The problem shown in the image is a linear programming problem. Here's the breakdown:

Objective Function:

$z = x_1 + 5x_2 \to \text{maximize.}$

Constraints:

1. $x_1 + x_2 \geq 3$,
2. $x_1 - 2x_2 \leq 0$,
3. $2x_1 - x_2 \geq 0$,
4. $x_1 \geq 0, \, x_2 \geq 0$ (non-negativity constraints).

Solution Approach:

The goal is to maximize $z = x_1 + 5x_2$ under the given constraints. To solve this:

1. Graph the inequalities: Represent the constraints in a 2D plane to identify the feasible region (the intersection of all the constraints).
2. Find the corner points of the feasible region: Linear programming solutions are located at the vertices of the feasible region.
3. Evaluate the objective function $z$ at each vertex: Determine which vertex gives the maximum value of $z$.

Would you like me to proceed with solving this step-by-step and provide a detailed explanation? I can also create a graph for visualization if needed.

Here are five questions to expand your understanding:

1. Do you know how to graph inequalities in a 2D plane?
2. Would you like to see a step-by-step solution for finding the feasible region?
3. Are you familiar with the concept of corner points in linear programming?
4. Would you like me to explain why solutions always occur at the vertices?
5. Are you interested in using an algebraic approach instead of graphical?

Tip: In linear programming problems, always check if the feasible region is bounded, as unbounded regions can sometimes lead to no maximum value for the objective function.