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TUTORIALS:
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Order and Inequalities
One important property of real numbers is that they can be ordered. If a and b are real
numbers, a is less than b if b - a is positive. This order is denoted by the
inequality a < b.
The statement “b is greater than a†is equivalent to saying that a is less than b. When
three real numbers a, b, and c are ordered such that a < b and b < c we say that b
is between a and c and a < b < c.
Geometrically, a < b if and only if a lies to the left of b on the real line (see
the figure below).
For example, 1 < 2 because 1 lies to the left of 2 on the real line.
The following properties are used in working with inequalities. Similar properties
are obtained if < is replaced by ≤ and > is replaced by
≥. (The symbols
≤ and ≥ mean less than or equal to and
greater than or equal to, respectively.)
Properties of Inequalities
Let a, b, c, d, and k be real numbers.
| 1. If a < b and b < c, then a < c. |
Transitive Property. |
| 2. If a < b and c < d, then a + c < b + d. |
Add inequalities. |
| 3. If a < b, then a + k < b + k. |
Add a constant. |
| 4. If a < b and k > 0, then ak < bk. |
Multiply by a positive constant. |
| 4. If a < b and k < 0, then ak > bk. |
Multiply by a negative constant. |
NOTE Note that you reverse the inequality when you multiply by a negative number. For
example, if x < 3 then -4x > -12. This also applies to division by a negative number. Thus,
if -2x > 4, then x < -2.
A set is a collection of elements. Two common sets are the set of real numbers
and the set of points on the real line. Many problems in calculus involve subsets
of
one of these two sets. In such cases it is convenient to use set notation of the form
{x: condition on x}, which is read as follows.
For example, you can describe the set of positive real numbers as
| {x: x > 0} |
Set of positive real numbers |
Similarly, you can describe the set of nonnegative real numbers as
| {x: x ≥ 0} |
Set of nonnegative real numbers |
The union of two sets A and B, denoted by is the set of elements that are
members of A or B or both. The intersection of two sets A and B, denoted by
is the set of elements that are members of A and B. Two sets are disjoint if they have
no elements in common.
The most commonly used subsets are intervals on the real line. For example, the
open interval
| (a, b) = {x: a < x < b} |
Open interval |
is the set of all real numbers greater than a and less than b, where a and b are the
endpoints of the interval. Note that the endpoints are not included in an open
interval. Intervals that include their endpoints are closed and are denoted by
| [a, b] = {x: a ≤ x
≤ b} |
Closed interval |
The nine basic types of intervals on the real line are shown in the table below.
The first four are bounded intervals and the remaining five are unbounded intervals.
Unbounded intervals are also classified as open or closed. The intervals (-∞,
b)
and (a, ∞) are open, the intervals (-∞,
b] and [a, ∞) are closed, and the
interval (-∞, ∞) is considered to be both open and closed.
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