Contents

Introduction

Properties

Linear Inequalities 

1  2

Quadratic Inequalities 

Cubic Inequalities

Fractional Inequalities

Modulus Inequalities

Problem Solving

Quiz 

Algebra Main Page

Linear Inequalities in One Variable

Given two or more linear inequalities which are connected by the word 'and', the solution(s) to each inequality must satisfy all the other simultaneously. In other words, only the common solutions of the inequalities should be considered.

Example 1

Find the integer value of x which 3x x + 6 and 2x + 4 < 3x +6

Solving the inequalities separately, we have

3x x + 6  and  2x +4 < 3x +6

3x - x 6         3x - 2x > 4 - 6

2x 6                  x > -2

                    x 3 

The solution satisfying both inequalities are x, such that -2<x3.

The integer value of x in this range are -1, 0, 1, 2 and 3.

Example 2

Solve the inequalities 3x - 1 < x + 5 4x +14.

3x - 1 < x + 5  and  x + 5 4x + 14

3x - x < 5 + 1         5 - 14 4x - x

2x < 6        -9 3x

x < 3        -3 x 

the solution is -3 x 3.

Example 3

Given that 3 x 7 and -5 y -1, find 

(a) the largest possible value of 2x - y.;

(b) the smallest possible value of xy;

(c) the largest possible value of x² + y²

(a) The largest possible value of 2x - y occurs when x is largest and y is smallest.

the largest possible value of 2x - y = 2(7) - (-5) = 14 + 5 = 19 

(b) The smallest possible of xy occurs when x and y are positive. In this case y is negative, thus the smallest possible value of xy occurs when xy is numerically the greatest.

the smallest value of xy = 7 x (-5) = -35

(c) The largest possible value of x² + y² occurs when x and y are numerically the greatest.

the largest value of x² + y² = 7² + (-5)² = 49 + 25 = 74