College Algebra 9th txtbk

SECTION 1–2 Linear Inequalities 57
Z Understanding Inequality and Interval Notation
The preceding mathematical statements use the inequality, or order, relations, more commonly
known as “greater than” and “less than.” Just as we use the symbol “” to replace
the words “is equal to,” we use the inequality symbols and to replace “is less than”
and “is greater than,” respectively.
You probably have a natural understanding of how to compare numbers using these
symbols, but to be precise about using inequality symbols, we should have a clear definition
of what they mean.
Z DEFINITION 1 a < b
and b > a
For two real numbers a and b, we say that a is less than b, and write a b, if
there is a positive real number p so that a p b. The statement b a, read
b is greater than a, means exactly the same as a b.
This definition basically says that if you add a positive number to any number, the sum is
larger than the original number.
When we write a b we mean a 6 b or a b and say a is less than or equal
to b. When we write a b we mean a 7 b or a b and say a is greater than or equal
to b.
The inequality symbols 6 and 7 have a very clear geometric interpretation on the real
number line. If a 6 b, then a is to the left of b; if c 7 d, then c is to the right of d (Fig. 1).
This is called a line graph.
a
d
b
c
Z Figure 1 a b, c d.
If we want to state that some number x is between a and b, we could use two inequalities:
x a and x b. Instead, we will write one double inequality, a x b. For example,
the inequality 2 x 5 indicates that x is between 2 and 5, and could be equal
to 5, but not 2. The set of all real numbers that satisfy this inequality is called an interval,
and is commonly represented by (2, 5]. In general,
(a, b] 5x ƒ a 6 x b6*
The number a is called the left endpoint of the interval, and the symbol “(” indicates that
a is not included in the interval. The number b is called the right endpoint of the interval,
and the symbol “]” indicates that b is included in the interval. An interval is closed if it
contains its endpoint(s) and open if it does not contain any endpoint. Other types of intervals
of real numbers are shown in Table 1.
Note that the symbol “ ,” read “infinity,” used in Table 1 is not a numeral. When we
write [b, ), we are simply referring to the interval starting at b and continuing indefinitely
to the right. We would never write [b, ] or b x , because cannot be used as an
endpoint of an interval. The interval (, ) represents the set of real numbers R, since its
graph is the entire real number line.
*In general, 5x ƒ P(x)6 represents the set of all x such that statement P(x) is true. To express this set verbally, just
read the vertical bar as “such that.”