Real Numbers - Absolute Value

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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.