hours worked by George')],
    'objective_function': '1*x1 + 9*x2',
    'constraints': [
        '4*x1 + 2*x2 >= 22',
        '7*x1 + 2*x2 >= 30',
        '3*x1 - 4*x2 >= 0',
        '4*x1 + 2*x2 <= 55',
        '7*x1 + 2*x2 <= 51',
        'x1 ∈ ℤ',
        'x2 ∈ ℤ'
    ]
}
```