## Step 1: Define the symbolic representation of the variables
The variables are ['hours worked by Peggy', 'hours worked by John', 'hours worked by Dale', 'hours worked by Bill']. Let's denote them as $x_1, x_2, x_3, x_4$ respectively.

## Step 2: Convert the natural language description into a symbolic representation
The symbolic representation of the variables is:
- $x_1$ : hours worked by Peggy
- $x_2$ : hours worked by John
- $x_3$ : hours worked by Dale
- $x_4$ : hours worked by Bill

## 3: Define the objective function in symbolic notation
The objective function to minimize is $9x_1 + 7x_2 + 4x_3 + 5x_4$.

## 4: Define the constraints in symbolic notation
The constraints are:
- $2x_3 + 15x_4 \geq 14$
- $11x_1 + 2x_3 \geq 20$
- $x_2 + 2x_3 \geq 17$
- $11x_1 + x_2 + 2x_3 + 15x_4 \geq 17$
- $8x_1 - 4x_2 \geq 0$
- $x_2 + 15x_4 \leq 38$

## 5: Consider the bounds of the variables
Since there are no explicit upper bounds given for $x_1, x_2, x_3, x_4$ except for the implicit bounds from the constraints and the fact that hours worked can be fractional, we consider them as continuous variables without explicit upper bounds.

## 6: Formulate the problem in Gurobi
We will use Gurobi to solve this linear programming problem.

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

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

# Define the variables
x1 = m.addVar(name="x1", lb=0)  # hours worked by Peggy
x2 = m.addVar(name="x2", lb=0)  # hours worked by John
x3 = m.addVar(name="x3", lb=0)  # hours worked by Dale
x4 = m.addVar(name="x4", lb=0)  # hours worked by Bill

# Define the objective function
m.setObjective(9 * x1 + 7 * x2 + 4 * x3 + 5 * x4, gp.GRB.MINIMIZE)

# Define the constraints
m.addConstr(2 * x3 + 15 * x4 >= 14)
m.addConstr(11 * x1 + 2 * x3 >= 20)
m.addConstr(x2 + 2 * x3 >= 17)
m.addConstr(11 * x1 + x2 + 2 * x3 + 15 * x4 >= 17)
m.addConstr(8 * x1 - 4 * x2 >= 0)
m.addConstr(x2 + 15 * x4 <= 38)

# Optimize the model
m.optimize()

# Print the solution
if m.status == gp.GRB.OPTIMAL:
    print("Objective: ", m.objVal)
    print("Hours worked by Peggy: ", x1.varValue)
    print("Hours worked by John: ", x2.varValue)
    print("Hours worked by Dale: ", x3.varValue)
    print("Hours worked by Bill: ", x4.varValue)
else:
    print("The model is infeasible")
```

## 8: Provide the symbolic representation of the problem
```json
{
    'sym_variables': [
        ('x1', 'hours worked by Peggy'), 
        ('x2', 'hours worked by John'), 
        ('x3', 'hours worked by Dale'), 
        ('x4', 'hours worked by Bill')
    ], 
    'objective_function': '9*x1 + 7*x2 + 4*x3 + 5*x4', 
    'constraints': [
        '2*x3 + 15*x4 >= 14', 
        '11*x1 + 2*x3 >= 20', 
        'x2 + 2*x3 >= 17', 
        '11*x1 + x2 + 2*x3 + 15*x4 >= 17', 
        '8*x1 - 4*x2 >= 0', 
        'x2 + 15*x4 <= 38'
    ]
}
```