Overview: Inequalities are almost as important as
equations in calculus. Many functions’ domains are
intervals, which are defined by inequalities. Inequalities are needed to study
where functions have positive
and negative values . They are also used in the definitions of limits and with
derivatives to study where
functions are increasing and decreasing and where their graphs are concave up
and concave down. In this
section we describe notation and terminoogy for intervals and other sets and
discuss the rules for solving
inequalities. These rules are similar to those for solving equations but are
somewhat more difficult to
apply.
Topics:
• Intervals and other sets of numbers
• The absolute value function
• Working with inequalities
Intervals and other sets of numbers
Intervals can be defined, as in the last section, by giving their defining
inequalities. With this approach
the interval in Figure 1 is called the interval 0 < x ≤ 2. The heavy line in the
drawing indicates that the
points x with 0 < x < 2 are in the interval; the dot shows that x = 2 is in the
interval; and the open
circle at x = 0 indicates that the point x = 0 is not.
We can also define intervals with set-builder notation:
The symbols {x : P} designate the set
of numbers x that satisfy condition P. With this notation the interval in Figure
1 can be defined by
{x : 0 < x ≤ 2}, which reads “the set of those numbers x such that 0 < x ≤ 2.”
We also refer to this
interval as (0, 2], where the parenthesis at the left indicates that the point x
= 0 is not in the interval and
the square bracket at the right indicates that the point x = 2 is in the
interval. Similarly, the interval in
Figure 2 can be referred to either as the interval x ≤ 2, as the interval {x : x
≤ 2}, or as the interval
(−∞, 2]; and the interval in Figure 3 can be given by
x > 1, by {x : x > 1}, or by (1,∞).
 |
 |
 |
The interval 0 < x ≤ 2 or
{x : 0 < x ≤ 2} = (0, 2]
FIGURE 1 |
The interval x ≤ 2 or
{x : x ≤ 2} = (−∞, 2]
FIGURE 2 |
The interval x > 1 or
{x : x > 1} = (1,∞)
FIGURE 3 |
To describe a set that consists of two intervals, we use
the union symbol ∪ with the convention
that A ∪ B designates the set consisting of the points
in set A combined with the points in set B:
A ∪ B = {x : x is in A or x is in B}.
Example 1
(a) Draw on an x-axis the set of points {x : −5 ≤ x < 2 or x > 4}.
(b) Express the set in part (a) as a union of intervals.
Solution
(a) We show the intervals on an x-axis by drawing solid lines from x = −5 to x =
2
and to the right of x = 4 and by putting a solid dot at x = −5 and small open
circles
at x = 2 and x = 4, as in Figure 4.
(b) Since {x : −5 ≤ x < 2 or x > 4} consists of the interval −5 ≤ x < 2 and the
interval x > 4, it is their union [−5, 2) ∪ (4,∞).
FIGURE 4
The absolute value function
When we want to consider the size of a number x without reference to whether
it is positive or negative,
we use its absolute value, denoted |x|. The absolute value of x equals x if x is ≥
0 and can obtained
by multiplying x by −1 if it is negative, so that |5| = 5 and | − 5| = −(−5) =
5. Thus, the absolutevalue
function y = |x| with variable x can be defined by
(1)
Definition (1) shows that the graph of the absolute value
function consists of the line y = x for
x ≥ 0 and the line y = −x for x < 0, as shown in Figure 5.
FIGURE 5
The absolute value of a number x can also be obtained with the formula
(2)
since, by convention,
denotes the nonnegative square root of x2. For example, if x = −5, then
and this is the absolute value of −5.
Frequently it is convenient to think of the absolute value
|x| of a number x as its distance from
the origin on an x-axis (Figure 6). We can also think of |a − b| as the distance
between the points a and
b on an x-axis (Figure 7).
Example 2 Solve the equation |2x − 8| = 6 for x.
Solution
One solution: |2x − 8| = 6 if 2x − 8 = 6 or if 2x − 8 = −6. We solve these
equations
by adding 8 to both sides of each and then dividing both sides of each by 2:
The solutions are x = 7 and x = 1. This is illustrated in Figure 8 by the curve
y = |2x − 8| and the line y = 6, which intersect at x = 1 and x = 7.
Alternate solution: Dividing both sides of |2x − 8| = 6 by
2 gives |x − 4| = 3, so
the solutions are the points x = 1 and x = 7 that are a distance 3 from x = 4,
as can
be seen in Figure 8.
FIGURE 8
Question 1 Verify the solutions in Example 2 by
substituting the values of x in the formula |2x−8|.
The set consisting of a finite number of real numbers
is denoted
Thus, the solution set (the set of solutions) of the equation in Example 2 is
the set {1, 7}.
Working with inequalities
An inequality is a statement such as x3−1 < 4x3−4 that involves an
inequality symbol <,>, ≤, or
≥. Two inequalities involving the variable x are equivalent if they are
satisfied by the same values of x.
We say that an inequality has been solved if it has been replaced by an
equivalent inequality in which
the variable appears alone on one side of the inequality sign and not on the
other, as in the statement
x > −1.
How does solving inequalities differ from solving
equations? If we want to solve the equation
x3 −1 = 4x3 +2, we can add 1 to both sides to obtain x3 = 4x3 +3; subtract
4x from both sides to have
−3x3 = 3; divide both sides by −3 to have x3 = −1; and finally take cube roots
of both sides to obtain
the solution x = −1. We can also perform such procedures on inequalities, but
with some modifications.
Here are some basic rules:
Theorem 1 (a) Adding a number to both sides of an
inequality, subtracting a number from both sides,
or multiplying or dividing both sides by a positive number yields an
equivalent inequality.
(b) Multiplying or dividing both sides of an inequality by a negative number
and reversing the
direction of the inequality sign yields an equivalent inequality.
(c) Taking an odd power or odd root of both sides of an inequality yields an
equivalent inequality.
(d) If the numbers on both sides of an inequality are nonnegative, then
taking an even power or
even root of both sides yields an equivalent inequality.
