Absolute Value: Definition, Formula, Properties, Questions & More

In mathematics, the absolute value  of a real number x, represented as |x|, is the non-negative value of x without regard to its sign.

Absolute value Formula: The absolute value formula or absolute value equation is an equation that contains an absolute value expression. It is represented as follows:

Thus,

|x| = x if x is positive,

|x| = −x if x is negative, and

|0| = 0.

→ Absolute value is also known as modulus.

→ |x| which is pronounced as 'Mod x' or 'Modulus of x'.

What is the meaning of absolute value?

The absolute value of a number represents its distance from 0 on a number line. We know that distance is always a non-negative quantity. That is why the absolute value is always non-negative. 

Basic properties of absolute value inequalities are:

Let x be a variable or an algebraic expression and a be the real number such that a>0. Then the following inequalities hold:

∣x∣ ≤a ⇔ −a ≤ x ≤ a

|x| ≥ a ⇔ x ≤ −a  or  x ≥ a

|x| < a ⇔ −a < x < a

|x| > a ⇔ x < −a  or  x > a

Some other properties of absolute value inequalities are:

|a + b| ≤ |a| + |b| if both a and b have the same sign, i.e. ab > 0

|a + b| ≤ |a| + |b| if both a and b have different sign, i.e. ab < 0

Some examples showing application of absolute value formula or absolute value inequalities are given below:

1. Solve |5 – 3x| = 12

Solution:

|5 – 3x| = 12

5 – 3x = 12    or    5 – 3x = –12

      –3x = 7      or    –3x = –7

     x = –7/3     or         x = 17/3

2. Solve |4x – 3|= |x + 6|

Solutions:

|4x – 3|= |x + 6|

4x – 3 = x + 6      or       4x – 3 = – (x + 6)

 3x = 9                     or       4x – 3 = – x – 6

  x = 3                      or         5x = –3

  x = 3                      or            x = –3/5

3. Solve |2x+3|<6

Solution:

|2x+3|<6

–6 < 2x+3 < 6

–6 –3 <2x + 3 – 3 < 6 –3

–9 < 2x < 3

−9/2 < x < 3/2

Thus, the solution to the given absolute value inequality is the interval (−9/2 < x < 3/2).

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