Linear Inequalities – Introduction

Meaning of Inequality:

An inequality is a sentence that says that two values are not equal. E.g. if a is not equal to b, we can write a ≠ b (where ≠ means “is not equal to”). Therefore, inequalities are used to compare the relative sizes of values like integers, variables, etc. 

Inequality Symbols:

The following are inequality symbols 

\(\scriptsize > \:\) (“is greater than”)

e.g

\( \scriptsize 5 > 3 \) means 5 “is greater than” 3 

\( \scriptsize y > 10 \) the values of y include 11, 12, 13, 14 ……

\(\scriptsize < \: \) (“is less than”)

e.g

\( \scriptsize 3 < 5 \) reads 3 “is less than” 5 

\( \scriptsize y < 10 \) the values of y include 9, 8, 7, 6 ……

\(\scriptsize = \: \) (“is equal to”)

e.g. 

\( \scriptsize 2 \: + \: 3 = 5 \)

\( \scriptsize \geq \: \) (“is greater than or equal to”)

e.g

If \(y \scriptsize \geq 10 \) the values of y include: 10, 11, 12, 13, … (i.e. the value 10 is included). 

\( \scriptsize \leq \: \) (“is less than or equal to”)

e.g

If \( y \scriptsize \leq 10 \) the values of y include: 10, 9, 8, 7,6, … (the value 10 is included).

Note: y ≥ 10 means values of y that are equal to 10 or greater than 10. 

Example 7.1.1:

Write the following inequalities in a statement and list the range of values. 

(a) \( \scriptsize x < 2 \)

(b) \( \scriptsize x > 2 \)

(c) \( \scriptsize x \leq 2 \)

(d) \( \scriptsize x \geq 2 \)

Solution 

(a) \( \scriptsize x < 2 \) reads x is less than 2 

Range of values x = 1, 0, -1, -2, -3, -4, …

(b) \( \scriptsize x > 2 \)reads x is greater than 2 

Range of values of x = 3, 4, 5, 6, 7, 8, …

(c) \( \scriptsize x \leq 2 \) reads x is less than or equal to 2 

Range of values of x = 2, 1, 0, -1, -2, -3, -4, …

(d) \( \scriptsize x \geq 2 \)reads x is greater than or equal to 2 

Range of values of x = 2, 3, 4, 5, 6, 7, 8, 9, 10, …