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7.4 Absolute value

Definition: If x is a real number, then | x | is called the absolute value of x and it is the distance between x and 0.

For example, | 0 | = 0, | 3 | = 3 and | -5 | = 5

Now consider the distance between -2 and 3 is either |3 - (-2 | = | -5 | = 5 . Thus if a and b are any two numbers, then | b - a | = | a - b |

Many equations and inequalities involving absolute values can be solved using this geometric aspect of absolute value .

Solving equations : To solve an equation involving absolute value

  1. Isolate the absolute value one side of the equation.
  2. Set its contents equal to both + and - the other side of the equation.
  3. Now solve both equations.

Example     Solve 2 | x | - 3 = 7

Solution :     i) Isolating absolute value as

2 | x | - 3 = 7

2 | x | = 7 + 3

\ 2 | x | = 10

\ | x | = 5

ii) Set the contents of the absolute value equal to +5 and - 5

i. e. x = + 5 and x = - 5 . . . required solution

Example    Solve 5 | 2 x + 3 | + 3 = 20

Solution :     i) Isolate the absolute value as

5 | 2 x + 3 | = 17

| 2 x + 3 | =

ii) Setting the contents of the absolute value equal to + and - we get,

2 x + 3 = and 2 x + 3 = -

2 x = -3 and 2 x = - -3

2 x = and 2 x =

x = and x = . . . required solution

Index

7.1 Definition
7.2 Simultaneous Equations
7.3 Inequations (Inequalities)
7.4 Absolute Values

Chapter 8

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