Contents

Introduction

Properties

Linear Inequalities 

1  2

Quadratic Inequalities 

Cubic Inequalities

Fractional Inequalities

Modulus Inequalities

Problem Solving

Quiz 

Algebra Main Page

Properties of Inequalities

1. For any three numbers x, y and z, if x>y and y>z, then x>z. This is known as the transitive property of inequalities

e.g. if x = 20, y = 10, z = 4, then 20>10, 10>4;

if x = 1, y = 0, z = -4, then 1>0, 0>-4 and 1>-4.

2. We can add or subtract a positive number from both sides of an inequality without having to change the inequality sign. For any number x and y, and a positive number p,

         if x>y (e.g. 5>3)

then, x-p > y-p (e.g. 5-2>3-2)

        x+p> y+p (e.g. 5+2>3+2)

This is also true for a negative number q;

        if x>y (e.g. 5 >3)

then, x - q> y - p (e.g. 5 + 2 >3 + 2)

        x+q> y+q (e.g. 5-2> 3-2)

3. We can multiply and divide both sides of an inequality by a positive number without having to change the equality sign. For any two numbers x and y, and a third number a>0,

        if x>y (e.g. 5>3)

then ax> ay (2 x 5>2 x 3)

This is not true for a negative number b;

if x>y (e.g. 5>3)

then bx < by (e.g. -2 x 5< -2 x 3)

Thus we have to change the inequality sign when we multiply or divide both sides of an inequality by a negative number.