The given problem is a linear programming problem that aims to maximize the objective function $z = x_1 + 5x_2$ subject to the following constraints:

1. $x_1 + x_2 \geq 3$
2. $x_1 - 2x_2 \leq 0$
3. $2x_1 - x_2 \geq 0$
4. $x_1 \geq 0, \, x_2 \geq 0$

To solve this problem, you can use the graphical method (if it's a 2-variable problem like this) or linear programming techniques such as the simplex method. Here's a brief outline of the graphical method:

1. Plot the constraints on a graph:

   • Each inequality represents a half-plane. Plot the lines $x_1 + x_2 = 3$, $x_1 - 2x_2 = 0$, and $2x_1 - x_2 = 0$, and determine the feasible region that satisfies all constraints.

2. Identify the feasible region:

   • The feasible region is the intersection area that satisfies all constraints.

3. Evaluate the objective function at each vertex of the feasible region:

   • Calculate $z = x_1 + 5x_2$ for each corner point of the feasible region.

4. Determine the maximum value:

   • The maximum value of $z$ will occur at one of the vertices of the feasible region.

By examining these steps, you can find the optimal solution to this linear programming problem.