To graph and determine the feasible region for the system of inequalities, we will break the problem into steps.

The system of inequalities:

1. $2x + y \leq 3$
2. $x - y \geq 3$

Steps to solve:

1. Graph the boundary lines:

   • Convert each inequality to an equation to plot the lines:
     • For $2x + y = 3$: Solve for $y$: $y = -2x + 3$
     • For $x - y = 3$: Solve for $y$: $y = x - 3$

2. Plot the lines:

   • The line $y = -2x + 3$ has a slope of -2 and a y-intercept of 3.
   • The line $y = x - 3$ has a slope of 1 and a y-intercept of -3.

3. Determine the shading direction:

   • For $2x + y \leq 3$, shade below the line $y = -2x + 3$.
   • For $x - y \geq 3$, rearranged as $y \leq x - 3$, shade below the line $y = x - 3$.

4. Find the intersection point:

   • Set $-2x + 3 = x - 3$ to find the intersection of the two lines. $-2x + 3 = x - 3$ Solving this: $-2x - x = -3 - 3$ $-3x = -6 \quad \Rightarrow \quad x = 2$ Substitute $x = 2$ into one of the equations to find $y$: $y = -2(2) + 3 = -4 + 3 = -1$ So, the lines intersect at $(2, -1)$.

5. Shade the feasible region:

   • The feasible region is the area where the shaded parts from both inequalities overlap, which will be below both lines.

Would you like me to generate the graph for this system? Also, let me know if you have any questions!

Related Questions:

1. How do you check if a point is in the feasible region?
2. What does the slope of each line represent in terms of the inequalities?
3. How do you handle systems of inequalities with more than two inequalities?
4. What is the significance of the intersection point in linear programming?
5. How can the feasible region change if we reverse the inequality symbols?

Tip:

When graphing inequalities, always remember to test a point not on the boundary to confirm where the shading should be placed!