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What is a Feasible Region on a Graph?
Grade Level:
Class 9
AI/ML, Data Science, Physics, Economics, Cryptography, Computer Science, Engineering
Definition
What is it?
A Feasible Region on a graph is the area that satisfies ALL the given conditions or inequalities at the same time. Think of it as the 'allowed' zone where all rules are followed. Any point within this region is a 'feasible solution'.
Simple Example
Quick Example
Imagine you want to buy samosas and jalebis. You have Rs 100, and you want to buy at least 2 samosas and 1 jalebi. If you plot these rules on a graph (cost, minimum quantity), the area on the graph where all these conditions are met (you have enough money, you bought enough samosas and jalebis) is your feasible region.
Worked Example
Step-by-Step
Let's find the feasible region for these conditions: x >= 0, y >= 0, and x + y <= 5.
STEP 1: Draw the x-axis and y-axis. Since x >= 0 and y >= 0, our feasible region will be in the first quadrant (top-right section of the graph).
---STEP 2: Consider the inequality x + y <= 5. First, draw the line x + y = 5. You can find two points: if x=0, y=5 (point (0,5)); if y=0, x=5 (point (5,0)). Draw a straight line connecting these two points.
---STEP 3: Now, we need to decide which side of the line x + y = 5 satisfies x + y <= 5. Pick a test point, like (0,0). Substitute it into the inequality: 0 + 0 <= 5, which is 0 <= 5. This is true.
---STEP 4: Since (0,0) satisfies the inequality, the feasible region for x + y <= 5 is the area on the same side of the line as (0,0).
---STEP 5: Combine all conditions. The region must be in the first quadrant (x>=0, y>=0) AND on the side of x+y=5 that includes (0,0).
---ANSWER: The feasible region is the triangle formed by the points (0,0), (5,0), and (0,5).
Why It Matters
Understanding feasible regions is super useful in fields like AI/ML and Data Science to find the best solutions for complex problems. Engineers use it to design optimal structures, and economists use it to maximize profits or minimize costs. It helps solve 'best choice' puzzles in many real-world careers.
Common Mistakes
MISTAKE: Only considering one inequality at a time and shading the region for each separately without finding the overlap. | CORRECTION: The feasible region is the area where ALL shaded regions (from all inequalities) overlap. It's the common area.
MISTAKE: Shading the wrong side of a boundary line. For example, shading above y=2 for y <= 2. | CORRECTION: Always pick a test point (like (0,0) if it's not on the line) and substitute its coordinates into the inequality to see which side satisfies it. Shade that side.
MISTAKE: Drawing a dashed line for inequalities like 'x >= 3' or 'y <= 5'. | CORRECTION: For inequalities with '=' (like >= or <=), the boundary line itself IS part of the feasible region, so it should be drawn as a solid line. Use a dashed line only for strict inequalities like '>' or '<'.
Practice Questions
Try It Yourself
QUESTION: On a graph, what is the feasible region for the conditions x >= 0, y >= 0, and x <= 3? | ANSWER: The region in the first quadrant (where x and y are positive) bounded by the y-axis, the x-axis, and the vertical line x=3. It's a square/rectangle from (0,0) to (3,0) to (3,infinity) to (0,infinity). If y also has an upper bound, it would be a rectangle.
QUESTION: Shade the feasible region for the conditions: y >= x, x >= 0, y <= 4. | ANSWER: The region is a triangle formed by the points (0,0), (0,4), and (4,4).
QUESTION: Find the feasible region for: x + y >= 6, x <= 4, y <= 5, x >= 0, y >= 0. | ANSWER: The feasible region is a quadrilateral (four-sided shape) with vertices at (1,5), (4,5), (4,2), and (1,5). (Note: The point (1,5) is from x+y=6 and y=5; (4,5) from x=4 and y=5; (4,2) from x=4 and x+y=6).
MCQ
Quick Quiz
Which of these describes a point within a feasible region?
It satisfies only one of the given conditions.
It satisfies none of the given conditions.
It satisfies all of the given conditions simultaneously.
It is always located on the boundary line.
The Correct Answer Is:
C
A point in the feasible region must satisfy ALL the conditions or inequalities at the same time. Options A and B are incorrect as they don't meet all conditions. Option D is incorrect because points inside the region also satisfy conditions, not just those on the boundary.
Real World Connection
In the Real World
Imagine a logistics company like Delhivery or Ecom Express planning delivery routes. They have conditions like 'delivery truck must not exceed 60 km/h', 'must deliver at least 50 packages', and 'fuel consumption must be below X litres'. When they plot these conditions on a graph, the feasible region helps them find the most efficient routes and delivery schedules that meet all these rules, ensuring timely and cost-effective service.
Key Vocabulary
Key Terms
INEQUALITY: A mathematical statement comparing two expressions using symbols like <, >, <=, or >=. | BOUNDARY LINE: The line drawn on a graph that separates the region satisfying an inequality from the region that doesn't. | OPTIMIZATION: The process of finding the best solution (e.g., maximum profit, minimum cost) within a feasible region. | QUADRANT: One of the four regions into which the x and y axes divide a coordinate plane.
What's Next
What to Learn Next
Great job understanding feasible regions! Next, you can learn about 'Linear Programming'. This builds on feasible regions by teaching you how to find the BEST possible point (like maximum profit or minimum cost) within that feasible region. It's like finding the 'sweet spot' in your 'allowed zone'!
