Exercise 89 Page 48

What does the absolute value of a real number represent?

See solution.

Practice makes perfect

Let's start by reviewing what the absolute value of a real number is, and what an absolute value inequality implies.

The absolute value of a real number |x| is the distance from 0 on the number line. For example, |x|=a. In this equation, the value of x is a or - a.

The inequality |x|

In the same way, we can think about the inequality |x|>a as referring to all those values whose distance from 0 in the number line is greater than a.

Therefore, the absolute value inequality |x|>a is equivalent to the compound inequality x>a or x<- a.

Considering how each absolute value inequality is rewritten as a compound inequality, we conclude the following.

If the absolute value is less than a value, the compound inequality is joined by the word and.
If the absolute value is greater than a value, the compound inequality is joined by the word or.

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