Real Number Line and Absolute Value

The real numbers can be represented on a number line. This number line can be helpful in determining whether or not one number is greater than another. One number that is to the right of another on the real line is the greater of the two.

Since 4 is to the right of −2, 4 is greater than −2. Symbolically, this is expressed as 4 > −2.

This type of inequality symbol is known as a strict inequality. Notice that the arrow points to the lesser number.

In later lessons, a weak inequality symbol will be used. This is an inequality sign with a line underneath: ≥ or ≤.

The line underneath allows for the possibility of both sides being equal.

 

Absolute Value

The method used to measure the distance of a number from zero (regardless of which side of zero the number appears on) is known as absolute value.

Since absolute value measures a distance, it is never equal to a negative number. Absolute value is denoted by placing the number inside | |, such as |3| or | − 5|.

 

Examples

| − 4| = 4 (since −4 sits 4 units from 0)

|6| = 6 (since 6 sits 6 units from 0)

Occasionally, there is another sign placed outside the absolute value sign, such as in −| − 7|.

This is actually indicating that two operations are taking place.

−| − 7| first, find the absolute value of − 7 : | − 7| = 7

second, find the opposite of 7 : −7

so, − | − 7| = −7

 

The formal definition of absolute value is the following: for a real number x, the absolute value of x is defined as follows:

 

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