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Definition of absolute value


Definition of absolute value: the absolute value of a number is the distance the number is from zero


Look at the following two graphs:

Absolute-value-image




The first one shows 6 located at a distance of 6 units from zero. We write |6| = 6

The second one shows -8 located at a distance of 8 units from zero. We write |-8| = 8

You can see from this that the absolute value of a number is always positive with the exception of taking the absolute value of 0 (|0| = 0)

Therefore, do not write |5| = -5 or |4| = -4 please!

|6| means distance from 0 to 6

|-8| means distance from 0 to -8

For |-8| = 8, you could also argue that to get 8, you have to take the negative of -8 since - - 8 = 8

So, | - 8| = - - 8 = 8

This observation helps us to come up with a formal definition of absolute value

| x | = x if x is positive or zero, but -x if x is negative

This definition is important to understand before solving absolute value equations or absolute value inequalities

Calculate the absolute value of the following numerical expressions

1) 42 − 4 × 2

2) -5 + 5 × 2 − 15

3) -8 + 2 × 5

1)

| -8 + 2 × 5 | = | -8 + 10 |

| -8 + 2 × 5 | = | 2 |

| -8 + 2 × 5 | = 2

2)

| 42 − 4 × 2| = | 16 − 4 × 2|

| 42 − 4 × 2| = | 16 − 8|

| 42 − 4 × 2| = | 8 |

| 42 − 4 × 2| = 8

3)

|-5 + 5 × 2 − 15| = | -5 + 10 − 15 |

|(-5 + 5 × 2 − 15)| = | 5 − 15 |

|(-5 + 5 × 2 − 15)| = | -15 |

|(-5 + 5 × 2 − 15)| = 15

Related topics

Solving absolute value equations

Solving absolute value inequalities





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