To graph the feasible region of the system of inequalities, we need to first analyze the inequalities provided:

1. $x + y \leq 10$
2. $-x + y \geq 2$
3. $x \geq 3$

Let's break this down:

Step 1: Rewrite the inequalities

1. $x + y \leq 10$ can be rewritten as $y \leq -x + 10$. This is the equation of a line with a slope of -1 and a y-intercept at 10.
2. $-x + y \geq 2$ can be rewritten as $y \geq x + 2$. This is the equation of a line with a slope of 1 and a y-intercept at 2.
3. $x \geq 3$ is a vertical line at $x = 3$.

Step 2: Graph the lines

• For $y = -x + 10$, plot the line with points (0, 10) and (10, 0).
• For $y = x + 2$, plot the line with points (0, 2) and (8, 10).
• For $x = 3$, plot the vertical line passing through $x = 3$.

Step 3: Shade the regions

• Shade below the line $y = -x + 10$ (because $y \leq -x + 10$).
• Shade above the line $y = x + 2$ (because $y \geq x + 2$).
• Shade to the right of the vertical line $x = 3$ (because $x \geq 3$).

Step 4: Find the feasible region

The feasible region is the area where all the shaded regions overlap. You will see that this forms a polygon bounded by the intersections of the lines.

Step 5: Solve for intersection points

• Intersection of $y = -x + 10$ and $y = x + 2$: $-x + 10 = x + 2 \quad \Rightarrow \quad 2x = 8 \quad \Rightarrow \quad x = 4$ Substituting $x = 4$ into either equation gives $y = 6$. So, the intersection point is $(4, 6)$.

• Intersection of $y = -x + 10$ and $x = 3$: $y = -(3) + 10 = 7$ So, the intersection point is $(3, 7)$.

• Intersection of $y = x + 2$ and $x = 3$: $y = 3 + 2 = 5$ So, the intersection point is $(3, 5)$.

Step 6: Plot the polygon

The vertices of the polygon are $(3, 5)$, $(3, 7)$, and $(4, 6)$.

Would you like further details or have any questions?

Here are 5 follow-up questions to deepen your understanding:

1. How do you determine the slope of a line from its equation?
2. What is the significance of the shading in an inequality graph?
3. Can you explain how to find the intersection point of two lines algebraically?
4. How would the graph change if the inequality signs were reversed?
5. How do you verify that a point lies within the feasible region?

Tip: Always check the boundary conditions by substituting points back into the inequalities to confirm the feasible region.