## Step 1: Define the symbolic representation of the problem
The problem involves two variables: 'hours worked by Ringo' and 'hours worked by Dale'. Let's denote 'hours worked by Ringo' as $x_1$ and 'hours worked by Dale' as $x_2$. The objective function to maximize is $8x_1 + 4x_2$.

## Step 2: List the constraints in symbolic notation
The constraints given are:
1. $10x_1 + 2x_2 \geq 15$ (total combined work quality rating)
2. $10x_1 + 7x_2 \geq 42$ (total combined computer competence rating)
3. $3x_1 + 4x_2 \geq 9$ (total combined likelihood to quit index)
4. $2x_1 - 9x_2 \geq 0$ (relationship between hours worked by Ringo and Dale)
5. $10x_1 + 2x_2 \leq 48$ (upper limit on total combined work quality rating)
6. $10x_1 + 7x_2 \leq 96$ (upper limit on total combined computer competence rating)
7. $3x_1 + 4x_2 \leq 31$ (upper limit on total combined likelihood to quit index)
8. $x_1$ is an integer (hours worked by Ringo)
9. $x_2$ can be a non-integer (hours worked by Dale)

## 3: Provide the symbolic representation in JSON format
```json
{
'sym_variables': [('x1', 'hours worked by Ringo'), ('x2', 'hours worked by Dale')],
'objective_function': '8*x1 + 4*x2',
'constraints': [
    '10*x1 + 2*x2 >= 15',
    '10*x1 + 7*x2 >= 42',
    '3*x1 + 4*x2 >= 9',
    '2*x1 - 9*x2 >= 0',
    '10*x1 + 2*x2 <= 48',
    '10*x1 + 7*x2 <= 96',
    '3*x1 + 4*x2 <= 31',
    'x1 == int(x1)',
    'x2 >= 0'
]
}
```

## 4: Write the Gurobi code to solve the optimization problem
```python
import gurobi

def solve_optimization_problem():
    # Create a new Gurobi model
    model = gurobi.Model()

    # Define the variables
    x1 = model.addVar(name="x1", vtype=gurobi.GRB.INTEGER)  # hours worked by Ringo
    x2 = model.addVar(name="x2")  # hours worked by Dale

    # Define the objective function
    model.setObjective(8 * x1 + 4 * x2, gurobi.GRB.MAXIMIZE)

    # Add constraints
    model.addConstr(10 * x1 + 2 * x2 >= 15, name="work_quality_rating")
    model.addConstr(10 * x1 + 7 * x2 >= 42, name="computer_competence_rating")
    model.addConstr(3 * x1 + 4 * x2 >= 9, name="likelihood_to_quit_index")
    model.addConstr(2 * x1 - 9 * x2 >= 0, name="relationship_hours_worked")
    model.addConstr(10 * x1 + 2 * x2 <= 48, name="upper_limit_work_quality")
    model.addConstr(10 * x1 + 7 * x2 <= 96, name="upper_limit_computer_competence")
    model.addConstr(3 * x1 + 4 * x2 <= 31, name="upper_limit_likelihood_to_quit")

    # Set bounds for x2 (not necessary but good practice)
    model.addConstr(x2 >= 0)

    # Solve the model
    model.optimize()

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

solve_optimization_problem()
```