How to Study Inequalities

INEQUALITIES

a > b if "a-b" is a "+" number

a < b if "a-b" is a "-" number

Fundamental Properties

1) An inequality is unchanged in sense if the same number is added or subtracted from both sides.

a > b

a + c > b + c

a - c > b - c

2) An inequality is unchanged in sense if both sides are multiplied or both divided by the same positive number.

3 > 1

(1/3) 3 > 1 (1/3)

1 > 1/3

3) An inequality is changed in sense if both sides are both multiplied or divided a negative number.

3 > 1

-1(3) > (1) - 1

-3 < -1

4) An inequality of positive quantities is unchanged in sense if the same positive power or positive root each side is taken.

3 > 2

3² > 2²

9 > 4

25 < 49

25^1/2 > 49^1/2
25 < 7

5) If the first of the three quantities is greater than the second, and the second is greater than the third, then the first is greatre than the third.

if  a > b

    b > c , then a > c

6) If inequalities are added to unequals in the same order, the sums are unequal in the same order.

a < b

c > d

  a < b

+d < c

a + d < b + c

7) If unequals are subtracted from equals, the results are unequal, the result are unequal in the opposite order.

a > b        c = d        5 > 3       9 = 9

  c = d
- a > b
c - < d - b

  9 = 9
- 5 > 3
9 - 5 < 9 - 3

8) If a > b ≥ c and c > d ≥ 0, then ac > bd.
* 7 > 3 > 1 , 1 > 1/2 > 0
   7 (1) > 3(1/2)
   7 > 1.5

9) If a, b, c are positive, then a² + b² + c² > ab + ac + bc.

7² + 3² + 1² > 7(3) + 3(1)
49 + 9 + 1 > 21 + 7 + 3
51 > 31

Example: Find the values of "x" for which 2 (3x-4) > (7x/2 + 2).

Solution:

2 (3x-4) > (7x/2 + 2)

-7x (6x-8) > (7x + 4) - 7x

 -x -8 > 4

-1 (-x) > (12) - 1

x < -12

Therefore: x = { set of real nos. < -12}

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