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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


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

2.1 Modulus
2.2 Inequalities
2.3 Limits Of Functions
2.4 Left Hand And Right Hand Limits
2.5 Theorems On The Algebra Of Limits
2.6 Evaluating Limits
2.7 Limits Of Trigonometric Functions
2.8 The Exponential Limits

Chapter 3





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