# Absolute Value Equations and Inequalities

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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