What is a System of Inequalities?

S6-SA1-0349

Grade Level:

Class 10

AI/ML, Physics, Biotechnology, Space Technology, Chemistry, Engineering, Medicine

Definition

What is it?

A system of inequalities is a set of two or more inequalities that you need to solve together. The solution to a system of inequalities is the set of all points that satisfy ALL the inequalities at the same time.

Simple Example

Quick Example

Imagine you want to buy a new mobile phone. The shopkeeper tells you the price (P) must be less than Rs 15,000 (P < 15000), but also more than Rs 8,000 (P > 8000) for a good quality model. These two conditions together form a system of inequalities for the phone's price.

Worked Example

Step-by-Step

Solve the following system of inequalities:
1) x + 3 > 5
2) x - 2 < 4

Step 1: Solve the first inequality.
x + 3 > 5
x > 5 - 3
x > 2

---
Step 2: Solve the second inequality.
x - 2 < 4
x < 4 + 2
x < 6

---
Step 3: Combine the solutions. We need 'x' to be greater than 2 AND less than 6.
So, 2 < x < 6

---
Step 4: The solution is all numbers between 2 and 6 (not including 2 and 6).

Answer: 2 < x < 6

Why It Matters

Understanding systems of inequalities is crucial in fields like AI/ML to define boundaries for data, in Physics to describe conditions for experiments, and in Engineering to design systems with multiple constraints. Engineers use them to make sure bridges can hold certain weights and stay within budget, or to optimize delivery routes for maximum efficiency.

Common Mistakes

MISTAKE: Solving each inequality separately but not finding the 'overlap' or common region for the solution. | CORRECTION: Always remember that the solution must satisfy ALL inequalities simultaneously. Think of it as finding the intersection of individual solutions.

MISTAKE: Confusing inequality signs (e.g., using > instead of >=). | CORRECTION: Pay close attention to whether the boundary value is included (>=, <=) or excluded (>, <). This impacts whether you use a solid or dashed line on a graph, or square vs. round brackets in interval notation.

MISTAKE: Incorrectly flipping the inequality sign when multiplying or dividing by a negative number. | CORRECTION: Remember to reverse the inequality sign (e.g., from < to >) whenever you multiply or divide both sides of an inequality by a negative number.

Practice Questions

Try It Yourself

QUESTION: Which of the following numbers is a solution to the system: x > 3 and x <= 7?
A) 2
B) 5
C) 8
D) 3 | ANSWER: B) 5

QUESTION: Find the solution to the system: 2x - 1 >= 5 and x + 4 < 10. | ANSWER: 3 <= x < 6

QUESTION: A baker wants to use at least 100 grams of flour (F) but no more than 250 grams. Also, the amount of sugar (S) must be less than half the amount of flour. Write a system of inequalities to represent these conditions. | ANSWER: 100 <= F <= 250 and S < F/2

MCQ

Quick Quiz

Which point (x, y) is a solution to the system:
x + y > 5
x - y < 1

(1, 2)

(3, 3)

(0, 0)

(2, 1)

The Correct Answer Is:

For (3,3): 3+3 = 6 > 5 (True) and 3-3 = 0 < 1 (True). Both inequalities are satisfied. For other options, at least one inequality fails.

Real World Connection

In the Real World

In logistics, companies like Delhivery or Ecom Express use systems of inequalities to plan delivery routes. They might have conditions like 'total distance must be less than 100 km' AND 'total delivery time must be less than 4 hours' AND 'each vehicle must carry at least 10 packages'. Solving this system helps them optimize their operations.

Key Vocabulary

Key Terms

INEQUALITY: A mathematical statement comparing two expressions using symbols like <, >, <=, or >=. | SYSTEM: A collection of equations or inequalities that are considered together. | SOLUTION SET: The set of all values that make all inequalities in the system true. | GRAPHING: Representing inequalities visually on a number line or coordinate plane to find common regions.

What's Next

What to Learn Next

Once you've mastered systems of inequalities, you can move on to 'Linear Programming'. This advanced topic uses systems of inequalities to find the best possible outcome (like maximum profit or minimum cost) in real-world problems. It's a powerful tool!