To solve the given optimization problem, we first need to translate the natural language description into a symbolic representation. Let's define the variables as follows:

- $x_1$ represents the number of low-quality face wash units produced.
- $x_2$ represents the number of medium-quality face wash units produced.
- $x_3$ represents the number of high-quality face wash units produced.

The objective function aims to maximize profits. Given that the profit per low-quality face wash is $3, the profit per medium-quality face wash is $7, and the profit per high-quality face wash is $9, we can write the objective function as:

Maximize: $3x_1 + 7x_2 + 9x_3$

The constraints are based on the availability of rare ingredients and water. Since a low-quality face wash contains 1 unit of rare ingredients and 4 units of water, a medium-quality face wash contains 3 units of rare ingredients and 2 units of water, and a high-quality face wash contains 4 units of rare ingredients and 1 unit of water, we have:

- For rare ingredients: $x_1 + 3x_2 + 4x_3 \leq 100$
- For water: $4x_1 + 2x_2 + x_3 \leq 200$

Additionally, since the number of units produced cannot be negative, we have non-negativity constraints:

- $x_1 \geq 0$
- $x_2 \geq 0$
- $x_3 \geq 0$

Now, let's represent this problem in the requested JSON format:

```json
{
    'sym_variables': [('x1', 'low-quality face wash units'), ('x2', 'medium-quality face wash units'), ('x3', 'high-quality face wash units')],
    'objective_function': '3*x1 + 7*x2 + 9*x3',
    'constraints': ['x1 + 3*x2 + 4*x3 <= 100', '4*x1 + 2*x2 + x3 <= 200', 'x1 >= 0', 'x2 >= 0', 'x3 >= 0']
}
```

To solve this problem using Gurobi in Python, we'll write the following code:

```python
from gurobipy import *

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

# Define variables
x1 = m.addVar(lb=0, vtype=GRB.CONTINUOUS, name="low_quality")
x2 = m.addVar(lb=0, vtype=GRB.CONTINUOUS, name="medium_quality")
x3 = m.addVar(lb=0, vtype=GRB.CONTINUOUS, name="high_quality")

# Set the objective function
m.setObjective(3*x1 + 7*x2 + 9*x3, GRB.MAXIMIZE)

# Add constraints
m.addConstr(x1 + 3*x2 + 4*x3 <= 100, "rare_ingredients")
m.addConstr(4*x1 + 2*x2 + x3 <= 200, "water")

# Optimize the model
m.optimize()

# Print the solution
for v in m.getVars():
    print(f"{v.varName}: {v.x}")

print("Objective:", m.objVal)
```