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What is the Feasible Region for Inequalities?
Grade Level:
Class 8
AI/ML, Data Science, Physics, Economics, Cryptography, Computer Science, Engineering
Definition
What is it?
The feasible region for inequalities is the area on a graph where all the given inequalities are true at the same time. Think of it as the 'solution zone' where every point satisfies all the conditions. This region shows all possible solutions that work for a set of rules.
Simple Example
Quick Example
Imagine you want to buy samosas and jalebis. You have Rs 100. Each samosa costs Rs 10 and each jalebi costs Rs 5. Also, you want to buy at least 3 samosas. The feasible region would show all the combinations of samosas and jalebis you can buy without spending more than Rs 100 and making sure you get at least 3 samosas.
Worked Example
Step-by-Step
Let's find the feasible region for these inequalities: 1) x >= 0, 2) y >= 0, 3) x + y <= 5.
STEP 1: Draw the x-axis and y-axis on a graph paper. This is your coordinate plane.
---STEP 2: For x >= 0, this means all points to the right of the y-axis (including the y-axis itself). Shade this area lightly.
---STEP 3: For y >= 0, this means all points above the x-axis (including the x-axis itself). Shade this area lightly. The common shaded area so far is the first quadrant.
---STEP 4: For 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 line connecting these two points.
---STEP 5: Now, decide which side of the line x + y = 5 to shade. Pick a test point, like (0,0). Substitute into x + y <= 5: 0 + 0 <= 5, which is 0 <= 5. This is TRUE. So, shade the region that includes the point (0,0), which is the area below the line x + y = 5.
---STEP 6: The feasible region is the area where ALL three shaded regions overlap. This will be a triangle with vertices at (0,0), (5,0), and (0,5). This triangle is the solution zone.
ANSWER: The feasible region is the triangular area bounded by the points (0,0), (5,0), and (0,5), including the boundaries.
Why It Matters
Understanding feasible regions helps solve real-world problems like how to best use limited resources or manage budgets. Engineers use it to design optimal structures, and economists use it to find the best production plans. It's a key skill for future careers in AI/ML, data science, and even in managing finances for a business.
Common Mistakes
MISTAKE: Shading the wrong side of a line. | CORRECTION: Always use a test point (like (0,0) if it's not on the line) to check which side of the line satisfies the inequality. If the test point makes the inequality true, shade that side; if false, shade the other side.
MISTAKE: Confusing strict inequalities (< or >) with non-strict inequalities (<= or >=). | CORRECTION: For strict inequalities, draw a dashed line to show that points on the line are NOT part of the solution. For non-strict inequalities, draw a solid line to show points on the line ARE part of the solution.
MISTAKE: Not finding the region where ALL inequalities overlap. | CORRECTION: After shading each inequality individually, the feasible region is only the area that has been shaded by every single inequality. It's the 'common ground' for all conditions.
Practice Questions
Try It Yourself
QUESTION: Which quadrant is always part of the feasible region if x >= 0 and y >= 0 are two of the inequalities? | ANSWER: The first quadrant.
QUESTION: Graph the feasible region for: y <= x + 1 and y >= 0. What are the vertices of the feasible region? | ANSWER: The feasible region is the area below y = x + 1 and above y = 0. It is an unbounded region, but if we consider the intersection with the y-axis and x-axis, it starts from (0,0) and goes along y=0 to the right, and along y=x+1 to the right, intersecting the y-axis at (0,1). The vertices are (0,0) and (0,1). (Note: This is an unbounded region, so 'vertices' might not be the best term for a student at this level, but (0,0) and (0,1) are key boundary points.)
QUESTION: A tiffin service makes parathas (P) and rotis (R). They can make at most 100 items total. They must make at least 20 parathas. Write the inequalities and describe the feasible region. | ANSWER: Inequalities: P + R <= 100, P >= 20, R >= 0. The feasible region is a triangle with vertices at (20,0), (100,0), and (20,80) on a graph where P is on the x-axis and R is on the y-axis. (The point (20,80) comes from 20+R=100 => R=80).
MCQ
Quick Quiz
Which of these describes the feasible region for a set of inequalities?
The area where only one inequality is true.
The area where all inequalities are true at the same time.
The entire graph paper.
The area outside all the shaded regions.
The Correct Answer Is:
B
The feasible region is defined as the set of all points that satisfy every single inequality simultaneously. Option B correctly captures this 'all at the same time' aspect.
Real World Connection
In the Real World
Imagine a food delivery app like Swiggy or Zomato. They need to deliver food quickly and efficiently. They use inequalities to define the 'feasible region' for delivery routes, considering factors like maximum delivery time, traffic conditions (which can be an inequality like 'travel_speed <= 30 kmph'), and customer location. This helps them optimize which delivery rider takes which order to ensure timely service.
Key Vocabulary
Key Terms
INEQUALITY: A mathematical statement showing that two quantities are not equal, using symbols like <, >, <=, or >=. | FEASIBLE REGION: The area on a graph where all given inequalities are simultaneously satisfied. | COORDINATE PLANE: A two-dimensional surface formed by the intersection of a horizontal x-axis and a vertical y-axis. | VERTEX (PLURAL: VERTICES): A point where two or more lines or edges meet, often marking a corner of the feasible region. | UNBOUNDED REGION: A feasible region that extends infinitely in at least one direction.
What's Next
What to Learn Next
Great job understanding feasible regions! Next, you can explore 'Linear Programming,' which uses feasible regions to find the best possible outcome (like maximum profit or minimum cost) in real-world situations. It's like finding the 'sweet spot' within your solution zone!
