Let's analyze the figure and solve each part of the question systematically.

(a) Find a system of inequalities for which $R$ is the feasible set.

We can observe from the graph that the region $R$ is a triangle formed by:

1. The horizontal axis ($y = 0$) which is the bottom boundary.
2. The vertical axis ($x = 0$) which is the left boundary.
3. A slanted line passing through points $(0, 0)$ and $(6, 3)$.

Let's analyze each part:

• The horizontal boundary is represented by $y \geq 0$, as the region is above or on the x-axis.
• The vertical boundary is represented by $x \geq 0$, as the region is to the right or on the y-axis.
• The slanted line can be derived from the two points, $(0, 0)$ and $(6, 3)$. The slope $m$ of the line is: $m = \frac{3 - 0}{6 - 0} = \frac{1}{2}$ Thus, the equation of the line is: $y = \frac{1}{2}x$ Since the region is below or on this line, the inequality is: $y \leq \frac{1}{2}x$

Thus, the system of inequalities that represents $R$ is: $x \geq 0, \quad y \geq 0, \quad y \leq \frac{1}{2}x$

(b) Sketch the subset of all points in $R$ with $x$-coordinate equal to 4.

When $x = 4$, the value of $y$ must satisfy the inequalities from part (a):

• From $y \geq 0$, we know that $y$ must be non-negative.
• From $y \leq \frac{1}{2}x$, substituting $x = 4$: $y \leq \frac{1}{2} \times 4 = 2$

Thus, the subset of all points in $R$ with $x = 4$ is all points where $0 \leq y \leq 2$.

(c) Use set-builder notation to describe the subset of all points in $R$ with $x$-coordinate equal to 4.

In set-builder notation, we describe this subset as: $\{ (4, y) \mid 0 \leq y \leq 2 \}$

Would you like a more detailed explanation for any part? Here are five questions for further exploration:

1. How would the system of inequalities change if the slope of the slanted line were steeper?
2. Can you explain why the region satisfies $y \leq \frac{1}{2}x$ but not $y \geq \frac{1}{2}x$?
3. What would the system of inequalities look like if the region were reflected over the x-axis?
4. How would you describe the solution in set-builder notation if the x-coordinate was equal to a different value?
5. How does the concept of slope apply to this feasible region?

Tip: When determining the equation of a line from two points, always use the slope formula $m = \frac{y_2 - y_1}{x_2 - x_1}$ and plug it into the point-slope form of a line $y - y_1 = m(x - x_1)$.