# 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.