## Step 1: Define the symbolic representation of the variables
The variables are 'hours worked by Bill' and 'hours worked by Hank', 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 $7x_1 + 2x_2$.

## Step 3: List all the constraints in symbolic notation
- $x_1 \leq 48$ (Bill's hours worked upper bound, but note this is a resource attribute and not directly a constraint on $x_1$)
- $x_2 \leq 48$ (Hank's hours worked upper bound, similarly)
- $1x_1 + 6x_2 \geq 10$ (total combined likelihood to quit index)
- $1x_1 + 6x_2 \geq 10$ is the same as the previous, so we keep it as is.
- $5x_1 + 4x_2 \geq 15$ (total combined computer competence rating)
- $-8x_1 + 8x_2 \geq 0$ ( constraint on hours worked)
- $1x_1 + 6x_2 \leq 45$ (total combined likelihood to quit index upper bound)
- $5x_1 + 4x_2 \leq 49$ (total combined computer competence rating upper bound)
- $x_1$ is an integer (Bill's hours worked must be a whole number)

## 4: Correct and refine the constraints based on the problem description
Upon review, we note:
- The upper bounds for $x_1$ and $x_2$ should be considered as $x_1 \leq 48$ and $x_2 \leq 48$ but these are not explicitly stated as constraints in the traditional sense but rather attributes.
- The actual constraints from the problem are:
  - $1x_1 + 6x_2 \geq 10$
  - $5x_1 + 4x_2 \geq 15$
  - $-8x_1 + 8x_2 \geq 0$
  - $1x_1 + 6x_2 \leq 45$
  - $5x_1 + 4x_2 \leq 49$
  - $x_1$ is an integer.

## 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
model = gurobi.Model()

# Define the variables
x1 = model.addVar(lb=0, ub=48, name="hours_worked_by_Bill", vtype=gurobi.GRB.INTEGER)
x2 = model.addVar(lb=0, ub=48, name="hours_worked_by_Hank")

# Set the objective function
model.setObjective(7 * x1 + 2 * x2, gurobi.GRB.MINIMIZE)

# Add constraints
model.addConstr(1 * x1 + 6 * x2 >= 10, name="likelihood_to_quit_index")
model.addConstr(5 * x1 + 4 * x2 >= 15, name="computer_competence_rating")
model.addConstr(-8 * x1 + 8 * x2 >= 0, name="hours_worked_constraint")
model.addConstr(1 * x1 + 6 * x2 <= 45, name="likelihood_to_quit_index_upper_bound")
model.addConstr(5 * x1 + 4 * x2 <= 49, name="computer_competence_rating_upper_bound")

# Optimize the model
model.optimize()

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

## 7: Symbolic representation of the problem
```json
{
    'sym_variables': [('x1', 'hours worked by Bill'), ('x2', 'hours worked by Hank')],
    'objective_function': '7*x1 + 2*x2',
    'constraints': [
        '1*x1 + 6*x2 >= 10',
        '5*x1 + 4*x2 >= 15',
        '-8*x1 + 8*x2 >= 0',
        '1*x1 + 6*x2 <= 45',
        '5*x1 + 4*x2 <= 49',
        'x1 is an integer'
    ]
}
```