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This problem is a **Linear Programming** (Quy hoạch tuyến tính) task where we are tasked with optimizing an objective function \( f(x) \) subject to a set of linear constraints. Let's break down the components:

### Objective Function:
\[
f(x) = x_1 - 2x_2 + 2x_3 + x_4
\]
We need to maximize or minimize this function, although the problem doesn't explicitly state which. Typically, the aim is to maximize the objective function unless otherwise stated.

### Constraints:
\[
\begin{aligned}
x_1 + x_2 + 4x_4 &= 6 \\
2x_2 + x_3 + 5x_4 &= 8
\end{aligned}
\]
These are the system of linear constraints that the variables \( x_1, x_2, x_3, x_4 \) must satisfy.

### Non-negativity Conditions:
\[
x_j \geq 0, \quad j = 1, 2, 3, 4
\]
Each variable must be non-negative.

### Approach:
This is a standard problem for the **Simplex Method** or other linear programming techniques, where we can either:
1. Convert the problem into standard form by adding slack variables.
2. Apply the Simplex algorithm to solve it.

Would you like a detailed step-by-step solution using the Simplex method or just a high-level solution?

### Related Questions:
1. What is the Simplex method and how does it work for solving linear programming problems?
2. Can we convert this problem into standard form by adding slack variables?
3. How do we know if this problem involves maximization or minimization?
4. How can duality theory be applied to this linear programming problem?
5. What are the typical software tools used to solve large-scale linear programming problems?

**Tip:** Always double-check whether the problem is asking for maximization or minimization before applying the solution method!

Giải bài toán Quy hoạch tuyến tính: f(x) = x_1 - 2x_2 + 2x_3 + x_4, với các ràng buộc: x_1 + x_2 + 4x_4 = 6, 2x_2 + x_3 + 5x_4 = 8, và x_j ≥ 0, j = 1, 2, 3, 4.

Learn how to solve a linear programming problem where the objective function f(x) = x_1 - 2x_2 + 2x_3 + x_4 is optimized under linear constraints. This problem involves concepts such as the Simplex method and non-negativity conditions. Suitable for advanced high school or undergraduate students.

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