To determine the feasible space, we need to interpret each of the given constraints, including the non-negativity conditions for $x_1$ and $x_2$ (i.e., $x_1 \geq 0$ and $x_2 \geq 0$), and graphically find the region where all constraints are satisfied simultaneously.

Let’s analyze the constraints one by one:

(a) $-3x_1 + x_2 \geq 6$

• This inequality represents a region above the line $-3x_1 + x_2 = 6$.
• To rewrite the inequality for clarity: $x_2 \geq 3x_1 + 6$.

(b) $x_1 - 2x_2 \leq 5$

• This inequality represents the region below the line $x_1 - 2x_2 = 5$.
• To rewrite the inequality: $x_1 \leq 2x_2 + 5$.

(c) $2x_1 - 3x_2 \geq 12$

• This inequality represents a region above the line $2x_1 - 3x_2 = 12$.
• To rewrite the inequality: $x_2 \leq \frac{2x_1 - 12}{3}$.

(d) $x_1 - x_2 \geq 0$

• This inequality represents the region above or on the line $x_1 = x_2$, or equivalently, $x_1 \geq x_2$.

(e) $-x_1 + x_2 \leq 0$

• This inequality represents the region below or on the line $x_1 = x_2$, or equivalently, $x_1 \geq x_2$.

Now, we can define the feasible space, which is the intersection of all these regions, along with the non-negativity constraints:

1. $x_1 \geq 0$
2. $x_2 \geq 0$

Visualizing the Feasible Space

• The feasible space is the region in the $x_1$-$x_2$ plane where all inequalities are simultaneously satisfied.
• This involves finding the intersection of the half-planes defined by the inequalities, keeping in mind the non-negativity constraints.

You can plot these constraints to identify the exact region of feasibility.

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Would you like a more detailed analysis of how to graph these constraints, or assistance with solving for specific values within the feasible space? Let me know!

Further Exploration:

1. How would you graph the line $x_1 = x_2$ on a coordinate plane?
2. How does the intersection of multiple linear inequalities form a feasible region?
3. What is the concept of a feasible region in linear programming?
4. How would you solve an optimization problem with these constraints?
5. Can you calculate the vertices of the feasible region from these constraints?

Tip:

When graphing constraints like these, it's helpful to first graph the boundary lines as equalities and then shade the region that satisfies the inequality.