https://file.aiquickdraw.com/mathbot/file/1726926331842hqftbazb.jpg

The system of inequalities from the image is:

\[
\begin{align*}
1. & \quad x + y \geq 3 \\
2. & \quad 3x - y \geq 6 \\
3. & \quad 2x + 2y < 16
\end{align*}
\]

Let's solve this system step by step:

### Step 1: Convert the inequalities into standard form.
We'll rewrite them in slope-intercept form (for graphing purposes) or standard form if needed:

1. \( x + y \geq 3 \)  
   Rearranged to:  
   \[ y \geq -x + 3 \]  
   (This is a line with slope -1 and y-intercept 3. The region above this line satisfies the inequality.)

2. \( 3x - y \geq 6 \)  
   Rearranged to:  
   \[ y \leq 3x - 6 \]  
   (This is a line with slope 3 and y-intercept -6. The region below this line satisfies the inequality.)

3. \( 2x + 2y < 16 \)  
   Simplified to:  
   \[ x + y < 8 \]  
   Rearranged to:  
   \[ y < -x + 8 \]  
   (This is a line with slope -1 and y-intercept 8. The region below this line satisfies the inequality.)

### Step 2: Graph the inequalities.
1. \( y \geq -x + 3 \) — Region above the line \( y = -x + 3 \).
2. \( y \leq 3x - 6 \) — Region below the line \( y = 3x - 6 \).
3. \( y < -x + 8 \) — Region below the line \( y = -x + 8 \).

### Step 3: Identify the solution region.
The solution is the region where all three inequalities overlap.

To summarize:

- The inequality \( x + y \geq 3 \) restricts the region to the half-plane above the line \( y = -x + 3 \).
- The inequality \( 3x - y \geq 6 \) restricts the region to the half-plane below the line \( y = 3x - 6 \).
- The inequality \( x + y < 8 \) restricts the region to the half-plane below the line \( y = -x + 8 \).

### Step 4: Find vertices (if needed for optimization).
We can find the intersection points of the lines:

- Intersection of \( y = -x + 3 \) and \( y = 3x - 6 \):
  Set the equations equal:
  \[ -x + 3 = 3x - 6 \]
  Solving for \( x \):
  \[ -x - 3x = -6 - 3 \]
  \[ -4x = -9 \]
  \[ x = \frac{9}{4} \]
  Substituting \( x = \frac{9}{4} \) into \( y = -x + 3 \):
  \[ y = -\frac{9}{4} + 3 = \frac{3}{4} \]
  So, the intersection is \( \left( \frac{9}{4}, \frac{3}{4} \right) \).

- Intersection of \( y = -x + 3 \) and \( y = -x + 8 \):
  Set the equations equal:
  \[ -x + 3 = -x + 8 \]
  This simplifies to a contradiction (no intersection), so these lines are parallel.

- Intersection of \( y = 3x - 6 \) and \( y = -x + 8 \):
  Set the equations equal:
  \[ 3x - 6 = -x + 8 \]
  Solving for \( x \):
  \[ 3x + x = 8 + 6 \]
  \[ 4x = 14 \]
  \[ x = \frac{14}{4} = 3.5 \]
  Substituting \( x = 3.5 \) into \( y = 3x - 6 \):
  \[ y = 3(3.5) - 6 = 10.5 - 6 = 4.5 \]
  So, the intersection is \( (3.5, 4.5) \).

Thus, the vertices of the solution region are important for understanding the boundary, but the region satisfying all three inequalities can be visualized through the graph.

Would you like more details or have any questions?

---

### Relative questions:
1. How do you graph the system of inequalities for a better visual understanding?
2. What is the importance of identifying vertices in a solution region?
3. Can this system of inequalities be used for optimization (e.g., linear programming)?
4. How do you interpret the solution region geometrically?
5. How do parallel lines influence the solution set in inequality systems?

### Tip:
Always sketch the graphs of inequalities when solving systems to better visualize the solution set.

Solve the system of inequalities: x + y ≥ 3, 3x - y ≥ 6, 2x + 2y < 16.

Learn how to solve the system of inequalities x + y ≥ 3, 3x - y ≥ 6, and 2x + 2y < 16 step by step, with a focus on graphical solutions and understanding the overlap of regions. Suitable for high school students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation