## Step 1: Define the symbolic representation of the variables
The variables are 'hours worked by George', 'hours worked by Bill', and 'hours worked by Hank', which can be represented symbolically as $x_0$, $x_1$, and $x_2$ respectively.

## Step 2: Define the objective function in symbolic notation
The objective function to minimize is $9x_0 + 5x_1 + 9x_2$.

## Step 3: Define the constraints in symbolic notation
The constraints given are:
- $1x_0 \geq 1$ (George's likelihood to quit index)
- $3x_0 \geq 3$ (George's dollar cost per hour, but this seems to be an equality or directly given condition rather than an optimization constraint)
- $3x_1 \geq 3$ (Bill's likelihood to quit index, but similar to George's, this seems directly given)
- $4x_1 \geq 4$ (Bill's dollar cost per hour, similar to above)
- $4x_2 \geq 4$ (Hank's likelihood to quit index, directly given)
- $1x_2 \geq 1$ (Hank's dollar cost per hour, directly given)
- $1x_0 + 4x_2 \geq 19$ (Total combined likelihood to quit index from George and Hank)
- $1x_0 + 3x_1 \geq 24$ (Total combined likelihood to quit index from George and Bill)
- $1x_0 + 3x_1 + 4x_2 \geq 24$ (Total combined likelihood to quit index from all)
- $3x_0 + 1x_2 \geq 14$ (Total combined dollar cost per hour from George and Hank)
- $3x_0 + 4x_1 \geq 6$ (Total combined dollar cost per hour from George and Bill)
- $4x_1 + 1x_2 \geq 6$ (Total combined dollar cost per hour from Bill and Hank)
- $3x_0 + 4x_1 + 1x_2 \geq 6$ (Total combined dollar cost per hour from all)
- $-8x_1 + 3x_2 \geq 0$ (Constraint involving Bill and Hank)
- $1x_0 + 4x_2 \leq 39$ (Total combined likelihood to quit index from George and Hank, upper bound)
- $3x_0 + 1x_2 \leq 53$ (Total combined dollar cost per hour from George and Hank, upper bound)
- $3x_0 + 4x_1 + 1x_2 \leq 38$ (Total combined dollar cost per hour from all, upper bound)

## 4: Correct and Simplify Constraints
Some constraints seem to directly describe the variables rather than being optimization constraints. The actual optimization constraints from the problem statement are:
- $x_0 + 4x_2 \geq 19$
- $x_0 + 3x_1 \geq 24$
- $x_0 + 3x_1 + 4x_2 \geq 24$
- $3x_0 + x_2 \geq 14$
- $3x_0 + 4x_1 \geq 6$
- $4x_1 + x_2 \geq 6$
- $3x_0 + 4x_1 + x_2 \geq 6$
- $-8x_1 + 3x_2 \geq 0$
- $x_0 + 4x_2 \leq 39$
- $3x_0 + x_2 \leq 53$
- $3x_0 + 4x_1 + x_2 \leq 38$

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

## 6: Write the Gurobi Code
```python
import gurobi

# Create a new model
m = gurobi.Model()

# Define the variables
x0 = m.addVar(name="x0", lb=0)  # hours worked by George
x1 = m.addVar(name="x1", lb=0)  # hours worked by Bill
x2 = m.addVar(name="x2", lb=0)  # hours worked by Hank

# Objective function
m.setObjective(9 * x0 + 5 * x1 + 9 * x2, gurobi.GRB.MINIMIZE)

# Constraints
m.addConstr(x0 + 4 * x2 >= 19)
m.addConstr(x0 + 3 * x1 >= 24)
m.addConstr(x0 + 3 * x1 + 4 * x2 >= 24)
m.addConstr(3 * x0 + x2 >= 14)
m.addConstr(3 * x0 + 4 * x1 >= 6)
m.addConstr(4 * x1 + x2 >= 6)
m.addConstr(3 * x0 + 4 * x1 + x2 >= 6)
m.addConstr(-8 * x1 + 3 * x2 >= 0)
m.addConstr(x0 + 4 * x2 <= 39)
m.addConstr(3 * x0 + x2 <= 53)
m.addConstr(3 * x0 + 4 * x1 + x2 <= 38)

# Optimize the model
m.optimize()

# Print the solution
if m.status == gurobi.GRB.OPTIMAL:
    print("Objective: ", m.objval)
    print("Hours worked by George: ", x0.varValue)
    print("Hours worked by Bill: ", x1.varValue)
    print("Hours worked by Hank: ", x2.varValue)
else:
    print("The model is infeasible")
```

## 7: Symbolic Representation
```json
{
    'sym_variables': [('x0', 'hours worked by George'), ('x1', 'hours worked by Bill'), ('x2', 'hours worked by Hank')],
    'objective_function': '9*x0 + 5*x1 + 9*x2',
    'constraints': [
        'x0 + 4*x2 >= 19',
        'x0 + 3*x1 >= 24',
        'x0 + 3*x1 + 4*x2 >= 24',
        '3*x0 + x2 >= 14',
        '3*x0 + 4*x1 >= 6',
        '4*x1 + x2 >= 6',
        '3*x0 + 4*x1 + x2 >= 6',
        '-8*x1 + 3*x2 >= 0',
        'x0 + 4*x2 <= 39',
        '3*x0 + x2 <= 53',
        '3*x0 + 4*x1 + x2 <= 38'
    ]
}
```