Absolute Value and Distance
The absolute value of a number a, denoted by |a|, is the distance from a to 0 on the real
Distance is always positive or zero, so we have |a| 0 for every number a. Remembering that
a is positive when a is negative,
Absolute Value Inequalities
EXAMPLE: Solve the inequality |x 5| < 2 and sketch the solution set.
Solution: The inequality |x 5| < 2 is equivalent to
2 < x 5 < 2
2 + 5 < x 5 + 5 < 2 + 5
The solution set is the open interval (3, 7).
EXAMPLE: Solve the
General Equation of a Line
EXAMPLE: Sketch the graph of the equation 2x 3y 12 = 0.
Solution 1: Since the equation is linear, its graph is a line. To draw the graph, it is enough to
find any two points on the line. The intercepts are the easiest points to
Section 1.2 Exponents and Radicals
A product of identical numbers is usually written in exponential notation. For example, 5 5 5
is written as 53 . In general, we have the following definition.
Section 1.1 Real Numbers
Types of Real Numbers
1. Natural numbers (N):
1, 2, 3, 4, 5, . . .
2. Integer numbers (Z):
0, 1, 2, 3, 4, 5, . . .
REMARK: Any natural number is an integer number, but not any integer number is a natural
3. Rational number
Sets and Intervals
A set is a collection of objects, and these objects are called the elements of the set. If S is
a set, the notation a S means that a is an element of S, and b 6 S means that b isnot an
element of S. For example, if Q represents the set
1. Solve each equation:
(a) x2 = 0
(b) x2 = 1
(c) x2 = 4
(d) x2 = 5
(a) We have x = 0.
(b) We have (see the Appendix) x = 1.
(c) We have (see the Appendix) x = 2 (which is 4).
(d) We have (see the Appendix) x = 5.
(e) We have
(x 4)2 = 7
Absolute Value Equations
1. Solve the equation |2x 5| = 3.
Solution: By the definition of absolute value, |2x 5| = 3 is equivalent to
2x 5 = 3
2x 5 = 3
2x = 8
2x = 2
The solutions are x = 4 and x = 1.
2. Solve the equation 5|2 4x| + 3
Addition and Subtraction
The number 0 is special for addition; it is called the additive identity because
for any real number a. Every real number a has a negative, a, that satisfies
a + (a) = 0
Subtraction is the operation that undoes addition; to
REMARK 1: With this definition it can be proved that the Laws of Exponents also hold for
REMARK 2: It is important that a 0 if n is even in the definition above. Otherwise
contradictions are possible. For example,
2 is irrational.
Proof: Assume to the contrary that
2 is rational, that is
where p and q are integers and q 6= 0. Without loss of generality we can assume that p and q have
no common divisor > 1. Then
2q 2 = p2