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Real Numbers - Absolute Value

Hello Readers!!
Go through initial notes of Real Numbers before surfing here!! ^_^
Absolute Value
    If x is an non- negative real number then its absolute is denoted by |x| which is defined as, 
                         


Properties of Absolute Value
1. |x| ≥ x & |x| ≥ -x
2. |x| ≥ 0 
3. 
|x|2 = (x)2
4. 

5. |xy| = |x|.|y|
6. |x + y| ≤ |x| + |y|
7. 
|x - y| ≤ |x| + |y|
8. 
|x - y| ≥ |x| - |y|


Some important proofs











  • x ∈ R & 'a' be any positive real number then prove that if  |x|<a then
    -a<x<a and conversely. 
    Soln :
        Here,  |x|<a.........(i)
        We know, 
                  -x ≤ |x|...........(ii)
          from eqn (i) and (ii)
                  -x ≤ |x| <a
             or, -x < a
             or,  x > -a ...........(A)
        Again, we know that,
                   x ≤ |x|........(iv)
        Again, combining eqn (i) and (iv)
                   x ≤ |x| <a
              or, x < a
              or, x < a ...........(B)            
        Finally, combining eqn (A) and (B)
               -a < x < a

    CONVERSELY,
        Given, 
               -a < x < a
    So, -a <x.....(1) and x<a.......(2)
    By the definition of absolute value,
      
    Case I, when x<0                                  
          |x| = -x                                                    
    or, -|x| = x
    or, -|x| > -a [ from eqn 1 ]
        ∴ 
    |x| < a
    Case II, when x ≥0,        
          |x|=x
        ∴|x| < a [ from eqn 2 ]
    Thus, for all x ∈ R, |x| < a if -a<x<a.