## Step 1: Define the symbolic representation of the problem
The problem involves two variables: 'hours worked by Bobby' and 'hours worked by Peggy'. Let's denote 'hours worked by Bobby' as $x_1$ and 'hours worked by Peggy' as $x_2$. The objective function to maximize is $1 \cdot x_1 + 1 \cdot x_2 = x_1 + x_2$.

## Step 2: List the constraints in symbolic notation
The constraints given are:
1. Bobby's work quality rating is 4: $4x_1$ (but this seems to be an attribute, not a constraint in the traditional sense, as it's a given condition).
2. Bobby has an organization score of 2: $2x_1$ (similarly, this is a given condition).
3. Peggy has a work quality rating of 5: $5x_2$ (given condition).
4. Peggy has an organization score of 8: $8x_2$ (given condition).
5. The total combined work quality rating must be no less than 6: $4x_1 + 5x_2 \geq 6$.
6. The total combined organization score must be equal to or greater than 10: $2x_1 + 8x_2 \geq 10$.
7. $-1 \cdot x_1 + 6 \cdot x_2 \geq 0$.
8. The total combined work quality rating must be at most 13: $4x_1 + 5x_2 \leq 13$.
9. The total combined organization score should be less than or equal to 31: $2x_1 + 8x_2 \leq 31$.
10. $x_1$ must be an integer (nonfractional).

## 3: Correctly frame the symbolic representation
Given the variables and constraints, the symbolic representation is:
- Variables: $x_1$ (hours worked by Bobby), $x_2$ (hours worked by Peggy)
- Objective function: Maximize $x_1 + x_2$
- Constraints:
  1. $4x_1 + 5x_2 \geq 6$
  2. $2x_1 + 8x_2 \geq 10$
  3. $-x_1 + 6x_2 \geq 0$
  4. $4x_1 + 5x_2 \leq 13$
  5. $2x_1 + 8x_2 \leq 31$
  6. $x_1 \in \mathbb{Z}$ (integer constraint)

## 4: Convert to Gurobi Python code
```python
import gurobi

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

# Define the variables
x1 = m.addVar(name="hours_worked_by_Bobby", vtype=gurobi.GRB.INTEGER)  # Integer variable for Bobby
x2 = m.addVar(name="hours_worked_by_Peggy")  # Continuous variable for Peggy

# Objective function: Maximize x1 + x2
m.setObjective(x1 + x2, gurobi.GRB.MAXIMIZE)

# Constraints
m.addConstr(4*x1 + 5*x2 >= 6, name="work_quality_rating_constraint")
m.addConstr(2*x1 + 8*x2 >= 10, name="organization_score_constraint")
m.addConstr(-x1 + 6*x2 >= 0, name="bobby_peggy_hours_constraint")
m.addConstr(4*x1 + 5*x2 <= 13, name="max_work_quality_rating_constraint")
m.addConstr(2*x1 + 8*x2 <= 31, name="max_organization_score_constraint")

# Optimize the model
m.optimize()

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

## 5: Symbolic Representation Output
```json
{
    'sym_variables': [('x1', 'hours worked by Bobby'), ('x2', 'hours worked by Peggy')],
    'objective_function': 'x1 + x2',
    'constraints': [
        '4*x1 + 5*x2 >= 6',
        '2*x1 + 8*x2 >= 10',
        '-x1 + 6*x2 >= 0',
        '4*x1 + 5*x2 <= 13',
        '2*x1 + 8*x2 <= 31',
        'x1 ∈ ℤ'
    ]
}
```