Laws of Inequality -
There are some following laws that hold for inequations. –
Laws.1) If the same quantity is added to or subtracted from both sides of an inequation, then the sign of inequality between the two sides does not change. Symbolically, we can express this as follows.
If, a > b then a + c > b + c, a – c > b – c
The above equation holds for other inequalities as well.
Example.1) If x – 7 > 5 then x – 7 + 7 > 5 + 7 or x > 12.
Example.2) If y – 5 > 11 then y – 5 + 5 > 11 + 5 or y > 16
In other words, the rules of transposition holds for inequations as well. Thus, we can shift a term from one side of an inequation to the other by changing its sign.
Laws.2) If both sides of an inequation are multiplied or divided by the same positive quantity, the symbol of inequality does not change. Symbolically, this can be expressed as follows. It holds for all inequalities.
If a < b then ac < bc and a/c < b/c, where ‘c’ is a positive integers or quantity.
Example.1) If a/4 > 5 then a/4 X 4 > 5 X 4 or a > 20
Example.2) If 5z < 15 then 5z/5 < 15/5 or z < 3
Laws.3) If both sides of an equality are multiplied or divided by the same negative quantity, the symbol of inequality is reversed. Symbolically this can be represented as follows. This holds for other inequalities as well.
If a > b then ac < bc and a/c < b/c where ‘c’ is negative quantity.
- x
Examples.1) If (–x) / 5 > 3 then ------- X (-5) < 3 X (-5)
5
or x < -15
-3x -6
Example.2) If, -3x ≤ - 6 then -------- ≥ -------- or x ≥ 2
-3 -3