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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