Absolute Value

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?)

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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