The problem presented is a Linear Programming (LP) problem. You are asked to maximize the objective function $Z = 3X_1 + 5X_2$, subject to the given constraints:

Objective Function:

$Z = 3X_1 + 5X_2$

Constraints:

1. $2X_1 \leq 8$
2. $3X_2 \leq 15$
3. $6X_1 + 5X_2 \leq 30$
4. $X_1 \geq 0$ and $X_2 \geq 0$ (Non-negativity constraints)

Steps to Solve:

1. Graphical Method:
   • Plot the lines corresponding to each inequality.
   • Identify the feasible region.
   • Determine the corner points of the feasible region.
   • Evaluate the objective function $Z$ at each corner point to find the maximum value.

Would you like a detailed step-by-step solution using the graphical method or another approach like the Simplex method? Let me know if you have any specific preferences!

Here are some follow-up questions:

1. Do you need help plotting the inequalities and finding the feasible region?
2. Would you prefer using the Simplex method to solve this?
3. Do you need an explanation of how linear programming works?
4. Should I calculate the corner points for you?
5. Would you like a summary of how constraints affect the solution?

Tip: When solving LP problems, always ensure that the feasible region is clearly identified, as the optimal solution will lie on the boundary or at a corner point.