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Introduction to Real Numbers and
Algebraic Expressions
9.1
9.2
9.3
9.4
9.5
9.6
9.7
9.8
CHAPTER
9
Introduction to Algebra
The Real Numbers
Addition of Real Numbers
Subtraction of Real Numbers
Multiplication of Real Numbers
Division of Real Numbers
Properties of Real Numbers
Simplifying Expressions; Order of Operations
Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.
Slide 2
9.7
Properties of Real Numbers
OBJECTIVES
a Find equivalent fraction expressions and simplify
fraction expressions.
b Use the commutative and associative laws to find
equivalent expressions.
c Use the distributive laws to multiply expressions like 8
and x – y.
d Use the distributive laws to factor expressions like
4x – 12 + 24y.
e Collect like terms.
Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.
Slide 3
9.7
Properties of Real Numbers
EQUIVALENT EXPRESSIONS
Two expressions that have the same value for all
allowable replacements are called equivalent.
Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.
Slide 4
9.7
Properties of Real Numbers
THE IDENTITY PROPERTY OF 0
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Slide 5
9.7
Properties of Real Numbers
THE IDENTITY PROPERTY OF 1
Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.
Slide 6
9.7
Properties of Real Numbers
Find equivalent fraction expressions and simplify
a
fraction expressions.
EXAMPLE 2
Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.
Slide 7
9.7
Properties of Real Numbers
THE COMMUTATIVE LAWS
Addition. For any numbers a and b,
(We can change the order when adding without
affecting the answer.)
Multiplication. For any numbers a and b,
(We can change the order when multiplying without
affecting the answer.)
Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.
Slide 8
9.7
Properties of Real Numbers
Use the commutative and associative laws to find
b
equivalent expressions.
EXAMPLE 6
Use the commutative laws to write an equivalent
expression:
Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.
Slide 9
9.7
Properties of Real Numbers
Use the commutative and associative laws to find
b
equivalent expressions.
EXAMPLE 6
a) An expression equivalent to y + 5 is 5 + y by the
commutative law of addition.
b) An expression equivalent to mn is nm by the
commutative law of multiplication.
c) An expression equivalent to 7 + xy is xy + 7 by the
commutative law of addition. Another expression
equivalent to 7 + xy is 7 + yx by the commutative law of
multiplication. Another equivalent expression is yx + 7.
Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.
Slide 10
9.7
Properties of Real Numbers
THE ASSOCIATIVE LAWS
Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.
Slide 11
9.7
Properties of Real Numbers
Use the commutative and associative laws to find
b
equivalent expressions.
EXAMPLE 9
Use an associative law to write an equivalent expression:
a) An expression equivalent to (y + z) + 3 is y + (z + 3) by
the associative law of addition.
b) An expression equivalent to 8(xy) is (8x)y by the
associative law of multiplication.
Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.
Slide 12
9.7
Properties of Real Numbers
Use the commutative and associative laws to find
b
equivalent expressions.
EXAMPLE 11
Use the commutative and associative laws to write at
least three expressions equivalent to (3x)y.
Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.
Slide 13
9.7
Properties of Real Numbers
Use the commutative and associative laws to find
b
equivalent expressions.
EXAMPLE 11
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Slide 14
9.7
Properties of Real Numbers
THE DISTRIBUTIVE LAW OF MULTIPLICATION OVER ADDITION
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9.7
Properties of Real Numbers
THE DISTRIBUTIVE LAW OF MULTIPLICATION OVER SUBTRACTION
Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.
Slide 16
9.7
c
Properties of Real Numbers
Use the distributive laws to multiply expressions like 8
and x – y.
Terms are separated by addition signs. If there are
subtraction signs, we can find an equivalent expression
that uses addition signs.
Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.
Slide 17
9.7
Properties of Real Numbers
Use the distributive laws to multiply expressions like 8
c
and x – y.
EXAMPLE 13
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Slide 18
9.7
Properties of Real Numbers
Use the distributive laws to multiply expressions like 8
c
and x – y.
EXAMPLE 16
Multiply.
Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.
Slide 19
9.7
Properties of Real Numbers
Use the distributive laws to multiply expressions like 8
c
and x – y.
EXAMPLE
Name the property or law illustrated by each equation.
Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.
Slide 20
9.7
Properties of Real Numbers
Use the distributive laws to multiply expressions like 8
c
and x – y.
EXAMPLE
Name the property or law illustrated by each equation.
Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.
Slide 21
9.7
Properties of Real Numbers
Use the distributive laws to factor expressions like
d
4x – 12 + 24y.
Factoring is the reverse of multiplying. To factor, we can
use the distributive laws in reverse.
Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.
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9.7
Properties of Real Numbers
FACTORING
To factor an expression is to find an equivalent
expression that is a product.
Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.
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9.7
Properties of Real Numbers
d Use the distributive laws to factor expressions like 4x – 12 + 24y.
EXAMPLE
Factor.
Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.
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9.7
Properties of Real Numbers
e Collect like terms.
Terms whose variable factors are exactly the same, terms
of constants, and terms whose variables are raised to the
same power are called like terms.
The process of collecting like terms is also based on the
distributive laws. We can apply a distributive law when a
factor is on the right because of the commutative law of
multiplication.
Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.
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9.7
Properties of Real Numbers
e Collect like terms.
EXAMPLE
Collect like terms. Try to write just the answer, if you can.
Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.
Slide 26