https://file.aiquickdraw.com/mathbot/file/1728801858689yq4kn1fj.jpg

Let's break this down and solve the system of inequalities from the image.

### System of inequalities given:
1. \(x + y \leq 11\)
2. \(-1x + y \geq 4\)
3. \(x \geq 0\) (ensures solutions are in the first quadrant)
4. \(y \geq 0\) (ensures solutions are in the first quadrant)

### Step-by-step analysis:

#### 1. Plot the boundary line: \(x + y = 11\)
- **Intercepts:**
  - If \(x = 0\), then \(y = 11\). So, the point is \((0, 11)\).
  - If \(y = 0\), then \(x = 11\). So, the point is \((11, 0)\).
- This line passes through the points \((0, 11)\) and \((11, 0)\).
- Since the inequality is \(x + y \leq 11\), the region lies **below or on** the line.

#### 2. Plot the boundary line: \(-x + y = 4\)
- **Intercepts:**
  - If \(x = 0\), then \(y = 4\). So, the point is \((0, 4)\).
  - If \(y = 0\), then \(-x = -4\), so \(x = 4\). The point is \((4, 0)\).
- This line passes through the points \((0, 4)\) and \((4, 0)\).
- Since the inequality is \(-x + y \geq 4\), the region lies **above or on** the line.

#### 3. First-quadrant restrictions: \(x \geq 0\) and \(y \geq 0\)
- These inequalities restrict the feasible region to the **first quadrant**.

---

### Intersection points:
- We need to find the intersection of the two lines \(x + y = 11\) and \(-x + y = 4\).

1. **Add the two equations:**
   \[
   (x + y) + (-x + y) = 11 + 4 \implies 2y = 15 \implies y = 7.5
   \]
2. **Plug \(y = 7.5\) into \(x + y = 11\):**
   \[
   x + 7.5 = 11 \implies x = 3.5
   \]
- So, the intersection point is \((3.5, 7.5)\).

---

### Feasible Region:
- The corner points of the feasible region are:
  1. \((0, 4)\)
  2. \((3.5, 7.5)\)
  3. \((11, 0)\)
  4. \((0, 0)\) (origin)

---

### Final Step:
1. Draw a polygon around the points:
   - \((0, 4)\), \((3.5, 7.5)\), \((11, 0)\), and \((0, 0)\).
2. Shade the interior region to mark the feasible area.

This is your solution! If you plot these points and form the polygon, you’ll correctly graph the feasible region.

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Do you need further assistance with drawing it, or have any questions?

### Related Questions:
1. How do you solve systems of inequalities graphically?
2. How do you find intersection points of two linear equations algebraically?
3. What is the difference between \(\leq\) and \(<\) in graphing inequalities?
4. What methods exist for solving systems of inequalities besides graphing?
5. How does the feasible region change if one inequality is reversed?

**Tip:** When graphing inequalities, always pay attention to whether the inequality is strict (\(<\) or \(>\)) or inclusive (\(\leq\) or \(\geq\)), as it affects the boundary lines.

Graph the feasible region for the following system of inequalities by drawing a polygon around the feasible region: x + y ≤ 11, -x + y ≥ 4, x ≥ 0, y ≥ 0.

Learn how to graph the feasible region for the system of inequalities: x + y ≤ 11, -x + y ≥ 4, x ≥ 0, y ≥ 0. This solution involves graphing lines, identifying intercepts, and finding intersection points. Suitable for Grades 9-12.

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