https://file.aiquickdraw.com/mathbot/file/1728851224384lt2hxp7v.jpg

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 \( y - y_1 = m(x - x_1) \).

Find a system of inequalities for which R is the feasible set, sketch the subset of all points in R with x-coordinate equal to 4, and use set builder notation to describe the subset of all points in R with x-coordinate equal to 4.

Learn how to find a system of inequalities for a feasible region, sketch the subset of points with specific x-coordinates, and express the solution in set-builder notation. This problem involves linear inequalities and coordinate geometry, suitable for Grades 9-11.

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