§ 1.1 Precalculus Review I
MAC 2233 - Calculus for Business
The Real Number Line
Intervals
Types of intervals.
Type of interval
Open interval
Half-open interval
Closed interval
All Real numbers
Remark. Intervals
Remark. Intervals
in…nite intervals.
Set Notation
Interval Notation
Graph
fx : x > ag
(a; 1)
fx : a < x < bg
(a; b)
fx : x < bg
( 1; b)
fx : x ag
[a; 1)
fx : a < x bg
(a; b]
fx : a x < bg
[a; b)
fx : x bg
( 1; b]
fx : a x bg
[a; b]
fx : x is a real numberg
( 1; 1)
of the form (a; b), (a; b], [a; b) or [a; b] are called …nite intervals.
of the form (a; 1), ( 1; b), [a; 1), ( 1; b] or ( 1; 1) are called
Properties of Inequalities
Properties of Inequalities.
(1) If a < b and b < c, then a < c.
(2) If a < b, then a + c < b + c.
(3) If a < b and c > 0, then ac < bc.
(4) If a < b and c < 0, then ac > bc.
Remark. Always reverse the direction of the inequality symbol when multiplying by a negative
number.
Remark. The symbol < between a and b can be replaced with >, , or .
Absolute Value
De…nition. The absolute value of a number a is denoted by jaj and is de…ned by
jaj =
a
a
if a 0
.
if a < 0
Remark. Geometrically, jaj is the distance between the origin and the point on the number line
that represents the number a.
Absolute Value Properties. If a and b are any real numbers, then
(5) j aj = jaj
(6) jabj = jaj jbj
jaj
a
=
where b 6= 0
(7)
b
jbj
(8) Triangle Inequality: ja + bj jaj + jbj.
Exponents and Radicals
De…nition. If b is any real number and n is a positive integer, then the expression bn (read "b to
the power n") is de…ned as the number bn = b| b b{z
}b. The number b is called the base, and
n factors
the script n is called the power of the exponential expression bn .
De…nition. If b 6= 0, we de…ne b0 = 1.
Remark. The expression 00 is unde…ned.
1
De…nition. If n is a positive integer, then the expression b n is de…ned to be the number that,
1
when raised to the nth power, is
equal to b. Thus, (b n )n = b. Such a number, if it exists, is called
p
n
the nth root of b, also written b.
p
p
De…nition. If
is a rational number in lowest terms, then the expression b q is de…ne as the
q
p
p
1
p
number (b q ) , or equivalently, q bp , whenever it exists, and the expression b q is de…ned as the
1
number p .
bq
Laws of Exponents.
(1) am an = am+n .
am
(2) n = am n where a 6= 0.
a
(3) (am )n = am n .
(4) (a b)n = an bn
a n an
(5)
= n where b 6= 0.
b
b
De…nition. The process of eliminating the radical from the numerator or denominator of an
algebraic expression is called rationalization.