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

## Step 2: Define the symbolic representation of the problem
The objective function to maximize is $9x_1 + 6x_2$.

## Step 3: List the constraints
The constraints given are:
- $14.74x_1 + 7.16x_2 \geq 35$
- $11.02x_1 + 19.32x_2 \geq 51$
- $-6x_1 + 9x_2 \geq 0$
- $14.74x_1 + 7.16x_2 \leq 98$
- $11.02x_1 + 19.32x_2 \leq 119$

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

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

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

# Define the variables
x1 = m.addVar(name="hours_worked_by_Peggy", lb=0)  # hours worked by Peggy
x2 = m.addVar(name="hours_worked_by_Laura", lb=0)  # hours worked by Laura

# Define the objective function
m.setObjective(9 * x1 + 6 * x2, gp.GRB.MAXIMIZE)

# Define the constraints
m.addConstr(14.74 * x1 + 7.16 * x2 >= 35, name="total_dollar_cost")
m.addConstr(11.02 * x1 + 19.32 * x2 >= 51, name="total_work_quality_rating")
m.addConstr(-6 * x1 + 9 * x2 >= 0, name="work_quality_rating_constraint")
m.addConstr(14.74 * x1 + 7.16 * x2 <= 98, name="total_dollar_cost_upper_bound")
m.addConstr(11.02 * x1 + 19.32 * x2 <= 119, name="total_work_quality_rating_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 Peggy: {x1.varValue}")
    print(f"Hours worked by Laura: {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 Peggy'), ('x2', 'hours worked by Laura')],
    'objective_function': '9*x1 + 6*x2',
    'constraints': [
        '14.74*x1 + 7.16*x2 >= 35',
        '11.02*x1 + 19.32*x2 >= 51',
        '-6*x1 + 9*x2 >= 0',
        '14.74*x1 + 7.16*x2 <= 98',
        '11.02*x1 + 19.32*x2 <= 119'
    ]
}
```