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Properties of Inequality

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Properties of Inequality

Study Guide

Key Definition

Properties of Inequality include the Anti-reflexive, Trichotomy, Transitive, Addition, Subtraction, Multiplication, and Division properties for real numbers.

Important Notes

When multiplying or dividing by a negative number, reverse the inequality
The Anti-reflexive (also called irreflexive) property states $x \not< x$ and $x \not> x$
If $x < y$ and $y < z$, then $x < z$ by Transitive property
Adding or subtracting the same number on both sides keeps the inequality direction unchanged
Multiplying by a positive number keeps the inequality direction, by a negative reverses it

Mathematical Notation

$\leq$ means less than or equal to

$\geq$ means greater than or equal to

$+$ represents addition

$-$ represents subtraction

$\times$ or $*$ represents multiplication

$\div$ or $/$ represents division

Remember to use proper notation when solving problems

Why It Works

Inequalities describe the relative sizes or order of two values. Applying operations while maintaining inequality direction is crucial for accurate solutions.

Remember

Always reverse inequality when multiplying or dividing by a negative number.

Quick Reference

Anti-reflexive Property:$x \not< x$ and $x \not> x$

Transitive Property:If $x < y$ and $y < z$, then $x < z$

Addition Property:If $x < y$, then $x + z < y + z$

Subtraction Property:If $x < y$, then $x - z < y - z$

Multiplication Property (positive multiplier):If $x < y$ and $z > 0$, then $xz < yz$

Multiplication Property (negative multiplier):If $x < y$ and $z < 0$, then $xz > yz$

Division Property (positive divisor):If $x < y$ and $z > 0$, then $x / z < y / z$

Division Property (negative divisor):If $x < y$ and $z < 0$, then $x / z > y / z$

Understanding Properties of Inequality

Choose your learning level

Watch & Learn

Video explanation of this concept

Beginner

Start here! Easy to understand

Beginner Explanation

Simple explanation with $x < y$ means $x$ is less than $y$

Practice Problems

Test your understanding with practice problems

Quick Quiz

Single Choice Quiz

Beginner

If $x < 3$, which is true for $x + 2$?

Real-World Problem

Question Exercise

Intermediate

Teenager Scenario

You have $\$10$ and want to buy an item that costs $\$x$ with a discount of $\$2$. If $x - 2 < 10$, what is the greatest integer less than 12?

Thinking Challenge

Thinking Exercise

Intermediate

Think About This

Consider $y > 3$ and $z < -2$. What can you conclude about $y \times z$?

Challenge Quiz

Single Choice Quiz

Advanced

Given $a < b$ and $c < 0$, which of the following is true?

Recap

Watch & Learn

Review key concepts and takeaways