## Step 1: Define the symbolic representation of the problem
Let's denote the number of desks made by Manufacturer X as $x_1$ and the number of chairs made by Manufacturer Y as $x_2$. However, to accurately represent the problem, we need to consider that there are two manufacturers for desks and chairs, but the problem description specifically limits Manufacturer X to desks and Manufacturer Y to chairs. Thus, we will have:
- $x_1$ as the number of desks made by Manufacturer X
- $x_2$ as the number of chairs made by Manufacturer Y

The objective is to maximize profit, given that the profit per desk is $140 and per chair is $120. So, the objective function is $140x_1 + 120x_2$.

## Step 2: List the constraints
1. Manufacturer X can make at most 22 desks a day: $x_1 \leq 22$
2. Manufacturer Y can make at most 28 chairs a day: $x_2 \leq 28$
3. The quality checking company can check at most 40 items per day: $x_1 + x_2 \leq 40$
4. Non-negativity constraints: $x_1 \geq 0, x_2 \geq 0$

## 3: Symbolic representation in the required format
```json
{
'sym_variables': [('x1', 'desks from Manufacturer X'), ('x2', 'chairs from Manufacturer Y')],
'objective_function': '140*x1 + 120*x2',
'constraints': [
    'x1 <= 22',
    'x2 <= 28',
    'x1 + x2 <= 40',
    'x1 >= 0',
    'x2 >= 0'
]
}
```

## 4: Convert the problem into Gurobi code
```python
import gurobi

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

    # Define the variables
    x1 = model.addVar(lb=0, ub=22, name="desks_from_Manufacturer_X")
    x2 = model.addVar(lb=0, ub=28, name="chairs_from_Manufacturer_Y")

    # Set the objective function
    model.setObjective(140 * x1 + 120 * x2, gurobi.GRB.MAXIMIZE)

    # Add constraints
    model.addConstr(x1 <= 22, name="desks_limit")
    model.addConstr(x2 <= 28, name="chairs_limit")
    model.addConstr(x1 + x2 <= 40, name="quality_check_limit")

    # Optimize the model
    model.optimize()

    # Print the solution
    if model.status == gurobi.GRB.OPTIMAL:
        print("Optimal solution found.")
        print(f"Number of desks to make: {x1.varValue}")
        print(f"Number of chairs to make: {x2.varValue}")
        print(f"Maximum profit: {model.objVal}")
    else:
        print("No optimal solution found.")

solve_optimization_problem()
```