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The problem is about solving a linear programming (LP) problem graphically. Let me break down the steps for solving this problem, using the information provided.

### Linear Programming Problem:
Maximize \( Z = X + Y \)

Subject to:
1. \( 6X - 2Y \geq -2 \) (Constraint 1)
2. \( 2X - Y \geq 4 \) (Constraint 2)
3. \( 4X + 8Y \leq 48 \) (Constraint 3)
4. \( X, Y \geq 0 \) (Non-negativity)

#### Part (a):
- You need to find the optimal solution and objective function value graphically.
- First, rewrite each constraint in slope-intercept form to plot the boundary lines.
  
1. \( 6X - 2Y \geq -2 \) becomes \( Y \leq 3X + 1 \).
2. \( 2X - Y \geq 4 \) becomes \( Y \leq 2X - 4 \).
3. \( 4X + 8Y \leq 48 \) becomes \( Y \leq -\frac{1}{2}X + 6 \).

Now, we plot the lines:
- Line 1: \( Y = 3X + 1 \) (above this line satisfies the constraint).
- Line 2: \( Y = 2X - 4 \) (above this line satisfies the constraint).
- Line 3: \( Y = -\frac{1}{2}X + 6 \) (below this line satisfies the constraint).

Next, determine the feasible region where all constraints overlap. You will use this to evaluate the corner points.

Finally, the objective function \( Z = X + Y \) needs to be optimized (maximized) by evaluating it at the corner points of the feasible region.

#### Part (b):
You can use Excel Solver to solve the LP problem, but this part requires more detail about the steps in Solver.

Would you like to go through the graphical method in detail to identify the corner points and solve it?

Maximize X + Y subject to 6X - 2Y >= -2, 2X - Y >= 4, 4X + 8Y <= 48, X >= 0, Y >= 0

Solve the linear programming problem: maximize X + Y subject to several constraints. This problem explores concepts in optimization, inequalities, and linear programming, including the graphical solution method. Ideal for college-level students or advanced high school students.

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