What is absolute value and all of its properties give 100 examples

Step-by-step explanation:

Absolute value refers to the distance of a point from zero or origin on the number line, regardless of the direction. The absolute value of a number is always positive.

Absolute value of a number is denoted by two vertical lines enclosing the number or expression. For example, the absolute value of number 5 is written as, |5| = 5. This mean that, distance from 0 is 5 units:

Similarly, the absolute value of a negative 5 is denoted as, |-5| = 5. This means that, distance from 0 is 5 units:

Not only does a number show the distance from the origin, but also is important for graphing the absolute value.

Consider an expression |x| > 5. To represent this, on a number line, you need all numbers whose absolute value is greater than 5. This is done graphically by placing an open dot on the number line.

Consider another case where |x| = 5. This includes all absolute values that are less than or equal to 5. This expression is graphed by placing a closed dot on the number line. The equal sign indicates that all values being compared are included in the graph.

An easy way of representing expression with inequalities is by following the following rules.

For |x| < 5, -5 < x < 5

For |x| = 5, -5 = x = 5

For |x + 6| < 5, -5 < x + 6 < 5

Properties of Absolute Value

Absolute value  has the following fundamental properties:

Non-negativity |a| ≥ 0

Positive-definiteness |a| = 0a = 0

Multiplicativity         |ab| = |a| |b|

Subadditivity |a + b| ≤ |a| + |b|

Idempotence ||a|| = |a|

Symmetry |−a| = |a|

Identity of indiscernible |a − b| = 0 ⇔ a = b

Triangle inequality |a − b| ≤ |a − c| + |c − b|

Preservation of division |a/b|=|a|/|b|  if  b ≠ 0

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