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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

1. | x - 3 | < 1
\ 3 - 1 < x < 3 + 1   i.e. 2 < x < 4

2. | x - 2 | < 0.01
\2 - 0.01 < x < 2 + 0.01   i.e. 1.99 < x < 2.01

3. 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

4. 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)

1. It means a - d < x < a + d

2. It means | x - a | < d

3. It means x lies in the open interval ( a - d, a + d )

4. I means x belongs to d - nbd of a

5. It means d is closed to a by less than d

Note that all these statements mean the same thing | x - a | < d Your browser does not support the IFRAME tag.

Example 1  Write down the 0.01 nbd of 3 in the interval form

Solution :  d - nbd of a means ( a-d , a + d )

Here a = 3 and d = 0.01

\ 0.01 nbd of 3 = ( 3-0.01, 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.

Index

Chapter 3 