To solve the given optimization problem using Gurobi, we first need to understand and possibly simplify or clarify the constraints provided. The objective function to maximize is \(7x_0 + 4x_1\), where \(x_0\) represents the hours worked by Peggy and \(x_1\) represents the hours worked by Ringo.

The constraints given are as follows:
- Peggy's likelihood to quit index is 1, and Ringo's is 5.
- Peggy's computer competence rating is 6, and Ringo's is 7.
- The total combined likelihood to quit index from both must be at least 21 and at most 44.
- The total combined computer competence rating from both must be at least 48 and at most 98.
- \(3x_0 - 8x_1 \geq 0\).
- \(x_0\) can be fractional, but \(x_1\) must be an integer.

Given these constraints, let's formulate the problem in a way that Gurobi can solve. We will define two variables: `peggy_hours` for the hours worked by Peggy and `ringo_hours` for the hours worked by Ringo. The likelihood to quit index and computer competence rating for each are given as constants.

```python
from gurobipy import *

# Create a model
m = Model("Optimization_Problem")

# Define variables
peggy_hours = m.addVar(vtype=GRB.CONTINUOUS, name="Peggy_Hours")
ringo_hours = m.addVar(vtype=GRB.INTEGER, name="Ringo_Hours")

# Objective function: Maximize 7*peggy_hours + 4*ringo_hours
m.setObjective(7*peggy_hours + 4*ringo_hours, GRB.MAXIMIZE)

# Constraints
# Total combined likelihood to quit index is at least 21 and at most 44
m.addConstr(1*peggy_hours + 5*ringo_hours >= 21, "Likelihood_Lower_Bound")
m.addConstr(1*peggy_hours + 5*ringo_hours <= 44, "Likelihood_Upper_Bound")

# Total combined computer competence rating is at least 48 and at most 98
m.addConstr(6*peggy_hours + 7*ringo_hours >= 48, "Competence_Lower_Bound")
m.addConstr(6*peggy_hours + 7*ringo_hours <= 98, "Competence_Upper_Bound")

# Additional constraint: 3*peggy_hours - 8*ringo_hours >= 0
m.addConstr(3*peggy_hours - 8*ringo_hours >= 0, "Additional_Constraint")

# Solve the model
m.optimize()

# Print solution
if m.status == GRB.OPTIMAL:
    print("Optimal solution found:")
    print(f"Peggy Hours: {peggy_hours.X}")
    print(f"Ringo Hours: {ringo_hours.X}")
else:
    print("No optimal solution found")
```