
7.1 
Inequalities 

Definition 
The relationship between two quantities that do not have the same value.



Symbol 
Meaning 
\(\gt\) 
Greater than 
\(\lt\) 
Less than 


Example 

From the number line above, \(–2\) lies to the left of \(3\).
So, \(–2\) is less than \(3\).
Thus, the inequality is \(–2 < 3\).



Example 

From the number line above, \(–2\) lies to the right of \(7\).
So, \(–2\) is greater than \(7\).
Thus, the inequality is \(–2 > 7\).



Describe inequality and form algebraic inequality: 

Example 

From the number line above, \(x\) is less than \(8\).
Thus, \(x\lt 8\).



Identify relationship: 

Symbol 
Meaning 
\(\geq\) 
Greater than or equal to 
\(\leq\) 
Less than or equal to 


Properties of inequalities: 

Converse property of inequality


Example 
State the converse property of inequality of \(23\gt32\).

Answer: \(32\lt23\)



Transitive property of inequality


Example 
State the transitive property of inequality of \(15\lt8\lt0\).

Answer: \(15\lt0\)



 The inequality symbol remains unchanged when adding or subtracting a positive or negative number to or from both sides of the inequality.


\(\begin{aligned} \text{If }\,a&\lt b, \\\\\text{then }\,a+c&\lt b+c.\\\\ \end{aligned}\) 
\(\begin{aligned} \text{If }\,a&\lt b, \\\\\text{then }\,ac&\lt bc. \\\\\end{aligned}\) 
\(\begin{aligned} \text{If }\,a&\lt b, \\\\\text{then }\,a+(c)&\lt b+(c).\\\\ \end{aligned}\) 
\(\begin{aligned} \text{If }\,a&\lt b, \\\\\text{then }\,a(c)&\lt b(c).\end{aligned}\) 


 The inequality symbol remains unchanged when multiplying or dividing both sides of the inequality by a positive number.


\(\begin{aligned} \text{If }\,a&\lt b, \\\\\text{then }\,a\times c&\lt b\times c.\\\\ \end{aligned}\) 
\(\begin{aligned} \text{If }\,a&\lt b, \\\\\text{then }\,\dfrac{a}{c}&\lt \dfrac{b}{c}.\end{aligned}\) 


 The direction of the inequality symbol is reversed when multiplying or dividing both sides of the inequality by a negative number.


\(\begin{aligned} \text{If }\,a&\lt b, \\\\\text{then }\,a\times (c)&\gt b\times (c).\\\\ \end{aligned}\) 
\(\begin{aligned} \text{If }\,a&\lt b, \\\\\text{then }\,\dfrac{a}{c}&\gt \dfrac{b}{c}.\end{aligned}\) 


Additive inverse
 When both sides of the inequality are multiplied by \(1\), the direction of the inequality symbol is reversed.


\(\begin{aligned} \text{If }\,a&\lt b, \\\\\text{then }\,a&\gt b.\end{aligned}\) 


Multiplicative inverse
 When performing reciprocal of both numbers on both sides of the inequality, the direction of the inequality symbol is reversed.


\(\begin{aligned} \text{If }\,a&\lt b, \\\\\text{then }\,\dfrac{1}{a}&\gt \dfrac{1}{b}.\end{aligned}\) 

