What is a System of Linear Inequalities?

S3-SA1-0180

Grade Level:

Class 7

AI/ML, Data Science, Physics, Economics, Cryptography, Computer Science, Engineering

Definition

What is it?

A system of linear inequalities is a set of two or more linear inequalities that you need to solve at the same time. The solution to such a system is the region that satisfies ALL the inequalities simultaneously.

Simple Example

Quick Example

Imagine you want to buy 'ladoos' and 'jalebis'. You have Rs. 100, and you want to buy at least 3 ladoos and at least 2 jalebis. If a ladoo costs Rs. 15 and a jalebi costs Rs. 10, these conditions together form a system of linear inequalities.

Worked Example

Step-by-Step

Let's find the solution for the system:
1) x > 2
2) x < 5

Step 1: Understand the first inequality, x > 2. This means x can be any number greater than 2 (like 3, 4, 4.5, etc.). On a number line, you'd draw an open circle at 2 and an arrow pointing to the right.
---Step 2: Understand the second inequality, x < 5. This means x can be any number less than 5 (like 4, 3, 2.5, etc.). On a number line, you'd draw an open circle at 5 and an arrow pointing to the left.
---Step 3: Look for the numbers that satisfy BOTH conditions. Numbers like 3 or 4 are greater than 2 AND less than 5.
---Step 4: The numbers that are both greater than 2 and less than 5 are the numbers between 2 and 5 (but not including 2 or 5).
---Answer: The solution is 2 < x < 5.

Why It Matters

Understanding systems of linear inequalities helps engineers design safe bridges and buildings by considering multiple stress limits. In business, it helps companies decide how much to produce to maximize profit while staying within budget and resource limits. Data scientists use it to make decisions in AI and machine learning models.

Common Mistakes

MISTAKE: Solving each inequality separately and not finding the common region | CORRECTION: Always find the overlapping region (intersection) where all inequalities are true at the same time.

MISTAKE: Confusing '<' and '>' with '<=' and '>=' (e.g., using a solid line for strict inequalities) | CORRECTION: Remember to use a dashed line for strict inequalities (< or >) and a solid line for inequalities that include equality (<= or >=).

MISTAKE: Shading the wrong side of the line when graphing | CORRECTION: Pick a test point (like (0,0) if it's not on the line) and substitute it into the inequality. If it makes the inequality true, shade the side containing the test point; otherwise, shade the other side.

Practice Questions

Try It Yourself

QUESTION: What are the integer solutions for the system: x >= 1 and x <= 4? | ANSWER: 1, 2, 3, 4

QUESTION: Which of the following numbers satisfies both y > 5 and y < 8: 4, 6, 9? | ANSWER: 6

QUESTION: A small tiffin service can make at most 20 rotis and at least 10 rotis. Write this as a system of inequalities for the number of rotis, 'r'. | ANSWER: r <= 20 and r >= 10 (or 10 <= r <= 20)

MCQ

Quick Quiz

Which point satisfies the system: x > 3 and y < 2?

(2, 1)

(4, 3)

(4, 1)

(3, 2)

The Correct Answer Is:

For (4,1), x=4 which is > 3, and y=1 which is < 2. So, (4,1) satisfies both inequalities. The other options fail one or both conditions.

Real World Connection

In the Real World

When a delivery app like Swiggy or Zomato decides which delivery person gets an order, they use systems of inequalities. They consider factors like 'delivery person must be within 2 km of the restaurant' and 'delivery person must have less than 3 active orders'. These conditions together form a system of inequalities to optimize delivery.

Key Vocabulary

Key Terms

INEQUALITY: A mathematical statement showing that two values are not equal, using symbols like <, >, <=, >= | SYSTEM: A collection of two or more inequalities considered together | SOLUTION REGION: The area on a graph where all inequalities in a system are true | LINEAR INEQUALITY: An inequality involving a variable raised to the power of 1 (like x or y)

What's Next

What to Learn Next

Next, you can learn about 'Graphing Systems of Linear Inequalities'. This will show you how to draw these systems on a graph and visually find their solution regions, which is super useful for real-world problems!