What exactly do we mean by Absolute Value/ Modulus? Lamely speaking, whatever comes out of the two parallel bars |y| => will always be non-negative (it may be 0 also).

Eg. |-7|= 7, likewise, |7| = 7.

Let us say we have |y|, where y is any random number on the number line. Then, |y| means the distance of y from the origin! Can we have a negative distance? Not really! Thus, the absolute value is always non-negative. It is always positive; the only different case is |0|= 0.

If we have |x|> 5, this inequality means the x’s distance from the origin is greater than 5. This distance may be on the left side of 0 or the right side of 0. The left side means x will lie farther (leftwards) than -5, and the right side means x will lie farther (rightwards) than 5. Thus, x < -5 or x > 5.

Typical Important Properties of Absolute Value:

** x/|x| **may provide only two values, 1 and -1.

If x > 0, then **x/|x|** = 1

If x < 0, then **x/|x|** = -1

Likewise, also for |x|/x = 1 or -1

** **If x and y have the same signs, (+,+) or (-,-), the distance of (x+y) from the origin is the same as the sum of the distance of x from the origin and the distance of y from the origin.

Eg. |3 + 5| = |3|+|5|= 8

|(-3)+(-5)|= |-3|+|-5|= 3+5=8

If x and y have opposite signs, (+, -) or (-, +), due to the opposite signs, either x or y will cancel out some value of the other. Thus, the distance between x from the origin and y from the origin is always greater than the distance of (x+y).

Eg. |5+(-3)|= |2|= 2 < (|5|+|-3|) i.e 8

Likewise, |-5+3|=|-2|=2 < (|-5|+|3|) i.e 8

** **If x and y have same signs, (+,+) or (-,-), provided

If x and y have opposite signs (+,-) or (-, +) and if same signs but, then,

More about the unique properties of Absolute Values in the next** GMAT **article. Until then, these are enough to chew upon for a more vivid sense of absolute values.