Chapter 9 linear inequaliTieS 329
The sales clerk would need an annual sales level of $125,000 or more
in order for the salary plus commission option to be at least as attrac-
tive as the straight salary option.
Bounded Intervals
A bounded interval has both a left endpoint and a right endpoint. These
intervals are represented by double inequalities. The double inequality
a < x < b means all real numbers between a and b. In fact, a < x < b stands
for two separate inequalities. The notation a < x < b stands for “a < x and
x < b.”
An inequality such as 10 < x < 5 is never true because no number x is both
larger than 10 and smaller than 5. In other words, an inequality in the form
“larger number < x < smaller number” is meaningless.
The inequality a < x < b is represented on the number line by shading the
region between a and b. The inequality is represented with interval notation
by (a, b). Figure 9-20 gives the number line region and interval notation for
each type of double inequality.
Inequality Interval
a < x < b
a ≤ x ≤ b
a < x ≤ b
a ≤ x < b
Region on the Number Line
ab
ab
ab
ab
All real numbers between a and b
but not including a and b
All real numbers between a and b,
including b but not a
All real numbers between a and b,
including a but not b
All real numbers between a and b,
including a and b
(a, b)
[a, b]
(a, b]
[a, b)
FIgure 9-20
