**An introduction to the concept of Absolute value. **In one word, Absolute value is “distance”.

The absolute value of a number is the distance of the number from zero. Now, guess smart, what can the term Absolute value of a number signify? A number without a sign! Yes, you guessed it right! Why so? Because its distance and distance are just a number, neither positive nor negative (i.e., a number without a + or — sign). Whether it is +3 or -3, the absolute value of both numbers is just 3. And again, since the distance is never expressed in negative terms, therefore the absolute value of a number is never negative (You do not say your school is – 3 miles away from your home, do you?) (Not even positive for that matter, do you feel the distance is positive, or just a number?)

**The denotation of Absolute value of a number:**

The vertical bars | | are used to denote the absolute value of a number. For example, *| 5 | = 5 and | — 5 | = 5*, i.e. the absolute value of both the numbers 5 (i.e. +5) and —5 is 5.

**Note:**

*— | 5 | = — 5,* because the negative sign is outside of the bars. But,* | —5 |≠ —5,* because the negative sign is inside the bars. Representing Absolute Value of a number on a Number Line. On the Number line above, the two numbers 5 and -5 are both a distance of 5 from zero. This distance from zero is what the Absolute value of a number denotes. The absolute value of a number is, therefore, the geometrical concept of distance; a distance of a number from zero.

**Example 1:**

**Evaluate the following:**

*1. | 4 | 2. | – 4 | 3. | -2 | + | -5 | 4. | -9 | – | 10 | 5. | 2 | × | -3 |*

*2. | 5 |/| -6 | 7. | 0 |*

**Answers:**

*1. | 4 | is the distance of 4 from zero, therefore | 4 | = 4*

*2. | -4 | is distance of -4 from zero, therefore | – 4 | = 4*

*3. | -2 | + | -5 | = 2 + 5 = 7*

*4. | -9 | – | 10 | = 9 – 10 = -1*

*5. | 2 | × | -3 | = 2 × 3 = 6*

*6. | 5 |/| -6 | = 5/6*

*7. | 0 | = 0*

**Important Note:**

**1. Why is | 0 | = 0?**

Recall what the absolute value of a number signifies? It is a distance of the number from 0.

As a consequence, what should be the distance of Zero from Zero? Of course, 0!

**2. Very important note:**

As a convention, *√x* stands for the non-negative root of the number x.

By the same vein, *√x2* denotes the non-negative square root of *x2*

Now, since | x | is non-negative (≥0), it becomes, therefore, possible to write: *√x2 = | x |*

**Example 2:**

Is the question below true or false? *| — 6 | = — 6*

**Answer:** False

**Example 3:**

Evaluate || — 6 ||

**Solution:**

Move out from inside: *| — 6 | = 6*

*|| — 6 || = | 6 | = 6*

**Example 4:**

*Is | 2 — x | = | x — 2 |?*

**Answer:** Yes

First of all, we can write *| 2 — x | as |— (x — 2) | {since, 2 —x = —(x—2)}*

*Now, as an example, since | — 6 | = | 6 | = 6, therefore, | 2 — x | = | (x — 2) |*

*Finally, | 2 — x | = | (x — 2) | = x — 2*

The Algebraic definition of Absolute value (A.V.) of any real number x

*| x | = x , if x ≥ 0*, (i.e., x is non-negative number) and

*| x | = — x, if x < 0* (i.e. if x is negative number)

At the outset, let there not be the question of the doubt as to how | x | can ever be negative.

For, though the above definition does indeed create such a doubt, it is only apparently, not actually as the following explanation will clarify:

**1. What is | x |, if x is 3?**

Since x is 3, and as 3 is a positive number (i.e. 3 > 0), therefore, the first part of the definition of A.V is used to write | 3 |

| 3 | = 3

**2. What is | x |, if x is —3?**

This time, since x is —3, and as —3 < 0, i.e. — 3 is a negative number, therefore, the second part of the definition of A.V. is used to write | —3 |

*| —3 | = — (—3) = 3*

Therefore, *| 3 | = | —3 | = 3*

Note:

| —3 | ≠ —3, since the absolute value of a number can never be a negative number.

Why? You know it, because absolute value of a number, from its definition, is the distance of a number from zero, and distance is never expressed as a negative number.

**Note:**

If | x | = a, then the equation has two solutions, i.e., there can exist two values of x which satisfy the absolute value equation.

And they are: x = a or x = —a

For example, the absolute value equation | x | = 5, has two solutions: one x = 5 and the other x = —5

Important: x is called the argument of the absolute value | x |

Again, note the following

**1. If | x |= | a |, then x = ± a**

For example, if *| x |= | 3 |, then x = ± 3, i.e. x = 3 or x = —3*

Problems:

**1. Solve for x: | x — 2| = 5**

**Solution:**

The argument x — 2 can be either 5 or —5, so

*x — 2 = 5 or x — 2 = —5, i.e.*

*x = 2 + 5 or x = 2 — 5*

*So, x = 7 or — 3*

**Important:**

What is the geometrical meaning of *| x— 2 | = 5?*

*|x — 2 | = 5* represents the distance of a number x from 2 is 5