30 2 37 - - K K 30 2 37 - - K K � EMBED Equation.3 ���-5 � EMBED Equation.3 ���5 Sum=b 0 Product=a x c 1x (-25) x2+0x-25=0 substitute the value of bx to facilitate factorization The denominator is a constant term (x2+5x)(-5x-25)=0 x (x+5)-5(x+5)=0 (x+5)(x-5) are the factors for the domain All values of x in the expression are included as dividing by the domain will yield a solution Thus D=� EMBED Equation.3 ��� � EMBED Equation.3 ���The Domain is Integer 2 which divides any factor in the numerator. We factorize the Numerator to obtain the factors. � EMBED Equation.3 ���-6 � EMBED Equation.3 ���5 Sum=b -1 Product=a x c 1x (-30) k2+5k-6k-30=0 substitute the value of bx to facilitate factorization (k2+5k)(-6k-30)=0 k (k+5)-6(k+5)=0 (k+5)(k-6) are the factors for the domain k=-5 and k=6 are the excluded values in the expression since they will equal 0 in any range of values. Thus D={K:K� EMBED Equation.3 ���� EMBED Equation.3 ���,K� EMBED Equation.3 ���(-5) ,k� EMBED Equation.3 ���6} L � EMBED Equation.3 ���Factorizing the domain in the form we find factors:  2 25 2 - x 30 2 37 - - K K 30 2 37 - - K K Î Â ¹ ¹ 30 2 37 - - K K _1071350467.unknown _1071351244.unknown _1071350591.unknown _1071350663.unknown _1071348848.unknown INSTRUCTOR GUIDANCE EXAMPLE: Week One Discussion Domains of Rational Expressions Students, you are perfectly welcome to format your math work just as I have done in these examples. However, the written parts of the assignment MUST be done about your own thoughts and in your own words. You are NOT to simply copy this wording into your posts! Here are my given rational expressions oh which to base my work. 25x2 – 4 67 5 – 9w 9w2 – 4 The domain of a rational expression is the set of all numbers which are allowed to substitute for the variable in the expression. It is possible that some numbers will not be allowed depending on what the denominator has in it. In our Real Number System division by zero cannot be done. There is no number (or any other object) which can be the answer to division by zero so we must simply call the attempt “undefined.” A denominator cannot be zero because in a rational number or expression the denominator divides the numerator. In my first expression, the denominator is a constant term, meaning there is no variable present. Since it is impossible for 67 to equal zero, there are no excluded values for the domain. We can say the domain (D) is the set of all Real Numbers, written in set notation that would look like this: D = {x| x ∈ ℜ} or even more simply as D = ℜ. For my second expression, I need to set the denominator equal to zero to find my excluded values for w. 9w2 – 4 = 0 I notice this is a difference of squares which I can factor. (3w – 2)(3w + 2) = 0 Set each factor equal to zero. 3w – 2 = 0 or 3w + 2 = 0 Add or subtract 2 from both sides.

1. 30
2
37
-
-
K
K
30
2
37
-
-
K
K
� EMBED Equation.3 ���-5
� EMBED Equation.3 ���5
Sum=b
0
Product=a x c

2. 1x (-25)
x2+0x-25=0 substitute the value of bx to facilitate factorization
The denominator is a constant term
(x2+5x)(-5x-25)=0
x (x+5)-5(x+5)=0
(x+5)(x-5) are the factors for the domain
All values of x in the expression are included as dividing by the
domain will yield a solution
Thus D=� EMBED Equation.3 ���
� EMBED Equation.3 ���The Domain is Integer 2 which
divides any factor in the numerator. We factorize the Numerator
to obtain the factors.

3. � EMBED Equation.3 ���-6
� EMBED Equation.3 ���5
Sum=b
-1
Product=a x c
1x (-30)
k2+5k-6k-30=0 substitute the value of bx to facilitate
factorization
(k2+5k)(-6k-30)=0
k (k+5)-6(k+5)=0
(k+5)(k-6) are the factors for the domain
k=-5 and k=6 are the excluded values in the expression since

4. they will equal 0 in any range of values.
Thus D={K:K� EMBED Equation.3 ���� EMBED
Equation.3 ���,K� EMBED Equation.3 ���(-5) ,k�
EMBED Equation.3 ���6}
L
� EMBED Equation.3 ���Factorizing the domain in the form
we find factors:
Â
2
25
2
-
x
30
2
37
-
-
K
K
30
2
37

5. -
-
K
K
Î
Â
¹
¹
30
2
37
-
-
K
K
_1071350467.unknown
_1071351244.unknown
_1071350591.unknown
_1071350663.unknown
_1071348848.unknown
INSTRUCTOR GUIDANCE EXAMPLE: Week One Discussion
Domains of Rational Expressions
Students, you are perfectly welcome to format your math work
just as I have done in
these examples. However, the written parts of the assignment
MUST be done about
your own thoughts and in your own words. You are NOT to
simply copy this wording
into your posts!
Here are my given rational expressions oh which to base my

6. work.
25x2 – 4
67
5 – 9w
9w2 – 4
The domain of a rational expression is the set of all numbers
which are allowed to
substitute for the variable in the expression. It is possible that
some numbers will not be
allowed depending on what the denominator has in it.
In our Real Number System division by zero cannot be done.
There is no number (or
any other object) which can be the answer to division by zero so
we must simply call the
attempt “undefined.” A denominator cannot be zero because in a
rational number or
expression the denominator divides the numerator.
In my first expression, the denominator is a constant term,
meaning there is no variable
present. Since it is impossible for 67 to equal zero, there are no
excluded values for the
domain. We can say the domain (D) is the set of all Real
Numbers, written in set
notation that would look like this:
D = {x| x ∈ ℜ} or even more simply as D = ℜ.
For my second expression, I need to set the denominator equal
to zero to find my
excluded values for w.

7. 9w2 – 4 = 0 I notice this is a difference of squares which I can
factor.
(3w – 2)(3w + 2) = 0 Set each factor equal to zero.
3w – 2 = 0 or 3w + 2 = 0 Add or subtract 2 from both
sides.
3w = 2 or 3w = -2 Divide both sides by 3.
w = 2/3 or w = -2/3 These are the excluded values for my
second expression.
The domain (D) for my second expression is the set of all Reals
excluding ±2/3. In set
notation, this can be written as D = {w| w ∈ ℜ, w ≠ ±2/3}
Now, both of my expressions do not have excluded values. In
one expression, I have no
excluded values because there is no variable in the denominator
and a non-zero number
will never just become zero. In the other expression, there are
two excluded numbers
because both, if inserted in place of the variable, would cause
the denominator to become
zero and thus the whole expression would become undefined.
Inserting Math Symbols
Throughout these courses MAT 221 and MAT 222 students are
required to submit written assignments
in Microsoft Word that require the use of Mathematical
symbols. Please follow the directions below in

8. order to insert mathematical symbols into your documents.
Inserting Symbols:
1. Click with your mouse in the space you would like the
mathematical symbol to appear.
2. Click with your mouse on the “Insert” tab on the upper
navigation toolbar.
3. Click with your mouse on the “Symbol” button.
4. In the dropdown menu, click with your mouse on “More
Symbols”.
5. In the “Symbol” window, click on the “Subset” dropdown
menu and choose the desired category
that fits the mathematical equation.
6. Select the desired symbol and click with your mouse “Insert”.
7. The symbol should appear in your document.
If you have any questions, please do not hesitate the contact
your instructor.
MAT222 Week 1 Discussion Grading Rubric
Grading Criteria

9. MAT222 Discussions Week 1
2 points possible
Content Criteria Weight
Student thoroughly addresses all elements of the discussion
prompt, and demonstrates an
advanced knowledge of the topic. Student clearly understands
the mathematical concepts in the
assignment, and demonstrates the prerequisite math knowledge
to do the work successfully.
0.8
Writing of Math Work
Student effectively shows how the solution(s) was achieved.
The math work is shown and follows
an accurate process to get to the solution.
0.8
Participation
Student responds with thorough and constructive analysis to at
least two classmates, relating
the response to relevant course concepts. Student may pose
pertinent follow-up thoughts or
questions about the topic, and demonstrates respect for the
diverse opinions of fellow learners.
Initial post was made by Day 3 (Thursday), and responses were
made by Day 7 (Monday).
0.2
Writing

10. Initial post contains very few, if any, errors related to grammar,
spelling, and sentence structure.
Post is easy to read and understand.
0.2
6
Rational Expressions
Advanced technical developments have made sports equipment
faster, lighter,
and more responsive to the human body. Behind the more
flexible skis, lighter
bats, and comfortable athletic shoes lies the science of
biomechanics, which is the
study of human movement and the factors that influence it.
Designing and testing an athletic shoe go hand in hand. While a
shoe is being
designed, it is tested in a multitude of ways, including long-
term wear, rear foot
stability, and strength of materials. Testing basketball shoes
usually includes an
evaluation of the force applied to the ground by the foot during
running, jumping,

11. and landing. Many biome-
chanics laboratories have a
special platform that can mea-
sure the force exerted when a
player cuts from side to side, as
well as the force against the
bottom of the shoe. Force
exerted in landing from a lay-
up shot can be as high as
14 times the weight of the
body. Side-to-side force is usu-
ally about 1 to 2 body weights
in a cutting movement.
C
h
a
p
t
e
r

12. In Exercises 53 and 54 of
Section 6.7 you will see how
designers of athletic shoes use
proportions to find the amount
of force on the foot and soles of
shoes for activities such as
running and jumping.
6.1
Reducing Rational
Expressions
6.2
Multiplication and
Division
6.3
Finding the Least
Common Denominator
6.4 Addition and Subtraction
6.5 Complex Fractions
6.6
Solving Equations with
Rational Expressions
6.7
Applications of Ratios
and Proportions

13. 6.8
Applications of Rational
Expressions
0
3
4
5
2
1
50 100 150 200 250 300
Weight (pounds)
Fo
rc
e
(t
ho
us
an
ds
o
f
po

14. un
ds
)
dug84356_ch06a.qxd 9/14/10 12:39 PM Page 381
382 Chapter 6 Rational Expressions 6-2
E X A M P L E 1
U Calculator Close-Up V
To evaluate the rational expression
in Example 1(a) with a calculator, first
use Y � to define the rational expres-
sion. Be sure to enclose both numera-
tor and denominator in parentheses.
Then find y1(�3).
Evaluating a rational expression
a) Find the value of �4
x
x
�
�
2
1
� for x � �3. b) If R(x) � �

15. 3
2
x
x
�
�
2
1
�, find R(4).
Solution
a) To find the value of �4
x
x
�
�
2
1

16. � for x � �3, replace x by �3 in the rational expression:
�
4(
�
�
3
3)
�
�
2
1
� � �
�
�
1

17. 1
3
� � 13
So the value of the rational expression is 13. The Calculator
Close-Up shows how to
evaluate the expression with a graphing calculator using a
variable. With a scientific or
graphing calculator you could also evaluate the expression by
entering (4(�3) � 1)�
(�3 � 2). Be sure to enclose the numerator and denominator in
parentheses.
b) R(4) is the value of the rational expression when x � 4. To
find R(4), replace x by 4
in R(x) � �
3
2
x
x
�

18. �
2
1
�:
R(4) � �
3
2
(
(
4
4
)
)
�
�
2
1
�

19. R(4) � �
1
7
4
� � 2
So the value of the rational expression is 2 when x � 4, or R(4)
� 2 (read “R of 4
is 2”).
Now do Exercises 1–6
In This Section
U1V Rational Expressions and
Functions
U2V Reducing to Lowest Terms
U3V Reducing with the Quotient
Rule for Exponents
U4V Dividing a � b by b � a

20. U5V Factoring Out the Opposite
of a Common Factor
U6V Writing Rational Expressions
6.1 Reducing Rational Expressions
Rational expressions in algebra are similar to the rational
numbers in arithmetic. In
this section, you will learn the basic ideas of rational
expressions.
U1V Rational Expressions and Functions
A rational number is the ratio of two integers with the
denominator not equal to 0. For
example,
�
3
4
�, �
�
�

21. 9
6
�, �
�
1
7
�, and �
0
2
�
are rational numbers. Of course, we usually write the last three
of these rational num-
bers in their simpler forms �3
2
�, �7, and 0. A rational expression is the ratio of two poly-
nomials with the denominator not equal to 0. Because an integer
is a monomial, a

22. rational number is also a rational expression. As with rational
numbers, if the denom-
inator is 1, it can be omitted. Some examples of rational
expressions are
�
x
x
2
�
�
8
1
�, �
3a2
a
�

23. �
5a
9
� 3
�, �
3
7
� , and 9x.
A rational expression involving a variable has no value unless
we assign a value
to the variable. If the value of a rational expression is used to
determine the value of
a second variable, then we have a rational function. For
example,
y � �
x
x
2

24. �
–
8
1
� and w � �
3a2
a
�
�
5a
9
– 3
�
are rational functions. We can evaluate a rational expression
with or without function
notation as we did for polynomials in Chapter 5.

25. dug84356_ch06a.qxd 9/14/10 12:39 PM Page 382
6-3 6.1 Reducing Rational Expressions 383
An expression such as �
5
0
� is undefined because the definition of rational numbers
does not allow zero in the denominator. When a variable occurs
in a denominator, any
real number can be used for the variable except numbers that
make the expression
undefined.
E X A M P L E 2 Ruling out values for x
Which numbers cannot be used in place of x in each rational
expression?
a) �
x
x

26. 2
�
�
8
1
� b) �
2
x
x
�
�
2
1
� c) �
x
x
2

27. �
�
5
4
�