Math Problem Statement
Solution
Let's break this down and solve the system of inequalities from the image.
System of inequalities given:
- (ensures solutions are in the first quadrant)
- (ensures solutions are in the first quadrant)
Step-by-step analysis:
1. Plot the boundary line:
- Intercepts:
- If , then . So, the point is .
- If , then . So, the point is .
- This line passes through the points and .
- Since the inequality is , the region lies below or on the line.
2. Plot the boundary line:
- Intercepts:
- If , then . So, the point is .
- If , then , so . The point is .
- This line passes through the points and .
- Since the inequality is , the region lies above or on the line.
3. First-quadrant restrictions: and
- These inequalities restrict the feasible region to the first quadrant.
Intersection points:
- We need to find the intersection of the two lines and .
- Add the two equations:
- Plug into :
- So, the intersection point is .
Feasible Region:
- The corner points of the feasible region are:
- (origin)
Final Step:
- Draw a polygon around the points:
- , , , and .
- 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.
Do you need further assistance with drawing it, or have any questions?
Related Questions:
- How do you solve systems of inequalities graphically?
- How do you find intersection points of two linear equations algebraically?
- What is the difference between and in graphing inequalities?
- What methods exist for solving systems of inequalities besides graphing?
- 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 ( or ), as it affects the boundary lines.
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Math Problem Analysis
Mathematical Concepts
Linear Inequalities
Graphing Systems of Inequalities
Feasible Region
Formulas
x + y ≤ 11
-x + y ≥ 4
x ≥ 0
y ≥ 0
Theorems
Intersection of Linear Equations
Graphing Inequalities
Feasible Region Theorem
Suitable Grade Level
Grades 9-12
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