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What is the Graphical Solution of a System of Inequalities?
Grade Level:
Class 10
AI/ML, Physics, Biotechnology, Space Technology, Chemistry, Engineering, Medicine
Definition
What is it?
The graphical solution of a system of inequalities is finding the region on a graph that satisfies ALL the given inequalities at the same time. This 'common region' shows all the possible values for the variables that make every inequality true.
Simple Example
Quick Example
Imagine you want to buy samosas and jalebis. You have Rs 50. Each samosa costs Rs 10, and each jalebi costs Rs 5. You also want to buy at least 2 samosas. If 'x' is the number of samosas and 'y' is the number of jalebis, then 10x + 5y <= 50 (money limit) and x >= 2 (samosa limit). Graphing these helps you see all the combinations of samosas and jalebis you can buy.
Worked Example
Step-by-Step
Let's find the graphical solution for the system: 1) x + y >= 4 and 2) x - y < 2.
STEP 1: Graph the boundary line for the first inequality, x + y = 4. When x=0, y=4. When y=0, x=4. Plot (0,4) and (4,0) and draw a solid line (because of '>=', the line is included).
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STEP 2: Test a point (like (0,0)) for x + y >= 4. Is 0 + 0 >= 4? No (0 is not >= 4). So, shade the region that does NOT contain (0,0) for this inequality. This means shading above the line.
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STEP 3: Graph the boundary line for the second inequality, x - y = 2. When x=0, y=-2. When y=0, x=2. Plot (0,-2) and (2,0) and draw a dashed line (because of '<', the line is NOT included).
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STEP 4: Test a point (like (0,0)) for x - y < 2. Is 0 - 0 < 2? Yes (0 is < 2). So, shade the region that CONTAINS (0,0) for this inequality. This means shading above the line.
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STEP 5: The region where the two shaded areas overlap is the solution. This common region is the graphical solution to the system of inequalities.
Why It Matters
This concept is crucial in fields like AI/ML for optimizing decisions, in Physics for understanding motion constraints, and in Engineering for designing systems with limits. Future engineers, data scientists, and even doctors use these principles to solve complex real-world problems involving multiple conditions.
Common Mistakes
MISTAKE: Drawing a dashed line for inequalities with '>=' or '<=' | CORRECTION: Use a solid line for inequalities with '>=' or '<=' (meaning the boundary is included in the solution) and a dashed line for '>' or '<' (meaning the boundary is NOT included).
MISTAKE: Incorrectly shading the region after drawing the boundary line | CORRECTION: Always pick a test point (like (0,0) if it's not on the line) and substitute its coordinates into the original inequality. If the inequality holds true, shade the region containing the test point. If false, shade the other region.
MISTAKE: Forgetting to find the common overlapping region | CORRECTION: The solution to a system of inequalities is ONLY the region where ALL individual inequality solutions overlap. Shade each inequality's solution distinctly, then identify the common area.
Practice Questions
Try It Yourself
QUESTION: Graph the solution for y > x + 1. | ANSWER: Draw a dashed line for y = x + 1. Test (0,0): 0 > 0 + 1 (False). Shade the region above the line.
QUESTION: Find the graphical solution for: x >= 0, y >= 0, and x + y <= 3. | ANSWER: The solution is the triangular region in the first quadrant (where x>=0, y>=0) bounded by the x-axis, y-axis, and the solid line x + y = 3. Test (0,0) for x+y<=3: 0+0<=3 (True), so shade towards the origin for this line.
QUESTION: A factory produces two types of toys, A and B. It takes 2 hours to make toy A and 1 hour to make toy B. The factory can work at most 8 hours a day. The factory must produce at least 2 toys of type A. If x is the number of toy A and y is the number of toy B, find the graphical solution for the system of inequalities representing these conditions. | ANSWER: The inequalities are: 2x + y <= 8 (time constraint) and x >= 2 (toy A constraint). Also, x >= 0, y >= 0 (cannot make negative toys). The solution is the region in the first quadrant bounded by x=2, y=0, and the line 2x+y=8, forming a trapezoidal region.
MCQ
Quick Quiz
Which type of line is used for the inequality 'y < 2x + 3'?
Solid line
Dashed line
Dotted line
Thick line
The Correct Answer Is:
B
A dashed line is used for strict inequalities like '<' or '>' because the points on the boundary line itself are not part of the solution. Solid lines are for '<=' or '>='.
Real World Connection
In the Real World
When planning delivery routes for apps like Zomato or Swiggy, companies use systems of inequalities to find the most efficient paths. They consider factors like traffic (time limits), delivery zones (area limits), and vehicle capacity (load limits) to ensure food reaches customers quickly. Even ISRO scientists use similar principles to calculate launch windows and trajectories for rockets, making sure they stay within safe operational limits.
Key Vocabulary
Key Terms
INEQUALITY: A mathematical statement showing that two expressions are not equal, using symbols like <, >, <=, >= | SYSTEM OF INEQUALITIES: A set of two or more inequalities that need to be solved simultaneously | BOUNDARY LINE: The line formed by changing an inequality into an equation (e.g., x + y = 5 for x + y < 5) | FEASIBLE REGION: The common shaded area on the graph that represents all solutions to a system of inequalities | TEST POINT: A point chosen to determine which side of a boundary line to shade for an inequality.
What's Next
What to Learn Next
Once you master graphical solutions, you can explore Linear Programming. This exciting topic uses systems of inequalities to find the best possible outcome (like maximum profit or minimum cost) in real-world scenarios, which is a core concept in many advanced fields!
