2.2 Inequalities
 x  = a means x =  a or x = + a
 x  < a means  a < x < a
 x  a  = d means x = a  d or x = a + d
 x  a  £ d means a  d £ x £ a + d
0 <  x  a  < d means a  d < x < a + d Except x = a
0 <  x  a  £ d means a  d £ x £ a + d Except x = a
Examples

 x  3  < 1
\ 3  1 <
x < 3 + 1 i.e.
2 < x <
4
 x  2  < 0.01 \2  0.01 < x < 2 + 0.01 i.e. 1.99 < x < 2.01
0 <  x  1  < 0.01 \ 1  0.01 < x < 1 + 0.01 except x = 1 i.e.0.99 < x < 1.01 except x = 1
0 <  x  5  £ 0.01 \ 4.99 £ x £ 5.01 except x = 5
d  neighborhood of a ( d  nbd of a )
If the open interval ( a  d, a + d ) for the variable x ( say)
It means a  d < x < a + d
It means  x  a  < d
It means x lies in the open interval ( a  d, a + d )
I means x belongs to d  nbd of a
It means d is closed to a by less than d
Note that all these statements mean the same thing  x  a  <
d
Example 1 Write down the 0.01 nbd of 3 in the interval form
Solution : d  nbd of a means ( ad , a + d )
Here a = 3 and d = 0.01
\ 0.01 nbd of 3 = ( 30.01, 3 + 0.01)
= ( 2.99, 3.01 )
Example 2 State any two values of x such that  x 5 < 0.001
Solution :  x  5 < 0.001 i.e. 5  0.01 < x < 5 + 0.001
i.e. 4.999 < x < 5.001
Thus any two values of x in ( 4.999 < 5.001 ) are 4.9999 and 5.0001
Example 3 State any two values of x such that x is closer to 1/3 by less than 0.01.
Solution : x is closer to 1/3 by less than 0.01 can be put in the form  x  1/3  < 0.01 i.e. x belongs to the interval ( 1/3  0.01, 1/3 + 0.01)
Any two values in this interval of x are 0.323 and 0.340
Example 4 Find three values of x, satisfying the inequality  2x  1  < 0.05
Solution :  2x  1  < 0.05
\ 2x belongs to the interval ( 1  0.05, 1 + 0.05 ) i.e. ( 0.95, 1.05 )
\x e i.e. ( 0.475, 0.525 )
any three values of x are 0.49, 0.50, 0.5001.
