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If x and y  are two quantities such that x  \neq y  ; then any of the following conditions could be true:

  1. x > y 
  2. x \geq y 
  3. x < y 
  4. x \leq y 

EG8

 

If a, b and c are real numbers, then each of the following is called a linear inequation in one variable:

  1. ax+b > c 
  2. ax+b<c 
  3. ax+b \geq c 
  4. ax+b \leq c 

As you would have noticed already, the signs >, <, \geq, \leq   are called signs of inequality.

 

Replacement Set and Solution Set in Set Notation

The set from which the value of the variable x is chosen is called Replacement Set and its subsets, whose elements satisfy the inequation are called solution sets.

Let’s take the following example:

If the inequation is x < 5 , if

  1. The replacement set  = N  \ i.e \ \{1, 2, 3, 4, 5, 6, ... \} , then the solution  set  = \{1, 2, 3, 4 \} 
  2. The replacement set  = W \ i.e \  \{0, 1, 2, 3, 4,5 ,6 ,7,... \} , then the solution set  = \{0, 1, 2, 3, 4 \} 

We can also describe the solution set in set builder form  i.e \{x:x \in N \ and \ x < 5 \}    or as in the second case \{x:x \in W \ and \ x <  5 \}  

 

If the inequation is x \leq 5 , if

  1. The replacement set  = N  \ i.e \   \{1, 2, 3, 4, 5, 6, ... \} , then the solution  set  = \{1, 2, 3, 4, 5 \} 
  2. The replacement set  = W \ i.e \  \{0, 1, 2, 3, 4,5 ,6 ,7,... \} , then the solution set  = \{0, 1, 2, 3, 4, 5 \} 

We can also describe the solution set in set builder form  i.e \{x:x \in N \ and \ x \leq 5 \}    or as in the second case \{x:x \in W \ and \ x \leq 5 \}  

 

EG12You could also represent the above solutions on a number line as well. Please note the following notations:

\circ marks the end of the range with a strict inequality ( < \  or \ > )

\bullet  marks the end of the range with a strict inequality ( \leq \  or \ \geq )

 

Therefore, for \{x:x \in I \ and \ x < 5 \}   we can draw as follows….

fig 1

Therefore for \{x:x \in I \ and \ x \leq 5 \}   we can represent as follows….

fig 2

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