Types of Inequalities

Understand the types of inequalities with simple notes: symbols, strict vs non-strict inequalities, linear inequalities, real-life examples, mistakes, and practice.

1. Introduction

An inequality compares two quantities and tells us which one is greater or smaller. Unlike an equation, which shows equality using an equals sign (=), an inequality shows a range or limit of possible values.

Inequalities appear everywhere: age limits, speed limits, marks needed to pass, budgets, and measurements.

Examples:

  • Age ≥ 18
  • Speed < 60 km/h
  • Marks > 35

2. Symbols of Inequalities

These are the main symbols used in inequalities:

  • < : less than
  • > : greater than
  • ≤ : less than or equal to
  • ≥ : greater than or equal to

Strict inequalities use < or >.

Non-strict inequalities use ≤ or ≥ because equality is allowed.

3. Types of Inequalities

Inequalities can be classified in different ways depending on their structure.

3.1. Linear Inequalities in One Variable

A linear inequality involves a single variable (usually x) and does not exceed degree 1.

Examples:

  • 2x + 5 > 11
  • x - 7 ≤ 10
  • 3 - x ≥ 0

3.2. Linear Inequalities in Two Variables

These involve two variables like x and y, and represent a region on the coordinate plane instead of a single range.

Examples:

  • x + y ≥ 6
  • 2x - 3y < 4

These are used in graphs, constraints, economics, and optimization problems.

3.3. Strict and Non-Strict Inequalities

  • Strict: < and > (values cannot be equal)
  • Non-Strict: ≤ and ≥ (values can be equal)

3.4. Open and Closed Inequalities

An inequality is open when the boundary is not included: (<, >)

It is closed when equality is included: (≤, ≥)

4. Real-Life Examples of Inequalities

  • Age Restrictions: You must be ≥ 18 to vote.
  • Weight Limits: Luggage must be ≤ 15 kg.
  • Speed Limits: Speed < 80 km/h.
  • Budget Constraints: You can spend ≤ ₹500 on stationery.
  • Exam Marks: Passing marks > 33.

5. Examples

Identify the type of inequality:

  • 5x + 2 ≥ 12 → linear inequality in one variable
  • x + y < 10 → linear inequality in two variables
  • t > 0 → strict inequality
  • p ≤ 9 → non-strict inequality

6. Common Mistakes

  • Confusing the direction of < and > symbols.
  • Thinking ≤ and ≥ mean the same as < and >.
  • Not recognizing that inequalities represent ranges, not single values.
  • Misclassifying two-variable inequalities as one-variable inequalities.

7. Quick Practice

Classify each inequality:

  1. x + 7 < 12
  2. 3y - 2 ≥ 10
  3. x + y > 5
  4. 4 < m < 9

8. Summary

  • Inequalities compare values using symbols.
  • Strict inequalities use < and >; non-strict use ≤ and ≥.
  • Linear inequalities can be in one or two variables.
  • Inequalities appear naturally in everyday restrictions and limits.