You are watching: The distance of a number from zero on a number line

## What Does Absolute Value Mean?

Absolute value describes the **distance from zero** that a number is on the number line,** **without considering direction. The absolute value of a number is never negative. Take a look at some examples.

The absolute value of –5 is 5. The distance from –5 to 0 is 5 units.

The absolute value of 2 + (–7) is 5. When representing the sum on a number line, the resulting point is 5 units from zero.

The absolute value of 0 is 0. (This is why we **don"t** say that the absolute value of a number is positive. Zero is neither negative nor positive.)

## Absolute Value Examples and Equations

The most common way to represent the absolute value of a number or expression is to surround it with the absolute value symbol: two vertical straight lines.|6| = 6*means “*the absolute value of 6 is 6.”|–6| = 6

*means “*the absolute value of –6 is 6.

*”*|–2 – x|

*means “*the absolute value of the expression –2 minus x.

*”*–|

*x*|

*means “*the negative of the absolute value of x.

*”*

The number line is not just a way to show distance from zero; it"s also a useful way to graph equalities and inequalities that contain expressions with absolute value.

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Consider the equation |*x*| = 2. To show *x* on the number line, you need to show every number whose absolute value is 2. There are exactly two places where that happens: at 2 and at –2:

Now consider |*x*| > 2. To show *x* on the number line, you need to show every number whose absolute value is greater than 2. When you graph this on a number line, use open dots at –2 and 2 to indicate that those numbers are not part of the graph:

**In general, you get two sets of values for any inequality | x| > k or |x| ≥ k, where k is any number.**

Now consider |*x*| ≤ 2. You are looking for numbers whose absolute values are less than or equal to 2. This is true for any number between 0 and 2, including both 0 and 2. It is also true for all of the opposite numbers between –2 and 0. When you graph this on a number line, the closed dots at –2 and 2 indicate that those numbers are included. This is due to the inequality using ≤ (less than *or equal to*) instead of

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