## Problem Description and Formulation

The consumer electronics business needs to determine the optimal production levels of two video game consoles, PX7 and Silent X, to maximize profit. The production of each console has the following characteristics:

- PX7: 3 hours of labor, $40 profit per unit
- Silent X: 7 hours of labor, $75 profit per unit

There are two constraints:
1. The business must produce at least twice as many PX7 consoles as Silent X consoles.
2. The total labor hours available for production is 48 hours per week.

## Mathematical Formulation

Let \(x\) be the number of PX7 consoles and \(y\) be the number of Silent X consoles.

The objective function to maximize profit is:
\[ \text{Maximize:} \quad 40x + 75y \]

Subject to:
1. \( x \geq 2y \) (at least twice as many PX7 consoles as Silent X consoles)
2. \( 3x + 7y \leq 48 \) (total labor hours constraint)
3. \( x \geq 0, y \geq 0 \) (non-negativity constraint, as the number of consoles cannot be negative)

## Gurobi Code

```python
import gurobi

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

    # Define variables
    x = model.addVar(name="PX7", vtype=gurobi.GRB.INTEGER, lb=0)  # Number of PX7 consoles
    y = model.addVar(name="Silent_X", vtype=gurobi.GRB.INTEGER, lb=0)  # Number of Silent X consoles

    # Objective function: Maximize profit
    model.setObjective(40 * x + 75 * y, sense=gurobi.GRB.MAXIMIZE)

    # Constraints
    model.addConstr(x >= 2 * y, name="PX7_to_SilentX_ratio")
    model.addConstr(3 * x + 7 * y <= 48, name="Labor_hours_constraint")

    # Solve the model
    model.optimize()

    # Print solution
    if model.status == gurobi.GRB.OPTIMAL:
        print(f"Optimal production levels: {x.varName} = {x.x}, {y.varName} = {y.x}")
        print(f"Maximum profit: ${40 * x.x + 75 * y.x:.2f}")
    else:
        print("The problem is infeasible.")

solve_production_problem()
```

This Gurobi code defines the optimization problem as described, with the objective to maximize profit under the given constraints. It then solves the problem and prints out the optimal production levels for PX7 and Silent X consoles, along with the maximum achievable profit. If the problem is infeasible, it indicates so.