Absolute Value Formula

The absolute value formula is used to find the non-negative value of a number without considering whether the number is positive or negative. On a number line, absolute value represents the distance of a number from zero, so it can never be negative.

What Is the Absolute Value Formula?

The absolute value of a number x is written as:

|x|

The standard formula is:

|x| = x, if x ≥ 0

|x| = −x, if x < 0

This means a positive number or zero stays unchanged, while a negative number becomes its positive opposite.

Absolute Value Examples

For a positive number:

|7| = 7

For a negative number:

|−7| = 7

For zero:

|0| = 0

Both 7 and −7 are 7 units away from zero on the number line, so their absolute values are the same.

Absolute Value as Distance

One of the easiest ways to understand absolute value is to think about distance. The expression |x| tells you how far x is from zero, regardless of its direction.

For example:

|−4| = 4

because −4 is 4 units away from zero.

Absolute Value of an Expression

The absolute value can also be applied to an algebraic expression. For example:

|x − 5|

This represents the distance between x and 5 on the number line. More generally, |a − b| represents the distance between two real numbers a and b.

For example, if x = 2:

|2 − 5| = |−3| = 3

So, the distance between 2 and 5 is 3 units.

Important Absolute Value Rules

Some useful properties of absolute value are:

|x| ≥ 0

The absolute value of a real number is always zero or positive.

|−x| = |x|

A number and its opposite have the same absolute value.

|xy| = |x||y|

The absolute value of a product equals the product of the absolute values.

|x/y| = |x|/|y|, when y ≠ 0.

The absolute value of a quotient equals the quotient of the absolute values.