Page 17 - Calculus Demystified

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EXAMPLE 1.2 CHAPTER 1 Basics
Find the set of points that satisfy x − 2 < 4 and exhibit it on a number line.
SOLUTION
We solve the inequality to obtain x< 6. The set of points satisfying this
inequality is exhibited in Fig. 1.4.

_ _ _
9 6 3 0 3 6 9
Fig. 1.4

EXAMPLE 1.3
Find the set of points that satisfy the condition
|x + 3|≤ 2 (*)

and exhibit it on a number line.
SOLUTION
In case x + 3 ≥ 0 then |x + 3|= x + 3 and we may write condition (∗) as
x + 3 ≤ 2
or
x ≤−1.
Combining x + 3 ≥ 0 and x ≤−1 gives −3 ≤ x ≤−1.
On the other hand, if x +3 < 0 then |x +3|=−(x +3). We may then write
condition (∗) as
−(x + 3) ≤ 2

or
−5 ≤ x.

Combining x + 3 < 0 and −5 ≤ x gives −5 ≤ x< −3.
We have found that our inequality |x + 3|≤ 2 is true precisely when either
−3 ≤ x ≤−1or −5 ≤ x< −3. Putting these together yields −5 ≤ x ≤−1.
We display this set in Fig. 1.5.

_ _ _
9 6 3 0 3 6 9
Fig. 1.5

You Try It: Solve the inequality |x−4| > 1. Exhibit your answer on a number line.

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You Try It: On a real number line, sketch the set {x: x − 1 < 3}.