
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.


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


7.2 
Linear Inequalities in One Variable 

Symbol \(\geq\) 
Symbol \(\leq\) 
 At least
 Not less than
 Minimum

 At most
 Not more than
 Maximum



Example 
Construct the linear inequality:
The price, \(\text{RM}x\), of a doublestorey terrace house is \(\text{RM}450\,000\) and above.

The linear inequality is,
\(x\geq 450\,000\).



Solve problems involving linear inequalities in one variable: 

 Linear inequality in one variable is an unequal relationship between a number and a variable with power of \(1\).
 Has more than one possible solution.


Example 
Calculate: \(165x\leq 4\)

\(\begin{aligned} 165x&\leq 4 \\\\165x16&\leq 416 \\\\5x&\leq 20 \\\\\dfrac{5x}{5}&\geq \dfrac{20}{5} \\\\x&\geq 4. \end{aligned}\)



Solve simultaneous linear inequalities in one variable: 

Example 
Solve:
\(8x+5\geq 5x13\) and
\(3x4\gt 9x+20\).

\(\begin{aligned}8x+5&\geq 5x13 \\\\8x5x&\geq135 \\\\3x&\geq18 \\\\x&\geq 6.\\\\ \end{aligned}\)
\(\begin{aligned}3x4&\gt 9x+20 \\\\3x9x&\gt20+4 \\\\6x&\gt24 \\\\x&\lt4.\\\\ \end{aligned}\)

Thus, the solution is
\(6\leq x \lt 4\).


