Transcript Beginning Algebra

Copyright 2013, 2009, 2005, 2002 Pearson, Education, Inc.
1.2
Algebraic
Expressions and
Sets of Numbers
Algebraic Expressions
A variable is a letter used to represent any number.
A constant is either a fixed number or a letter that
represents a fixed number.
An algebraic expression is formed by numbers
and variables connected by the operation of
addition, subtraction, multiplication, division, raising
to powers, and/or taking roots.
3x  5
5a  6a  40
2
y  y  15
4y
4
2
Evaluating Algebraic Expressions
To evaluate an algebraic expression,
substitute the numerical value for each variable
into the expression and simplify the result.
Example
Evaluate each expression for the given value.
a) 5x – 2 for x = 8
= 5(8) – 2
= 40 – 2
= 38
b) 3a2 + 2a + 4 for a = – 4
= 3(– 4)2 + 2(– 4) + 4
= 3(16) + (– 8) + 4
= 44
Set of Numbers
• Natural numbers – {1, 2, 3, 4, 5, 6 . . .}
• Whole numbers – {0, 1, 2, 3, 4 . . .}
• Integers – {. . . –3, -2, -1, 0, 1, 2, 3 . . .}
• Rational numbers – the set of all numbers
that can be expressed as a quotient of
integers, with denominator  0
• Irrational numbers – the set of all numbers
that can NOT be expressed as a quotient of
integers
• Real numbers – the set of all rational and
irrational numbers combined
The Number Line
A number line is a line on which each point is
associated with a number.
–5 –4 –3 –2 –1
– 4.8
Negative
numbers
0
1
2
3
4
1.5
Positive
numbers
5
Vocabulary
Variable is a symbol used to represent a number.
Algebraic expressions are a collection of
numbers, variables, operations, grouping symbols,
but NO = or inequalities.
In describing some of the previous sets, we used a
. . . symbol, called an ellipsis. It means to continue
in the same pattern.
The members of a set are called its elements.
When we list (or attempt to list with an ellipsis) the
elements of a set, the set is written in roster form.
Set Builder Notation
A set can also be written in set builder notation.
This notation describes the members of a set, but
does not list them.
Example:
{ x | x is an even natural number less than 10}
The set of such
all x
that
x is an even natural number less than 10
Example
Given the set of numbers
1
3
1 , 2.5, ,  3.75, 15,
4
2
2, 0, 5
list the numbers in this set that belong to the set of:
a.
b.
c.
d.
e.
f.
Natural numbers: 15
Whole numbers: 0, 15
Integers: 0, 15,  5
1
3
Rational numbers: 0, 15,  5, 1 , 3.75, 2.5,
4
2
Irrational numbers: 2
Real numbers: all the numbers in the set
Absolute Value
The absolute value of a real number a, denoted
by |a|, is the distance between a and 0 on the
number line.
| – 4| = 4
|5| = 5
Symbol
for
absolute
value
Distance of 4 Distance of 5
–5 –4 –3 –2 –1
0
1
2
3
4
5
Example
Find each absolute value.
a. 9  9
b. 6  6
c.  4  4
5 5
d. 0  0
Finding the Opposite of a
Number
Opposite
The opposite of a number a is the number a.
Double Negative Property
For every real number a, (a) = a.
Example
Write the opposite of each number.
a. 12
The opposite of 12 is 12.
2
b.
3
The opposite of 2/3 is 2/3.
c. 10.4
The opposite of 10.4 is 10.4.
Translating Phrases
Addition
(+)
Subtraction
(–)
sum
plus
added to
more than
increased by
total
difference
minus
subtract
less than
decreased by
less
Multiplicatio
n
(·)
product
times
multiply
multiplied by
of
double/triple
Division
()
quotient
divide
shared
equally
among
divided by
divided into
Example
Write as an algebraic expression. Let x represent
the unknown number.
a. 5 decreased by a number
5–x
b. The quotient of a number and 12
x
12  x or
12
Example
Translate the following phrases into algebraic
expressions.
a. The sum of 4 and twice y
4 + 2y
b. x less than the product of y and z
yz  x