Absolute value is represented as | x | which implies that it always gives a positive number.
For example: - | -1 | = 1 and | 1 | = 1
When we represent an absolute value in the form of equation and inequalities then it is known as absolute
value equation and inequalities.
Properties of absolute value equations and inequalities: -
If x and y are two real numbers then
i) | x + y | ≤ | x | + | y |
ii) | x – y | ≤ | x | + | y |
iii) | x y | = | x | | y |
iv) If x = y then | x – y | = 0 and conversely, | x – y | = 0
v) If | x | ≤ y then – y ≤ x ≤ y
vi If | x | ≥ y then either x ≥ y or, x ≤ - y.
vii) | x – a | = b implies that either x = a + b or, x = a – b
Question 1: - if | x | > 1, find x.
Solution: - | x | < 1
Case 1: - + x < 1 So x < 1 … (1)
Case 2: - - x < 1 so x > -1 … (2)
From (1) and (2), we can write
-1 < x < 1
Question 2: - If |a|≤ 5, find a.
Solution: - -5≤a≤ 5