To solve this system of inequalities and graph the feasible region, let’s break down the steps:

System of Inequalities

1. $x + y \leq 6$
2. $x - 2y \geq -4$

Step-by-Step Solution

Step I: Understanding the Problem

We need to find the solution set (feasible region) for the system by graphing these inequalities and shading the area that satisfies both conditions.

Step II: Make a Plan

1. Rewrite each inequality as an equation to plot the boundary lines:

   • For $x + y = 6$: Rearrange to find points for plotting.
   • For $x - 2y = -4$: Similarly, rearrange to get points.

2. Determine the boundary lines:

   • The line $x + y = 6$ can be plotted by finding intercepts.
   • The line $x - 2y = -4$ can also be plotted with intercepts.

3. Identify shading directions for each inequality by testing points (such as the origin) or observing the inequality signs.

Step III: Carry out the Plan

1. Graph each boundary line:

   • For $x + y = 6$:
     • When $x = 0$, $y = 6$ (point: $(0,6)$)
     • When $y = 0$, $x = 6$ (point: $(6,0)$)
   • For $x - 2y = -4$:
     • When $x = 0$, $y = 2$ (point: $(0,2)$)
     • When $y = 0$, $x = -4$ (point: $(-4,0)$)

2. Shade the regions:

   • For $x + y \leq 6$, shade below the line $x + y = 6$.
   • For $x - 2y \geq -4$, shade above the line $x - 2y = -4$.

3. Find the intersection region where both shadings overlap.

Step IV: Look Back

Verify the solution by selecting points within the shaded region to ensure they satisfy both inequalities.

Would you like more details on graphing or additional explanation of any of the steps?

Additional Practice Questions

1. How would the feasible region change if the inequality $x + y \leq 6$ were replaced with $x + y < 6$?
2. What if we had $x - 2y < -4$ instead of $x - 2y \geq -4$?
3. How can we interpret the feasible region in terms of real-world problems?
4. How would the solution set change if we added a new inequality, such as $y \leq 3$?
5. Could we solve this system algebraically to find intersection points without graphing?

Tip

When graphing inequalities, always start by converting them into equations to find boundary lines, and then determine the shading area based on the inequality direction.