To solve the optimization problem described, we first need to translate the natural language description into a symbolic representation. This involves defining variables, an objective function, and constraints using mathematical notation.

Let's define:
- $x_1$ as the hours worked by Laura,
- $x_2$ as the hours worked by George.

The objective function is to maximize: $1 \cdot x_1 + 3 \cdot x_2$.

Given constraints are:
1. Organization score constraint for Laura: $0.82 \cdot x_1$ (no direct constraint given, but used in total organization score calculations).
2. Productivity rating constraint for Laura: $1.78 \cdot x_1$ (similar to above, no direct constraint).
3. Organization score constraint for George: $0.01 \cdot x_2$.
4. Productivity rating constraint for George: $1.34 \cdot x_2$.
5. Total combined organization score must be 7 or more: $0.82x_1 + 0.01x_2 \geq 7$.
6. Total combined productivity rating must be at least 6: $1.78x_1 + 1.34x_2 \geq 6$.
7. The specific linear inequality constraint: $7x_1 - 2x_2 \geq 0$.
8. Maximum total organization score should not exceed 19: $0.82x_1 + 0.01x_2 \leq 19$.
9. Maximum productivity rating constraint: $1.78x_1 + 1.34x_2 \leq 22$.

Symbolic representation:
```json
{
    'sym_variables': [('x1', 'hours worked by Laura'), ('x2', 'hours worked by George')],
    'objective_function': 'maximize x1 + 3*x2',
    'constraints': [
        '0.82*x1 + 0.01*x2 >= 7',
        '1.78*x1 + 1.34*x2 >= 6',
        '7*x1 - 2*x2 >= 0',
        '0.82*x1 + 0.01*x2 <= 19',
        '1.78*x1 + 1.34*x2 <= 22'
    ]
}
```

Now, let's implement this problem using Gurobi Python:

```python
from gurobipy import *

# Create a new model
m = Model("Work Hours Optimization")

# Define variables
x1 = m.addVar(vtype=GRB.CONTINUOUS, name="hours_worked_by_Laura")
x2 = m.addVar(vtype=GRB.INTEGER, name="hours_worked_by_George")

# Objective function: Maximize hours worked by Laura plus 3 times hours worked by George
m.setObjective(x1 + 3*x2, GRB.MAXIMIZE)

# Constraints
m.addConstr(0.82*x1 + 0.01*x2 >= 7, "Total organization score constraint")
m.addConstr(1.78*x1 + 1.34*x2 >= 6, "Total productivity rating minimum constraint")
m.addConstr(7*x1 - 2*x2 >= 0, "Linear inequality constraint")
m.addConstr(0.82*x1 + 0.01*x2 <= 19, "Maximum total organization score constraint")
m.addConstr(1.78*x1 + 1.34*x2 <= 22, "Maximum productivity rating constraint")

# Optimize model
m.optimize()

# Print solution
if m.status == GRB.OPTIMAL:
    print("Optimal solution found:")
    print(f"Hours worked by Laura: {x1.x}")
    print(f"Hours worked by George: {x2.x}")
else:
    print("No optimal solution found")
```