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Published on December 29th, 2010 In category Education | Maths

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Concept of Modulus and inequalities

An inequality is mentioned with either of the following signs , < , > <= or >=.

Meaning is as follows:

  • <  is less than
  • > is greater than
  • < = is equal to or less than
  • > = is equal to or greater than

The inequality will remain same if

  • If you add or multiply same positive number both the side
  • If you subtract or divide the same from both the side

The inequality will change if

  • if a negative number is multiplied or divided from both the side

Why this concept is important : Inequality helps us to set up conditions and give longer space of operations than only equal to equations; consider this  equation  x + 3 > 7 , this simple equation tells us  x + 3 – 3 > 7 -3  , x > 4 tells us the variable x is always greater than 4 that is it can assume any value between 5 to infinity.

The inequality also brings problem with itself like one may like to think  x2 > x is eqaul to x > 1 if we divide both the side with x, however that is not true as this does not consider the situation where x = 0

Modulus of any number is its absolute value for ex. | x | = 3 , this mean x = + 3  or – 3

Modulus is often expressed as inequality in equations  | x | <  4  can be represented as  – 4 < x < 4, this give a range of the variable

Inequalities can exists as quadratic equations like  x2 -3x + 2 > 0 can easily expressed as (x-2)(x-1) > 0 ,  if this would have been x2 -3x + 2 = 0, we could have easily said x = 2 and x = 1,  however inequality establishes few more variables hence there would be 2 cases for this when equation is satisfied , i.e.  when (x – 2) and ( x -1) are positive and also when (x – 2) and ( x -1) are negative

Results from x – 2 > 0 and x -1 > 0  both conditions are satisfied when x > 2

Results from x – 2 < 0 and x -1 < 0 both the condition are satisfied when x < 1 hence the solution is x > 2 and x < 1, on the graph line it mean x can’t be  2 or 1

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