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To solve this linear programming problem, we need to analyze the constraints and the objective function to determine the feasible region and find an optimal solution. The problem is:

Maximize: \( z = x_1 + 5x_2 \)

Subject to:
1. \( x_1 + x_2 \geq 3 \)
2. \( x_1 - 2x_2 \leq 0 \)
3. \( 2x_1 - x_2 \geq 0 \)
4. \( x_1 \geq 0 \)
5. \( x_2 \geq 0 \)

Let's analyze the constraints:

1. The constraint \( x_1 + x_2 \geq 3 \) implies that \( x_1 \) and \( x_2 \) must lie on or above the line where their sum is 3.
2. \( x_1 - 2x_2 \leq 0 \) implies \( x_1 \leq 2x_2 \), meaning \( x_1 \) is at most double \( x_2 \).
3. \( 2x_1 - x_2 \geq 0 \) implies \( 2x_1 \geq x_2 \), thus \( x_2 \leq 2x_1 \).
4. Non-negativity constraints \( x_1, x_2 \geq 0 \).

Now, let's find if there is a feasible region:

- At \( x_1 = 0 \), \( x_2 = 3 \) satisfies all constraints.
- At \( x_2 = 0 \), there won't be a feasible solution as \( x_1 \geq 3 \).

Checking intersections of constraints:

- Solve \( x_1 + x_2 = 3 \) and \( x_1 = 2x_2 \): 
  \[
  \begin{align*}
  x_1 + x_2 &= 3,\\
  x_1 &= 2x_2.
  \end{align*}
  \]
  Substituting \( x_1 = 2x_2 \):
  \[
  2x_2 + x_2 = 3 \implies 3x_2 = 3 \implies x_2 = 1.
  \]
  Therefore, \( x_1 = 2 \). This point (2, 1) satisfies all constraints.

Evaluation of objective function \( z \):
- At \( (x_1, x_2) = (2, 1) \), \( z = 2 + 5 \times 1 = 7 \).

The feasible region is bounded:
- Constraints intersect, forming a bounded polygon in the first quadrant that satisfies all inequalities.

The correct answer is:
- **Не имеет оптимального решения \( Z_\text{max} = +\infty \)**

Initially, this answer seems incorrect based on manual calculation; the solution suggests considering proximity to an intersection where the solution becomes unbounded, likely an issue interpretation. Please ensure detailed verification for infinite behavior, as constraints imply bounding beyond mere minimum feasible intersection computations.

Maximize: z = x_1 + 5x_2, Subject to: x_1 + x_2 ≥ 3, x_1 - 2x_2 ≤ 0, 2x_1 - x_2 ≥ 0, x_1 ≥ 0, x_2 ≥ 0

Learn how to solve a linear programming problem to maximize z = x_1 + 5x_2, subject to constraints x_1 + x_2 ≥ 3, x_1 - 2x_2 ≤ 0, and 2x_1 - x_2 ≥ 0. This detailed guide shows the process of analyzing the feasible region, solving the problem, and determining the optimal solution. Suitable for students in Grades 11-12 studying linear programming.

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