To solve this optimization problem, we first need to translate the natural language description into a symbolic representation. Let's denote:

- $x_1$ as the number of first-class tickets sold.
- $x_2$ as the number of economy-class tickets sold.

The objective is to maximize profit. The profit made on each first-class ticket is $150, and on each economy-class ticket is $350. Therefore, the objective function can be represented algebraically as:

\[ \text{Maximize:} \quad 150x_1 + 350x_2 \]

The constraints given in the problem are:
1. The total number of tickets sold cannot exceed 150.
2. At least 20 tickets must be reserved for first-class.
3. At least three times as many people prefer to travel by economy-class than by first-class.

These constraints can be represented algebraically as:
1. $x_1 + x_2 \leq 150$
2. $x_1 \geq 20$
3. $x_2 \geq 3x_1$

Additionally, since we cannot sell a negative number of tickets, we also have:
4. $x_1 \geq 0$ and $x_2 \geq 0$

Therefore, the symbolic representation of the problem is:

```json
{
    'sym_variables': [('x1', 'number of first-class tickets'), ('x2', 'number of economy-class tickets')],
    'objective_function': '150*x1 + 350*x2',
    'constraints': [
        'x1 + x2 <= 150',
        'x1 >= 20',
        'x2 >= 3*x1',
        'x1 >= 0',
        'x2 >= 0'
    ]
}
```

To solve this problem using Gurobi in Python, we can use the following code:

```python
from gurobipy import *

# Create a new model
m = Model("Flight_Ticket_Optimization")

# Add variables
x1 = m.addVar(lb=0, vtype=GRB.INTEGER, name="first_class_tickets")
x2 = m.addVar(lb=0, vtype=GRB.INTEGER, name="economy_class_tickets")

# Set the objective function
m.setObjective(150*x1 + 350*x2, GRB.MAXIMIZE)

# Add constraints
m.addConstr(x1 + x2 <= 150, "total_tickets")
m.addConstr(x1 >= 20, "min_first_class")
m.addConstr(x2 >= 3*x1, "economy_vs_first_class")
m.addConstr(x1 >= 0, "non_negative_first_class")
m.addConstr(x2 >= 0, "non_negative_economy")

# Optimize the model
m.optimize()

# Print the results
if m.status == GRB.OPTIMAL:
    print("Optimal solution found:")
    print(f"First-class tickets: {x1.x}")
    print(f"Economy-class tickets: {x2.x}")
    print(f"Total profit: ${150*x1.x + 350*x2.x}")
else:
    print("No optimal solution found.")
```