**Unformatted text preview: **ﬁnition, we have
√
|5| = (5)2 = 25 = 5 and | − 5| = (−5)2 = 25 = 5. The long and short of both of these
procedures is that |x| takes negative real numbers and assigns them to their positive counterparts
while it leaves positive numbers alone. This last description is the one we shall adopt, and is
summarized in the following deﬁnition.
Definition 2.4. The absolute value of a real number x, denoted |x|, is given by
|x| = −x, if x < 0
x, if x ≥ 0 In Deﬁnition 2.4, we deﬁne |x| using a piecewise-deﬁned function. (See page 49 in Section 1.5.) To
check that this deﬁnition agrees with what we previously understood as absolute value, note that
since 5 ≥ 0, to ﬁnd |5| we use the rule |x| = x, so |5| = 5. Similarly, since −5 < 0, we use the
rule |x| = −x, so that | − 5| = −(−5) = 5. This is one of the times when it’s best to interpret the
expression ‘−x’ as ‘the opposite of x’ as opposed to ‘negative x.’ Before we embark on studying
absolute value functions, we remind ourselves of the properties of absolute value.
Theo...

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