James Stewart-Calculus_ Early Transcendentals-Cengage Learning (2015)

A4
appendix A Numbers, Inequalities, and Absolute Values
We also need to consider infinite intervals such as
sa, `d − hx | x . aj
This does not mean that ` (“infinity”) is a number. The notation sa, `d stands for the
set of all numbers that are greater than a, so the symbol ` simply indicates that the interval
extends indefinitely far in the positive direction.
1 Table of Intervals
Notation Set description Picture
Table 1 lists the nine possible types of
intervals. When these intervals are discussed,
it is always assumed that a , b.
sa, bd
fa, bg
fa, bd
sa, bg
sa, `d
fa, `d
s2`, bd
s2`, bg
s2`, `d
hx | a , x , bj
hx | a < x < bj
hx | a < x , bj
hx | a , x < bj
hx | x . aj
hx | x > aj
hx | x , bj
hx | x < bj
R (set of all real numbers)
a
a
a
a
a
a
b
b
b
b
b
b
Inequalities
When working with inequalities, note the following rules.
2 Rules for Inequalities
1. If a , b, then a 1 c , b 1 c.
2. If a , b and c , d, then a 1 c , b 1 d.
3. If a , b and c . 0, then ac , bc.
4. If a , b and c , 0, then ac . bc.
5. If 0 , a , b, then 1ya . 1yb.
Rule 1 says that we can add any number to both sides of an inequality, and Rule 2 says
that two inequalities can be added. However, we have to be careful with multiplication.
Rule 3 says that we can multiply both sides of an inequality by a positive number, but
Rule 4 says that if we multiply both sides of an inequality by a negative number, then
we reverse the direction of the inequality. For example, if we take the inequality 3 , 5
and multiply by 2, we get 6 , 10, but if we multiply by 22, we get 26 . 210. Finally,
Rule 5 says that if we take reciprocals, then we reverse the direction of an inequality
(provided the numbers are positive).
Example 1 Solve the inequality 1 1 x , 7x 1 5.
SOLUTIOn The given inequality is satisfied by some values of x but not by others. To
solve an inequality means to determine the set of numbers x for which the inequality is
true. This is called the solution set.
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