1. Basic Concept of Inequalities

2. What are Inequalities?
The symbols ‘>’ and ‘<’ are used to compare two numbers.
For example,
(a) ‘7 is greater than 4’ can be written as
(b) ‘2 is less than 5’ can be written as ‘7
>
4’. ‘2
<
5’.
Both ‘>’ and ‘<’ are inequality signs.

3. This is known as the law of trichotomy.
Besides ‘>’ and ‘<’, there are other three common inequality
signs: ‘>’, ‘<’ and ‘=’.

5. When two expressions are connected with an inequality sign, an inequality is formed.
All the examples in the previous table are inequalities.
Note: For any two numbers x and y, the meanings of the following pairs of inequalities are the same.

6. Use an inequality to represent the statement in each of the following.
(a) 12 is less than x.
(b) y is not greater than 40.
(c) The weight of a box (w kg) is not less than 5 kg.
(a) 12 x
<
(b) 40 3 + y
(c) 5 w

7. Follow-up question 10. than less not is times 3 (b) 2. greater (a) .
statements
following
the of
each
for
inequality an up
Set y x 2 x
> 10 3 y
(c) 7 is less than or equal to b minus 5. 5 7 - b

8. Solutions of Inequalities and their Graphical Representations
For an inequality in one unknown, the values of the unknown
that satisfy the inequality are called the solutions of the
inequality.
Let us consider the inequality x > 0.
When x = 1, the inequality becomes 1 > 0, which is true. ∴
x = 1 is a solution of the inequality.

9. In fact, there is an infinite number of solutions to the
inequality x > 0. –2 –1 1 2 3 4 5
Consider the above number line.
All the numbers to the right of 0 on the number line are greater than 0.
Therefore, they are the solutions of the inequality x > 0 (i.e. solutions are 1, 2, 3, 4, 5, etc.).
Actually, we can represent the solutions of inequalities graphically.

10. o For example, of Solutions 1. > x 0. of solutions are 0, excludingof
Solutions 1.
> x 0. of
solutions
are 0,
excluding
right
the to
values
All
> x
included
not is ’ ‘
that
means
circle
hollow
The
(ii) 0. of
right
the to
numbers
all
are
solutions
indicates
pointing
arrow
(i) :
Note o
> x
in the solution.

11. · of Solutions 2. ³ x 0. of solutions are 0, including right the toof
Solutions 2. x 0. of
solutions
are 0,
including
right
the to
values
All x
included is ’ ‘
that
means
circle
solid
The :
Note
in the solution.

12. of
Solutions 3.
< x 0. of
solutions
are 0,
excluding
left
the to
values
All
< x 0. of
left
the to
numbers
all
are
solutions
that
indicates
pointing
arrow
The :
Note
< x

13. of
Solutions 4. x 0. of
solutions
are 0,
including
left
the to
values
All x

14. Follow-up question
1. Determine whether x = –2 is a solution of the inequality
2x –5 .
Solution
When x = –2,
L.H.S. = 2(–2) + 3 = –1
R.H.S. = –5
The inequality is satisfied. ∵
–1 –5 is true.
x = –2 is a solution of the inequality 2x –5 . ∴

15. Follow-up question (cont’d)
2. Represent the solutions of the following inequalities graphically.